id	sid	tid	token	lemma	pos
ejpam-5709	1	1	european	european	PROPN
ejpam-5709	1	2	journal	journal	PROPN
ejpam-5709	1	3	of	of	ADP
ejpam-5709	1	4	pure	pure	ADJ
ejpam-5709	1	5	and	and	CCONJ
ejpam-5709	1	6	applied	applied	ADJ
ejpam-5709	1	7	mathematics	mathematic	NOUN
ejpam-5709	1	8	2025	2025	NUM
ejpam-5709	1	9	,	,	PUNCT
ejpam-5709	1	10	vol	vol	NOUN
ejpam-5709	1	11	.	.	PROPN
ejpam-5709	1	12	18	18	NUM
ejpam-5709	1	13	,	,	PUNCT
ejpam-5709	1	14	issue	issue	NOUN
ejpam-5709	1	15	1	1	NUM
ejpam-5709	1	16	,	,	PUNCT
ejpam-5709	1	17	article	article	NOUN
ejpam-5709	1	18	number	number	NOUN
ejpam-5709	1	19	5709	5709	NUM
ejpam-5709	1	20	issn	issn	PROPN
ejpam-5709	1	21	1307	1307	NUM
ejpam-5709	1	22	-	-	SYM
ejpam-5709	1	23	5543	5543	NUM
ejpam-5709	1	24	–	–	PUNCT
ejpam-5709	1	25	ejpam.com	ejpam.com	X
ejpam-5709	1	26	published	publish	VERB
ejpam-5709	1	27	by	by	ADP
ejpam-5709	1	28	new	new	PROPN
ejpam-5709	1	29	york	york	PROPN
ejpam-5709	1	30	business	business	PROPN
ejpam-5709	1	31	global	global	ADJ
ejpam-5709	1	32	base	base	NOUN
ejpam-5709	1	33	-	-	PUNCT
ejpam-5709	1	34	β(c	β(c	NUM
ejpam-5709	1	35	)	)	PUNCT
ejpam-5709	1	36	representations	representation	NOUN
ejpam-5709	1	37	and	and	CCONJ
ejpam-5709	1	38	generalizations	generalization	NOUN
ejpam-5709	1	39	of	of	ADP
ejpam-5709	1	40	irreducibility	irreducibility	NOUN
ejpam-5709	1	41	criteria	criterion	NOUN
ejpam-5709	1	42	for	for	ADP
ejpam-5709	1	43	polynomials	polynomial	NOUN
ejpam-5709	1	44	over	over	ADP
ejpam-5709	1	45	any	any	DET
ejpam-5709	1	46	imaginary	imaginary	ADJ
ejpam-5709	1	47	quadratic	quadratic	ADJ
ejpam-5709	1	48	fields	field	NOUN
ejpam-5709	1	49	phitthayathon	phitthayathon	PROPN
ejpam-5709	1	50	phetnun1	phetnun1	NOUN
ejpam-5709	1	51	,	,	PUNCT
ejpam-5709	1	52	narakorn	narakorn	ADJ
ejpam-5709	1	53	rompurk	rompurk	NOUN
ejpam-5709	1	54	kanasri1,∗	kanasri1,∗	NOUN
ejpam-5709	1	55	1	1	NUM
ejpam-5709	1	56	department	department	NOUN
ejpam-5709	1	57	of	of	ADP
ejpam-5709	1	58	mathematics	mathematic	NOUN
ejpam-5709	1	59	,	,	PUNCT
ejpam-5709	1	60	faculty	faculty	NOUN
ejpam-5709	1	61	of	of	ADP
ejpam-5709	1	62	science	science	NOUN
ejpam-5709	1	63	,	,	PUNCT
ejpam-5709	1	64	khon	khon	PROPN
ejpam-5709	1	65	kaen	kaen	PROPN
ejpam-5709	1	66	university	university	PROPN
ejpam-5709	1	67	,	,	PUNCT
ejpam-5709	1	68	khon	khon	PROPN
ejpam-5709	1	69	kaen	kaen	PROPN
ejpam-5709	1	70	,	,	PUNCT
ejpam-5709	1	71	40002	40002	NUM
ejpam-5709	1	72	,	,	PUNCT
ejpam-5709	1	73	thailand	thailand	PROPN
ejpam-5709	1	74	abstract	abstract	PROPN
ejpam-5709	1	75	.	.	PUNCT
ejpam-5709	2	1	let	let	VERB
ejpam-5709	2	2	k	k	PRON
ejpam-5709	2	3	be	be	AUX
ejpam-5709	2	4	an	an	DET
ejpam-5709	2	5	imaginary	imaginary	ADJ
ejpam-5709	2	6	quadratic	quadratic	ADJ
ejpam-5709	2	7	field	field	NOUN
ejpam-5709	2	8	with	with	ADP
ejpam-5709	2	9	the	the	DET
ejpam-5709	2	10	ring	ring	NOUN
ejpam-5709	2	11	of	of	ADP
ejpam-5709	2	12	integers	integer	NOUN
ejpam-5709	2	13	ok	ok	INTJ
ejpam-5709	2	14	.	.	PUNCT
ejpam-5709	3	1	in	in	ADP
ejpam-5709	3	2	the	the	DET
ejpam-5709	3	3	authors	author	NOUN
ejpam-5709	3	4	’	'	PUNCT
ejpam-5709	3	5	earlier	early	ADJ
ejpam-5709	3	6	work	work	NOUN
ejpam-5709	3	7	,	,	PUNCT
ejpam-5709	3	8	the	the	DET
ejpam-5709	3	9	so	so	ADV
ejpam-5709	3	10	-	-	PUNCT
ejpam-5709	3	11	called	call	VERB
ejpam-5709	3	12	base	base	NOUN
ejpam-5709	3	13	-	-	PUNCT
ejpam-5709	3	14	β(c	β(c	NUM
ejpam-5709	3	15	)	)	PUNCT
ejpam-5709	3	16	representation	representation	NOUN
ejpam-5709	3	17	for	for	ADP
ejpam-5709	3	18	nonzero	nonzero	ADJ
ejpam-5709	3	19	elements	element	NOUN
ejpam-5709	3	20	of	of	ADP
ejpam-5709	3	21	ok	ok	ADJ
ejpam-5709	3	22	was	be	AUX
ejpam-5709	3	23	determined	determine	VERB
ejpam-5709	3	24	,	,	PUNCT
ejpam-5709	3	25	where	where	SCONJ
ejpam-5709	3	26	c	c	PROPN
ejpam-5709	3	27	is	be	AUX
ejpam-5709	3	28	a	a	DET
ejpam-5709	3	29	complete	complete	ADJ
ejpam-5709	3	30	residue	residue	NOUN
ejpam-5709	3	31	system	system	NOUN
ejpam-5709	3	32	modulo	modulo	VERB
ejpam-5709	3	33	β	β	NOUN
ejpam-5709	3	34	.	.	PUNCT
ejpam-5709	4	1	using	use	VERB
ejpam-5709	4	2	such	such	DET
ejpam-5709	4	3	a	a	DET
ejpam-5709	4	4	representation	representation	NOUN
ejpam-5709	4	5	,	,	PUNCT
ejpam-5709	4	6	irreducibility	irreducibility	NOUN
ejpam-5709	4	7	criteria	criterion	NOUN
ejpam-5709	4	8	for	for	ADP
ejpam-5709	4	9	polynomials	polynomial	NOUN
ejpam-5709	4	10	in	in	ADP
ejpam-5709	4	11	ok	ok	ADJ
ejpam-5709	4	12	[	[	X
ejpam-5709	4	13	x	x	X
ejpam-5709	4	14	]	]	X
ejpam-5709	4	15	were	be	AUX
ejpam-5709	4	16	established	establish	VERB
ejpam-5709	4	17	.	.	PUNCT
ejpam-5709	5	1	in	in	ADP
ejpam-5709	5	2	this	this	DET
ejpam-5709	5	3	paper	paper	NOUN
ejpam-5709	5	4	,	,	PUNCT
ejpam-5709	5	5	we	we	PRON
ejpam-5709	5	6	provide	provide	VERB
ejpam-5709	5	7	the	the	DET
ejpam-5709	5	8	explicit	explicit	ADJ
ejpam-5709	5	9	shapes	shape	NOUN
ejpam-5709	5	10	of	of	ADP
ejpam-5709	5	11	all	all	DET
ejpam-5709	5	12	baseβ(c	baseβ(c	NOUN
ejpam-5709	5	13	)	)	PUNCT
ejpam-5709	5	14	representations	representation	NOUN
ejpam-5709	5	15	for	for	ADP
ejpam-5709	5	16	nonzero	nonzero	ADJ
ejpam-5709	5	17	elements	element	NOUN
ejpam-5709	5	18	of	of	ADP
ejpam-5709	5	19	ok	ok	INTJ
ejpam-5709	5	20	.	.	PUNCT
ejpam-5709	6	1	generalizations	generalization	NOUN
ejpam-5709	6	2	of	of	ADP
ejpam-5709	6	3	such	such	ADJ
ejpam-5709	6	4	irreducibility	irreducibility	NOUN
ejpam-5709	6	5	criteria	criterion	NOUN
ejpam-5709	6	6	for	for	ADP
ejpam-5709	6	7	polynomials	polynomial	NOUN
ejpam-5709	6	8	in	in	ADP
ejpam-5709	6	9	ok	ok	ADJ
ejpam-5709	6	10	[	[	X
ejpam-5709	6	11	x	x	X
ejpam-5709	6	12	]	]	X
ejpam-5709	6	13	under	under	ADP
ejpam-5709	6	14	a	a	DET
ejpam-5709	6	15	certain	certain	ADJ
ejpam-5709	6	16	condition	condition	NOUN
ejpam-5709	6	17	are	be	AUX
ejpam-5709	6	18	also	also	ADV
ejpam-5709	6	19	established	establish	VERB
ejpam-5709	6	20	.	.	PUNCT
ejpam-5709	7	1	2020	2020	NUM
ejpam-5709	7	2	mathematics	mathematics	PROPN
ejpam-5709	7	3	subject	subject	NOUN
ejpam-5709	7	4	classifications	classification	NOUN
ejpam-5709	7	5	:	:	PUNCT
ejpam-5709	7	6	11r04	11r04	NUM
ejpam-5709	7	7	,	,	PUNCT
ejpam-5709	7	8	11r09	11r09	NUM
ejpam-5709	7	9	,	,	PUNCT
ejpam-5709	7	10	11r11	11r11	DET
ejpam-5709	7	11	key	key	ADJ
ejpam-5709	7	12	words	word	NOUN
ejpam-5709	7	13	and	and	CCONJ
ejpam-5709	7	14	phrases	phrase	NOUN
ejpam-5709	7	15	:	:	PUNCT
ejpam-5709	7	16	imaginary	imaginary	ADJ
ejpam-5709	7	17	quadratic	quadratic	ADJ
ejpam-5709	7	18	field	field	NOUN
ejpam-5709	7	19	,	,	PUNCT
ejpam-5709	7	20	ring	ring	NOUN
ejpam-5709	7	21	of	of	ADP
ejpam-5709	7	22	integers	integer	NOUN
ejpam-5709	7	23	,	,	PUNCT
ejpam-5709	7	24	complete	complete	ADJ
ejpam-5709	7	25	residue	residue	NOUN
ejpam-5709	7	26	system	system	NOUN
ejpam-5709	7	27	,	,	PUNCT
ejpam-5709	7	28	prime	prime	ADJ
ejpam-5709	7	29	element	element	NOUN
ejpam-5709	7	30	,	,	PUNCT
ejpam-5709	7	31	irreducible	irreducible	ADJ
ejpam-5709	7	32	polynomial	polynomial	ADJ
ejpam-5709	7	33	1	1	NUM
ejpam-5709	7	34	.	.	PUNCT
ejpam-5709	7	35	introduction	introduction	NOUN
ejpam-5709	7	36	an	an	DET
ejpam-5709	7	37	old	old	ADJ
ejpam-5709	7	38	and	and	CCONJ
ejpam-5709	7	39	interesting	interesting	ADJ
ejpam-5709	7	40	problem	problem	NOUN
ejpam-5709	7	41	in	in	ADP
ejpam-5709	7	42	mathematics	mathematics	NOUN
ejpam-5709	7	43	is	be	AUX
ejpam-5709	7	44	determining	determine	VERB
ejpam-5709	7	45	irreducible	irreducible	ADJ
ejpam-5709	7	46	polynomials	polynomial	NOUN
ejpam-5709	7	47	over	over	ADP
ejpam-5709	7	48	a	a	DET
ejpam-5709	7	49	field	field	NOUN
ejpam-5709	7	50	.	.	PUNCT
ejpam-5709	8	1	a	a	DET
ejpam-5709	8	2	polynomial	polynomial	NOUN
ejpam-5709	8	3	with	with	ADP
ejpam-5709	8	4	integer	integer	NOUN
ejpam-5709	8	5	coefficients	coefficient	NOUN
ejpam-5709	8	6	is	be	AUX
ejpam-5709	8	7	said	say	VERB
ejpam-5709	8	8	to	to	PART
ejpam-5709	8	9	be	be	AUX
ejpam-5709	8	10	irreducible	irreducible	ADJ
ejpam-5709	8	11	in	in	ADP
ejpam-5709	8	12	z[x	z[x	NOUN
ejpam-5709	8	13	]	]	PUNCT
ejpam-5709	8	14	if	if	SCONJ
ejpam-5709	8	15	it	it	PRON
ejpam-5709	8	16	is	be	AUX
ejpam-5709	8	17	an	an	DET
ejpam-5709	8	18	irreducible	irreducible	ADJ
ejpam-5709	8	19	element	element	NOUN
ejpam-5709	8	20	of	of	ADP
ejpam-5709	8	21	the	the	DET
ejpam-5709	8	22	polynomial	polynomial	ADJ
ejpam-5709	8	23	ring	ring	NOUN
ejpam-5709	8	24	z[x	z[x	NOUN
ejpam-5709	8	25	]	]	PUNCT
ejpam-5709	8	26	.	.	PUNCT
ejpam-5709	9	1	a	a	DET
ejpam-5709	9	2	polynomial	polynomial	ADJ
ejpam-5709	9	3	f(x	f(x	PROPN
ejpam-5709	9	4	)	)	PUNCT
ejpam-5709	9	5	of	of	ADP
ejpam-5709	9	6	positive	positive	ADJ
ejpam-5709	9	7	degree	degree	NOUN
ejpam-5709	9	8	in	in	ADP
ejpam-5709	9	9	z[x	z[x	NOUN
ejpam-5709	9	10	]	]	PUNCT
ejpam-5709	9	11	is	be	AUX
ejpam-5709	9	12	primitive	primitive	ADJ
ejpam-5709	9	13	if	if	SCONJ
ejpam-5709	9	14	the	the	DET
ejpam-5709	9	15	greatest	great	ADJ
ejpam-5709	9	16	common	common	ADJ
ejpam-5709	9	17	divisor	divisor	NOUN
ejpam-5709	9	18	of	of	ADP
ejpam-5709	9	19	its	its	PRON
ejpam-5709	9	20	coefficients	coefficient	NOUN
ejpam-5709	9	21	is	be	AUX
ejpam-5709	9	22	1	1	NUM
ejpam-5709	9	23	.	.	PUNCT
ejpam-5709	10	1	it	it	PRON
ejpam-5709	10	2	is	be	AUX
ejpam-5709	10	3	well	well	ADV
ejpam-5709	10	4	known	know	VERB
ejpam-5709	10	5	that	that	SCONJ
ejpam-5709	10	6	a	a	DET
ejpam-5709	10	7	nonconstant	nonconstant	ADJ
ejpam-5709	10	8	polynomial	polynomial	NOUN
ejpam-5709	10	9	in	in	ADP
ejpam-5709	10	10	z[x	z[x	NOUN
ejpam-5709	10	11	]	]	PUNCT
ejpam-5709	10	12	is	be	AUX
ejpam-5709	10	13	irreducible	irreducible	ADJ
ejpam-5709	10	14	in	in	ADP
ejpam-5709	10	15	z[x	z[x	NOUN
ejpam-5709	10	16	]	]	PUNCT
ejpam-5709	10	17	if	if	SCONJ
ejpam-5709	10	18	and	and	CCONJ
ejpam-5709	10	19	only	only	ADV
ejpam-5709	10	20	if	if	SCONJ
ejpam-5709	10	21	it	it	PRON
ejpam-5709	10	22	is	be	AUX
ejpam-5709	10	23	both	both	CCONJ
ejpam-5709	10	24	irreducible	irreducible	ADJ
ejpam-5709	10	25	over	over	ADP
ejpam-5709	10	26	q	q	NOUN
ejpam-5709	10	27	and	and	CCONJ
ejpam-5709	10	28	primitive	primitive	ADJ
ejpam-5709	10	29	in	in	ADP
ejpam-5709	10	30	z[x	z[x	NOUN
ejpam-5709	10	31	]	]	PUNCT
ejpam-5709	10	32	.	.	PUNCT
ejpam-5709	11	1	among	among	ADP
ejpam-5709	11	2	many	many	ADJ
ejpam-5709	11	3	irreducibility	irreducibility	NOUN
ejpam-5709	11	4	criteria	criterion	NOUN
ejpam-5709	11	5	for	for	ADP
ejpam-5709	11	6	polynomials	polynomial	NOUN
ejpam-5709	11	7	in	in	ADP
ejpam-5709	11	8	z[x	z[x	NOUN
ejpam-5709	11	9	]	]	PUNCT
ejpam-5709	11	10	,	,	PUNCT
ejpam-5709	11	11	we	we	PRON
ejpam-5709	11	12	are	be	AUX
ejpam-5709	11	13	interested	interested	ADJ
ejpam-5709	11	14	in	in	ADP
ejpam-5709	11	15	a	a	DET
ejpam-5709	11	16	classical	classical	ADJ
ejpam-5709	11	17	result	result	NOUN
ejpam-5709	11	18	of	of	ADP
ejpam-5709	11	19	a.	a.	PROPN
ejpam-5709	11	20	cohn	cohn	PROPN
ejpam-5709	11	21	,	,	PUNCT
ejpam-5709	11	22	given	give	VERB
ejpam-5709	11	23	by	by	ADP
ejpam-5709	11	24	pólya	pólya	PROPN
ejpam-5709	11	25	and	and	CCONJ
ejpam-5709	11	26	szegö	szegö	VERB
ejpam-5709	12	1	[	[	X
ejpam-5709	12	2	10	10	NUM
ejpam-5709	12	3	]	]	X
ejpam-5709	12	4	:	:	PUNCT
ejpam-5709	12	5	if	if	SCONJ
ejpam-5709	12	6	a	a	DET
ejpam-5709	12	7	prime	prime	NOUN
ejpam-5709	12	8	p	p	NOUN
ejpam-5709	12	9	is	be	AUX
ejpam-5709	12	10	expressed	express	VERB
ejpam-5709	12	11	in	in	ADP
ejpam-5709	12	12	the	the	DET
ejpam-5709	12	13	decimal	decimal	ADJ
ejpam-5709	12	14	representation	representation	NOUN
ejpam-5709	12	15	as	as	ADP
ejpam-5709	12	16	p	p	PROPN
ejpam-5709	12	17	=	=	SYM
ejpam-5709	12	18	an10	an10	PROPN
ejpam-5709	12	19	n	n	PROPN
ejpam-5709	12	20	+	+	CCONJ
ejpam-5709	12	21	an−110	an−110	PROPN
ejpam-5709	12	22	n−1	n−1	PROPN
ejpam-5709	12	23	+	+	PROPN
ejpam-5709	12	24	·	·	PUNCT
ejpam-5709	12	25	·	·	PUNCT
ejpam-5709	12	26	·	·	PUNCT
ejpam-5709	13	1	+	+	CCONJ
ejpam-5709	13	2	a110	a110	PROPN
ejpam-5709	13	3	+	+	CCONJ
ejpam-5709	13	4	a0	a0	PROPN
ejpam-5709	13	5	,	,	PUNCT
ejpam-5709	13	6	then	then	ADV
ejpam-5709	13	7	the	the	DET
ejpam-5709	13	8	polynomial	polynomial	ADJ
ejpam-5709	13	9	f(x	f(x	PROPN
ejpam-5709	13	10	)	)	PUNCT
ejpam-5709	13	11	=	=	SYM
ejpam-5709	14	1	anx	anx	ADJ
ejpam-5709	14	2	n	n	PROPN
ejpam-5709	14	3	+	+	CCONJ
ejpam-5709	14	4	an−1x	an−1x	PROPN
ejpam-5709	14	5	n−1	n−1	PROPN
ejpam-5709	14	6	+	+	CCONJ
ejpam-5709	14	7	·	·	PUNCT
ejpam-5709	14	8	·	·	PUNCT
ejpam-5709	14	9	·	·	PUNCT
ejpam-5709	15	1	+	+	NUM
ejpam-5709	15	2	a1x+	a1x+	NOUN
ejpam-5709	15	3	a0	a0	NOUN
ejpam-5709	15	4	is	be	AUX
ejpam-5709	15	5	irreducible	irreducible	ADJ
ejpam-5709	15	6	in	in	ADP
ejpam-5709	15	7	z[x	z[x	NOUN
ejpam-5709	15	8	]	]	PUNCT
ejpam-5709	15	9	.	.	PUNCT
ejpam-5709	16	1	this	this	DET
ejpam-5709	16	2	result	result	NOUN
ejpam-5709	16	3	was	be	AUX
ejpam-5709	16	4	subsequently	subsequently	ADV
ejpam-5709	16	5	generalized	generalize	VERB
ejpam-5709	16	6	to	to	ADP
ejpam-5709	16	7	any	any	DET
ejpam-5709	16	8	base	base	NOUN
ejpam-5709	16	9	b	b	NOUN
ejpam-5709	16	10	by	by	ADP
ejpam-5709	16	11	brillhart	brillhart	NOUN
ejpam-5709	16	12	et	et	PROPN
ejpam-5709	16	13	al	al	PROPN
ejpam-5709	16	14	.	.	PUNCT
ejpam-5709	17	1	[	[	X
ejpam-5709	17	2	2	2	X
ejpam-5709	17	3	]	]	PUNCT
ejpam-5709	17	4	and	and	CCONJ
ejpam-5709	17	5	murty	murty	NOUN
ejpam-5709	18	1	[	[	X
ejpam-5709	18	2	5	5	NUM
ejpam-5709	18	3	]	]	PUNCT
ejpam-5709	18	4	.	.	PUNCT
ejpam-5709	19	1	in	in	ADP
ejpam-5709	19	2	∗corresponding	∗corresponde	VERB
ejpam-5709	19	3	author	author	NOUN
ejpam-5709	19	4	.	.	PUNCT
ejpam-5709	20	1	doi	doi	NOUN
ejpam-5709	20	2	:	:	PUNCT
ejpam-5709	20	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5709	https://doi.org/10.29020/nybg.ejpam.v18i1.5709	NUM
ejpam-5709	20	4	email	email	NOUN
ejpam-5709	20	5	addresses	address	VERB
ejpam-5709	20	6	:	:	PUNCT
ejpam-5709	21	1	phitthayathon@kkumail.com	phitthayathon@kkumail.com	X
ejpam-5709	21	2	(	(	PUNCT
ejpam-5709	21	3	p.	p.	NOUN
ejpam-5709	21	4	phetnun	phetnun	PROPN
ejpam-5709	21	5	)	)	PUNCT
ejpam-5709	21	6	,	,	PUNCT
ejpam-5709	21	7	naraka@kku.ac.th	naraka@kku.ac.th	PROPN
ejpam-5709	21	8	(	(	PUNCT
ejpam-5709	21	9	n.r	n.r	PROPN
ejpam-5709	21	10	.	.	PROPN
ejpam-5709	21	11	kanasri	kanasri	PROPN
ejpam-5709	21	12	)	)	PUNCT
ejpam-5709	21	13	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5709	22	1	1	1	NUM
ejpam-5709	22	2	copyright	copyright	NOUN
ejpam-5709	22	3	:	:	PUNCT
ejpam-5709	22	4	©	©	PROPN
ejpam-5709	22	5	2025	2025	NUM
ejpam-5709	22	6	the	the	DET
ejpam-5709	22	7	author(s	author(s	NOUN
ejpam-5709	22	8	)	)	PUNCT
ejpam-5709	22	9	.	.	PUNCT
ejpam-5709	23	1	(	(	PUNCT
ejpam-5709	23	2	cc	cc	NOUN
ejpam-5709	23	3	by	by	ADP
ejpam-5709	23	4	-	-	PUNCT
ejpam-5709	23	5	nc	nc	PROPN
ejpam-5709	23	6	4.0	4.0	NUM
ejpam-5709	23	7	)	)	PUNCT
ejpam-5709	23	8	p.	p.	NOUN
ejpam-5709	23	9	phetnun	phetnun	PROPN
ejpam-5709	23	10	,	,	PUNCT
ejpam-5709	23	11	n.r	n.r	PROPN
ejpam-5709	23	12	.	.	PROPN
ejpam-5709	23	13	kanasri	kanasri	PROPN
ejpam-5709	23	14	/	/	SYM
ejpam-5709	23	15	eur	eur	PROPN
ejpam-5709	23	16	.	.	PUNCT
ejpam-5709	24	1	j.	j.	PROPN
ejpam-5709	24	2	pure	pure	PROPN
ejpam-5709	24	3	appl	appl	PROPN
ejpam-5709	24	4	.	.	PROPN
ejpam-5709	24	5	math	math	PROPN
ejpam-5709	24	6	,	,	PUNCT
ejpam-5709	24	7	18	18	NUM
ejpam-5709	24	8	(	(	PUNCT
ejpam-5709	24	9	1	1	NUM
ejpam-5709	24	10	)	)	PUNCT
ejpam-5709	24	11	(	(	PUNCT
ejpam-5709	24	12	2025	2025	NUM
ejpam-5709	24	13	)	)	PUNCT
ejpam-5709	24	14	,	,	PUNCT
ejpam-5709	24	15	5709	5709	NUM
ejpam-5709	24	16	2	2	NUM
ejpam-5709	24	17	of	of	ADP
ejpam-5709	24	18	15	15	NUM
ejpam-5709	24	19	1982	1982	NUM
ejpam-5709	24	20	,	,	PUNCT
ejpam-5709	24	21	filaseta	filaseta	PROPN
ejpam-5709	25	1	[	[	X
ejpam-5709	25	2	3	3	NUM
ejpam-5709	25	3	]	]	PUNCT
ejpam-5709	25	4	generalized	generalize	VERB
ejpam-5709	25	5	this	this	DET
ejpam-5709	25	6	result	result	NOUN
ejpam-5709	25	7	in	in	ADP
ejpam-5709	25	8	another	another	DET
ejpam-5709	25	9	way	way	NOUN
ejpam-5709	25	10	by	by	ADP
ejpam-5709	25	11	considering	consider	VERB
ejpam-5709	25	12	wp	wp	PRON
ejpam-5709	25	13	instead	instead	ADV
ejpam-5709	25	14	of	of	ADP
ejpam-5709	25	15	p	p	X
ejpam-5709	25	16	:	:	PUNCT
ejpam-5709	25	17	if	if	SCONJ
ejpam-5709	25	18	w	w	PROPN
ejpam-5709	25	19	and	and	CCONJ
ejpam-5709	25	20	b	b	NOUN
ejpam-5709	25	21	are	be	AUX
ejpam-5709	25	22	positive	positive	ADJ
ejpam-5709	25	23	integers	integer	NOUN
ejpam-5709	25	24	with	with	ADP
ejpam-5709	25	25	w	w	PROPN
ejpam-5709	25	26	<	<	X
ejpam-5709	25	27	b	b	PROPN
ejpam-5709	25	28	and	and	CCONJ
ejpam-5709	25	29	wp	wp	NOUN
ejpam-5709	25	30	=	=	NOUN
ejpam-5709	25	31	bmbm	bmbm	NOUN
ejpam-5709	26	1	+	+	CCONJ
ejpam-5709	26	2	bm−1b	bm−1b	VERB
ejpam-5709	26	3	m−1	m−1	PROPN
ejpam-5709	26	4	+	+	CCONJ
ejpam-5709	26	5	·	·	PUNCT
ejpam-5709	26	6	·	·	PUNCT
ejpam-5709	26	7	·	·	PUNCT
ejpam-5709	26	8	+	+	NUM
ejpam-5709	26	9	b1b+	b1b+	PROPN
ejpam-5709	26	10	b0	b0	NOUN
ejpam-5709	26	11	is	be	AUX
ejpam-5709	26	12	the	the	DET
ejpam-5709	26	13	base	base	NOUN
ejpam-5709	26	14	-	-	PUNCT
ejpam-5709	26	15	b	b	NOUN
ejpam-5709	26	16	representation	representation	NOUN
ejpam-5709	26	17	of	of	ADP
ejpam-5709	26	18	wp	wp	PROPN
ejpam-5709	26	19	,	,	PUNCT
ejpam-5709	26	20	where	where	SCONJ
ejpam-5709	26	21	p	p	NOUN
ejpam-5709	26	22	is	be	AUX
ejpam-5709	26	23	a	a	DET
ejpam-5709	26	24	prime	prime	NOUN
ejpam-5709	26	25	,	,	PUNCT
ejpam-5709	26	26	then	then	ADV
ejpam-5709	26	27	the	the	DET
ejpam-5709	26	28	polynomial	polynomial	ADJ
ejpam-5709	26	29	f(x	f(x	PROPN
ejpam-5709	26	30	)	)	PUNCT
ejpam-5709	27	1	=	=	PRON
ejpam-5709	28	1	bmxm+	bmxm+	X
ejpam-5709	28	2	bm−1x	bm−1x	NOUN
ejpam-5709	28	3	m−1	m−1	PROPN
ejpam-5709	28	4	+	+	CCONJ
ejpam-5709	28	5	·	·	PUNCT
ejpam-5709	28	6	·	·	PUNCT
ejpam-5709	28	7	·	·	PUNCT
ejpam-5709	29	1	+	+	NUM
ejpam-5709	29	2	b1x+	b1x+	NOUN
ejpam-5709	29	3	b0	b0	NOUN
ejpam-5709	29	4	is	be	AUX
ejpam-5709	29	5	irreducible	irreducible	ADJ
ejpam-5709	29	6	over	over	ADP
ejpam-5709	29	7	q.	q.	NOUN
ejpam-5709	29	8	in	in	ADP
ejpam-5709	29	9	another	another	DET
ejpam-5709	29	10	direction	direction	NOUN
ejpam-5709	29	11	,	,	PUNCT
ejpam-5709	29	12	let	let	VERB
ejpam-5709	29	13	k	k	PROPN
ejpam-5709	29	14	=	=	PUNCT
ejpam-5709	29	15	q	q	ADJ
ejpam-5709	29	16	(	(	PUNCT
ejpam-5709	29	17	√	√	NUM
ejpam-5709	29	18	m	m	PRON
ejpam-5709	29	19	)	)	PUNCT
ejpam-5709	29	20	with	with	ADP
ejpam-5709	29	21	a	a	DET
ejpam-5709	29	22	unique	unique	ADJ
ejpam-5709	29	23	squarefree	squarefree	NOUN
ejpam-5709	29	24	integer	integer	NOUN
ejpam-5709	29	25	m	m	PROPN
ejpam-5709	29	26	̸=	̸=	PROPN
ejpam-5709	29	27	1	1	NUM
ejpam-5709	29	28	,	,	PUNCT
ejpam-5709	29	29	be	be	AUX
ejpam-5709	29	30	a	a	DET
ejpam-5709	29	31	quadratic	quadratic	ADJ
ejpam-5709	29	32	field	field	NOUN
ejpam-5709	29	33	and	and	CCONJ
ejpam-5709	29	34	ok	ok	ADJ
ejpam-5709	29	35	denotes	denote	NOUN
ejpam-5709	29	36	the	the	DET
ejpam-5709	29	37	set	set	NOUN
ejpam-5709	29	38	of	of	ADP
ejpam-5709	29	39	algebraic	algebraic	ADJ
ejpam-5709	29	40	integers	integer	NOUN
ejpam-5709	29	41	that	that	PRON
ejpam-5709	29	42	lie	lie	VERB
ejpam-5709	29	43	in	in	ADP
ejpam-5709	29	44	k	k	PROPN
ejpam-5709	29	45	,	,	PUNCT
ejpam-5709	29	46	called	call	VERB
ejpam-5709	29	47	the	the	DET
ejpam-5709	29	48	ring	ring	NOUN
ejpam-5709	29	49	of	of	ADP
ejpam-5709	29	50	integers	integer	NOUN
ejpam-5709	29	51	of	of	ADP
ejpam-5709	29	52	k.	k.	NOUN
ejpam-5709	29	53	we	we	PRON
ejpam-5709	29	54	note	note	VERB
ejpam-5709	29	55	that	that	SCONJ
ejpam-5709	29	56	ok	ok	INTJ
ejpam-5709	29	57	is	be	AUX
ejpam-5709	29	58	an	an	DET
ejpam-5709	29	59	integral	integral	ADJ
ejpam-5709	29	60	domain	domain	NOUN
ejpam-5709	29	61	and	and	CCONJ
ejpam-5709	29	62	k	k	PROPN
ejpam-5709	29	63	is	be	AUX
ejpam-5709	29	64	its	its	PRON
ejpam-5709	29	65	quotient	quotient	NOUN
ejpam-5709	29	66	field	field	NOUN
ejpam-5709	29	67	.	.	PUNCT
ejpam-5709	30	1	moreover	moreover	ADV
ejpam-5709	30	2	,	,	PUNCT
ejpam-5709	30	3	the	the	DET
ejpam-5709	30	4	set	set	NOUN
ejpam-5709	30	5	of	of	ADP
ejpam-5709	30	6	units	unit	NOUN
ejpam-5709	30	7	in	in	ADP
ejpam-5709	30	8	the	the	DET
ejpam-5709	30	9	polynomial	polynomial	ADJ
ejpam-5709	30	10	ring	ring	NOUN
ejpam-5709	30	11	ok	ok	INTJ
ejpam-5709	31	1	[	[	X
ejpam-5709	31	2	x	x	X
ejpam-5709	31	3	]	]	X
ejpam-5709	31	4	is	be	AUX
ejpam-5709	31	5	u(ok	u(ok	NOUN
ejpam-5709	31	6	)	)	PUNCT
ejpam-5709	31	7	,	,	PUNCT
ejpam-5709	31	8	the	the	DET
ejpam-5709	31	9	group	group	NOUN
ejpam-5709	31	10	of	of	ADP
ejpam-5709	31	11	units	unit	NOUN
ejpam-5709	31	12	in	in	ADP
ejpam-5709	31	13	ok	ok	ADJ
ejpam-5709	31	14	.	.	PUNCT
ejpam-5709	32	1	the	the	DET
ejpam-5709	32	2	quadratic	quadratic	ADJ
ejpam-5709	32	3	field	field	NOUN
ejpam-5709	32	4	k	k	NOUN
ejpam-5709	32	5	is	be	AUX
ejpam-5709	32	6	said	say	VERB
ejpam-5709	32	7	to	to	PART
ejpam-5709	32	8	be	be	AUX
ejpam-5709	32	9	real	real	ADJ
ejpam-5709	32	10	if	if	SCONJ
ejpam-5709	32	11	m	m	VERB
ejpam-5709	32	12	>	>	X
ejpam-5709	32	13	0	0	PUNCT
ejpam-5709	32	14	and	and	CCONJ
ejpam-5709	32	15	imaginary	imaginary	ADJ
ejpam-5709	32	16	if	if	SCONJ
ejpam-5709	32	17	m	m	VERB
ejpam-5709	32	18	<	<	X
ejpam-5709	32	19	0	0	NUM
ejpam-5709	32	20	.	.	PUNCT
ejpam-5709	32	21	by	by	ADP
ejpam-5709	32	22	letting	let	VERB
ejpam-5709	32	23	the	the	DET
ejpam-5709	32	24	following	follow	VERB
ejpam-5709	32	25	notation	notation	NOUN
ejpam-5709	32	26	σm	σm	X
ejpam-5709	32	27	=	=	PUNCT
ejpam-5709	32	28			PUNCT
ejpam-5709	32	29	√	√	INTJ
ejpam-5709	32	30	m	m	VERB
ejpam-5709	32	31	if	if	SCONJ
ejpam-5709	32	32	m	m	VERB
ejpam-5709	32	33	̸≡	̸≡	VERB
ejpam-5709	32	34	1	1	NUM
ejpam-5709	32	35	(	(	PUNCT
ejpam-5709	32	36	mod	mod	PROPN
ejpam-5709	32	37	4	4	NUM
ejpam-5709	32	38	)	)	PUNCT
ejpam-5709	32	39	,	,	PUNCT
ejpam-5709	32	40	1	1	NUM
ejpam-5709	32	41	+	+	CCONJ
ejpam-5709	32	42	√	√	NUM
ejpam-5709	32	43	m	m	VERB
ejpam-5709	32	44	2	2	NUM
ejpam-5709	32	45	if	if	SCONJ
ejpam-5709	32	46	m	m	VERB
ejpam-5709	32	47	≡	≡	PROPN
ejpam-5709	32	48	1	1	NUM
ejpam-5709	32	49	(	(	PUNCT
ejpam-5709	32	50	mod	mod	NOUN
ejpam-5709	32	51	4	4	NUM
ejpam-5709	32	52	)	)	PUNCT
ejpam-5709	32	53	,	,	PUNCT
ejpam-5709	32	54	we	we	PRON
ejpam-5709	32	55	have	have	VERB
ejpam-5709	32	56	ok	ok	ADJ
ejpam-5709	32	57	=	=	SYM
ejpam-5709	32	58	{	{	PUNCT
ejpam-5709	32	59	a+	a+	PUNCT
ejpam-5709	32	60	bσm	bσm	NOUN
ejpam-5709	32	61	|	|	ADV
ejpam-5709	32	62	a	a	PRON
ejpam-5709	32	63	,	,	PUNCT
ejpam-5709	32	64	b	b	X
ejpam-5709	32	65	∈	∈	PROPN
ejpam-5709	32	66	z	z	NOUN
ejpam-5709	32	67	}	}	PUNCT
ejpam-5709	32	68	.	.	PUNCT
ejpam-5709	33	1	more	more	ADV
ejpam-5709	33	2	specifically	specifically	ADV
ejpam-5709	33	3	,	,	PUNCT
ejpam-5709	33	4	we	we	PRON
ejpam-5709	33	5	see	see	VERB
ejpam-5709	33	6	that	that	SCONJ
ejpam-5709	33	7	the	the	DET
ejpam-5709	33	8	ring	ring	NOUN
ejpam-5709	33	9	of	of	ADP
ejpam-5709	33	10	gaussian	gaussian	ADJ
ejpam-5709	33	11	integers	integer	NOUN
ejpam-5709	33	12	z[i	z[i	NOUN
ejpam-5709	33	13	]	]	X
ejpam-5709	33	14	is	be	AUX
ejpam-5709	33	15	the	the	DET
ejpam-5709	33	16	ring	ring	NOUN
ejpam-5709	33	17	of	of	ADP
ejpam-5709	33	18	integers	integer	NOUN
ejpam-5709	33	19	of	of	ADP
ejpam-5709	33	20	q(i	q(i	NOUN
ejpam-5709	33	21	)	)	PUNCT
ejpam-5709	34	1	[	[	X
ejpam-5709	34	2	1	1	NUM
ejpam-5709	34	3	]	]	PUNCT
ejpam-5709	34	4	.	.	PUNCT
ejpam-5709	35	1	it	it	PRON
ejpam-5709	35	2	is	be	AUX
ejpam-5709	35	3	important	important	ADJ
ejpam-5709	35	4	to	to	PART
ejpam-5709	35	5	note	note	VERB
ejpam-5709	35	6	that	that	SCONJ
ejpam-5709	35	7	every	every	DET
ejpam-5709	35	8	prime	prime	ADJ
ejpam-5709	35	9	element	element	NOUN
ejpam-5709	35	10	of	of	ADP
ejpam-5709	35	11	ok	ok	ADJ
ejpam-5709	35	12	is	be	AUX
ejpam-5709	35	13	irreducible	irreducible	ADJ
ejpam-5709	35	14	;	;	PUNCT
ejpam-5709	35	15	the	the	DET
ejpam-5709	35	16	converse	converse	NOUN
ejpam-5709	35	17	is	be	AUX
ejpam-5709	35	18	not	not	PART
ejpam-5709	35	19	generally	generally	ADV
ejpam-5709	35	20	true	true	ADJ
ejpam-5709	35	21	but	but	CCONJ
ejpam-5709	35	22	holds	hold	VERB
ejpam-5709	35	23	if	if	SCONJ
ejpam-5709	35	24	ok	ok	ADJ
ejpam-5709	35	25	is	be	VERB
ejpam-5709	35	26	a	a	DET
ejpam-5709	35	27	unique	unique	ADJ
ejpam-5709	35	28	factorization	factorization	NOUN
ejpam-5709	35	29	domain	domain	NOUN
ejpam-5709	35	30	.	.	PUNCT
ejpam-5709	36	1	we	we	PRON
ejpam-5709	36	2	say	say	VERB
ejpam-5709	36	3	that	that	SCONJ
ejpam-5709	36	4	a	a	DET
ejpam-5709	36	5	nonzero	nonzero	ADJ
ejpam-5709	36	6	polynomial	polynomial	ADJ
ejpam-5709	36	7	p(x	p(x	PROPN
ejpam-5709	36	8	)	)	PUNCT
ejpam-5709	36	9	∈	∈	NOUN
ejpam-5709	36	10	ok	ok	INTJ
ejpam-5709	37	1	[	[	X
ejpam-5709	37	2	x	x	X
ejpam-5709	37	3	]	]	X
ejpam-5709	37	4	is	be	AUX
ejpam-5709	37	5	irreducible	irreducible	ADJ
ejpam-5709	37	6	in	in	ADP
ejpam-5709	37	7	ok	ok	ADJ
ejpam-5709	37	8	[	[	X
ejpam-5709	37	9	x	x	X
ejpam-5709	37	10	]	]	X
ejpam-5709	37	11	if	if	SCONJ
ejpam-5709	37	12	it	it	PRON
ejpam-5709	37	13	is	be	AUX
ejpam-5709	37	14	an	an	DET
ejpam-5709	37	15	irreducible	irreducible	ADJ
ejpam-5709	37	16	element	element	NOUN
ejpam-5709	37	17	of	of	ADP
ejpam-5709	37	18	ok	ok	ADJ
ejpam-5709	37	19	[	[	X
ejpam-5709	37	20	x	x	X
ejpam-5709	37	21	]	]	X
ejpam-5709	37	22	,	,	PUNCT
ejpam-5709	37	23	in	in	ADP
ejpam-5709	37	24	other	other	ADJ
ejpam-5709	37	25	words	word	NOUN
ejpam-5709	37	26	,	,	PUNCT
ejpam-5709	37	27	p(x	p(x	PROPN
ejpam-5709	37	28	)	)	PUNCT
ejpam-5709	37	29	is	be	AUX
ejpam-5709	37	30	not	not	PART
ejpam-5709	37	31	a	a	DET
ejpam-5709	37	32	unit	unit	NOUN
ejpam-5709	37	33	and	and	CCONJ
ejpam-5709	37	34	p(x	p(x	PROPN
ejpam-5709	37	35	)	)	PUNCT
ejpam-5709	37	36	=	=	PUNCT
ejpam-5709	37	37	f(x)g(x	f(x)g(x	X
ejpam-5709	37	38	)	)	PUNCT
ejpam-5709	37	39	in	in	SCONJ
ejpam-5709	37	40	ok	ok	ADJ
ejpam-5709	37	41	[	[	X
ejpam-5709	37	42	x	x	X
ejpam-5709	37	43	]	]	X
ejpam-5709	37	44	implies	imply	VERB
ejpam-5709	37	45	f(x	f(x	PROPN
ejpam-5709	37	46	)	)	PUNCT
ejpam-5709	37	47	or	or	CCONJ
ejpam-5709	37	48	g(x	g(x	NOUN
ejpam-5709	37	49	)	)	PUNCT
ejpam-5709	37	50	is	be	AUX
ejpam-5709	37	51	a	a	DET
ejpam-5709	37	52	unit	unit	NOUN
ejpam-5709	37	53	in	in	ADP
ejpam-5709	37	54	ok	ok	PROPN
ejpam-5709	37	55	.	.	PUNCT
ejpam-5709	38	1	polynomials	polynomial	NOUN
ejpam-5709	38	2	that	that	PRON
ejpam-5709	38	3	are	be	AUX
ejpam-5709	38	4	not	not	PART
ejpam-5709	38	5	irreducible	irreducible	ADJ
ejpam-5709	38	6	are	be	AUX
ejpam-5709	38	7	said	say	VERB
ejpam-5709	38	8	to	to	PART
ejpam-5709	38	9	be	be	AUX
ejpam-5709	38	10	reducible	reducible	ADJ
ejpam-5709	38	11	.	.	PUNCT
ejpam-5709	39	1	for	for	ADP
ejpam-5709	39	2	β	β	X
ejpam-5709	39	3	=	=	SYM
ejpam-5709	39	4	a+	a+	PUNCT
ejpam-5709	39	5	bσm	bσm	NOUN
ejpam-5709	39	6	∈	∈	NOUN
ejpam-5709	39	7	ok	ok	INTJ
ejpam-5709	39	8	,	,	PUNCT
ejpam-5709	39	9	we	we	PRON
ejpam-5709	39	10	denote	denote	VERB
ejpam-5709	39	11	the	the	DET
ejpam-5709	39	12	norm	norm	NOUN
ejpam-5709	39	13	of	of	ADP
ejpam-5709	39	14	β	β	NOUN
ejpam-5709	39	15	by	by	ADP
ejpam-5709	39	16	n(β	n(β	PROPN
ejpam-5709	39	17	)	)	PUNCT
ejpam-5709	40	1	=	=	PUNCT
ejpam-5709	40	2			PUNCT
ejpam-5709	40	3	a2	a2	PROPN
ejpam-5709	40	4	−mb2	−mb2	INTJ
ejpam-5709	40	5	if	if	SCONJ
ejpam-5709	40	6	m	m	VERB
ejpam-5709	40	7	̸≡	̸≡	VERB
ejpam-5709	40	8	1	1	NUM
ejpam-5709	40	9	(	(	PUNCT
ejpam-5709	40	10	mod	mod	PROPN
ejpam-5709	40	11	4	4	NUM
ejpam-5709	40	12	)	)	PUNCT
ejpam-5709	40	13	,	,	PUNCT
ejpam-5709	40	14	a2	a2	PROPN
ejpam-5709	40	15	+	+	CCONJ
ejpam-5709	40	16	ab+	ab+	PROPN
ejpam-5709	40	17	b2	b2	NOUN
ejpam-5709	40	18	(	(	PUNCT
ejpam-5709	40	19	1−m	1−m	NUM
ejpam-5709	40	20	4	4	NUM
ejpam-5709	40	21	)	)	PUNCT
ejpam-5709	40	22	if	if	SCONJ
ejpam-5709	40	23	m	m	VERB
ejpam-5709	40	24	≡	≡	PROPN
ejpam-5709	40	25	1	1	NUM
ejpam-5709	40	26	(	(	PUNCT
ejpam-5709	40	27	mod	mod	NOUN
ejpam-5709	40	28	4	4	NUM
ejpam-5709	40	29	)	)	PUNCT
ejpam-5709	40	30	.	.	PUNCT
ejpam-5709	41	1	we	we	PRON
ejpam-5709	41	2	have	have	AUX
ejpam-5709	41	3	seen	see	VERB
ejpam-5709	41	4	from	from	ADP
ejpam-5709	41	5	[	[	X
ejpam-5709	41	6	1	1	X
ejpam-5709	41	7	]	]	PUNCT
ejpam-5709	41	8	that	that	SCONJ
ejpam-5709	41	9	if	if	SCONJ
ejpam-5709	41	10	n(β	n(β	NUM
ejpam-5709	41	11	)	)	PUNCT
ejpam-5709	41	12	=	=	SYM
ejpam-5709	41	13	±p	±p	NOUN
ejpam-5709	41	14	,	,	PUNCT
ejpam-5709	41	15	where	where	SCONJ
ejpam-5709	41	16	p	p	NOUN
ejpam-5709	41	17	is	be	AUX
ejpam-5709	41	18	a	a	DET
ejpam-5709	41	19	rational	rational	ADJ
ejpam-5709	41	20	prime	prime	NOUN
ejpam-5709	41	21	,	,	PUNCT
ejpam-5709	41	22	then	then	ADV
ejpam-5709	41	23	β	β	X
ejpam-5709	41	24	is	be	AUX
ejpam-5709	41	25	an	an	DET
ejpam-5709	41	26	irreducible	irreducible	ADJ
ejpam-5709	41	27	element	element	NOUN
ejpam-5709	41	28	.	.	PUNCT
ejpam-5709	42	1	in	in	ADP
ejpam-5709	42	2	addition	addition	NOUN
ejpam-5709	42	3	,	,	PUNCT
ejpam-5709	42	4	ifk	ifk	PROPN
ejpam-5709	42	5	is	be	AUX
ejpam-5709	42	6	an	an	DET
ejpam-5709	42	7	imaginary	imaginary	ADJ
ejpam-5709	42	8	quadratic	quadratic	ADJ
ejpam-5709	42	9	field	field	NOUN
ejpam-5709	42	10	,	,	PUNCT
ejpam-5709	42	11	then	then	ADV
ejpam-5709	42	12	|β|2	|β|2	PROPN
ejpam-5709	42	13	=	=	SYM
ejpam-5709	42	14	n(β	n(β	NUM
ejpam-5709	42	15	)	)	PUNCT
ejpam-5709	42	16	∈	∈	PROPN
ejpam-5709	42	17	n	n	X
ejpam-5709	42	18	for	for	ADP
ejpam-5709	42	19	all	all	DET
ejpam-5709	42	20	β	β	X
ejpam-5709	42	21	∈	∈	NOUN
ejpam-5709	42	22	ok\{0	ok\{0	PROPN
ejpam-5709	42	23	}	}	PUNCT
ejpam-5709	42	24	and	and	CCONJ
ejpam-5709	42	25	|β|	|β|	NOUN
ejpam-5709	42	26	=	=	SYM
ejpam-5709	42	27	1	1	NUM
ejpam-5709	42	28	for	for	ADP
ejpam-5709	42	29	all	all	DET
ejpam-5709	42	30	β	β	X
ejpam-5709	42	31	∈	∈	NOUN
ejpam-5709	42	32	u(ok	u(ok	NOUN
ejpam-5709	42	33	)	)	PUNCT
ejpam-5709	42	34	.	.	PUNCT
ejpam-5709	43	1	let	let	VERB
ejpam-5709	43	2	us	we	PRON
ejpam-5709	43	3	recall	recall	VERB
ejpam-5709	43	4	the	the	DET
ejpam-5709	43	5	divisibility	divisibility	NOUN
ejpam-5709	43	6	and	and	CCONJ
ejpam-5709	43	7	congruence	congruence	NOUN
ejpam-5709	43	8	for	for	ADP
ejpam-5709	43	9	elements	element	NOUN
ejpam-5709	43	10	of	of	ADP
ejpam-5709	43	11	ok	ok	NOUN
ejpam-5709	43	12	as	as	SCONJ
ejpam-5709	43	13	follows	follow	VERB
ejpam-5709	43	14	:	:	PUNCT
ejpam-5709	43	15	for	for	ADP
ejpam-5709	43	16	α	α	NOUN
ejpam-5709	43	17	,	,	PUNCT
ejpam-5709	43	18	β	β	X
ejpam-5709	43	19	∈	∈	NOUN
ejpam-5709	43	20	ok	ok	ADJ
ejpam-5709	43	21	with	with	ADP
ejpam-5709	43	22	α	α	PROPN
ejpam-5709	43	23	̸=	̸=	PROPN
ejpam-5709	43	24	0	0	NUM
ejpam-5709	43	25	,	,	PUNCT
ejpam-5709	43	26	we	we	PRON
ejpam-5709	43	27	say	say	VERB
ejpam-5709	43	28	that	that	SCONJ
ejpam-5709	43	29	α	α	PROPN
ejpam-5709	43	30	divides	divide	VERB
ejpam-5709	43	31	β	β	X
ejpam-5709	43	32	,	,	PUNCT
ejpam-5709	43	33	denoted	denote	VERB
ejpam-5709	43	34	by	by	ADP
ejpam-5709	43	35	α	α	PROPN
ejpam-5709	43	36	|	|	ADV
ejpam-5709	43	37	β	β	X
ejpam-5709	43	38	,	,	PUNCT
ejpam-5709	43	39	if	if	SCONJ
ejpam-5709	43	40	there	there	PRON
ejpam-5709	43	41	exists	exist	VERB
ejpam-5709	43	42	δ	δ	PROPN
ejpam-5709	43	43	∈	∈	PROPN
ejpam-5709	43	44	ok	ok	INTJ
ejpam-5709	43	45	such	such	ADJ
ejpam-5709	43	46	that	that	SCONJ
ejpam-5709	43	47	β	β	X
ejpam-5709	43	48	=	=	SYM
ejpam-5709	43	49	αδ	αδ	PROPN
ejpam-5709	43	50	.	.	PUNCT
ejpam-5709	44	1	for	for	ADP
ejpam-5709	44	2	α	α	PROPN
ejpam-5709	44	3	,	,	PUNCT
ejpam-5709	44	4	β	β	X
ejpam-5709	44	5	,	,	PUNCT
ejpam-5709	44	6	γ	γ	PROPN
ejpam-5709	44	7	∈	∈	PROPN
ejpam-5709	44	8	ok	ok	ADJ
ejpam-5709	44	9	with	with	ADP
ejpam-5709	44	10	γ	γ	PROPN
ejpam-5709	44	11	̸=	̸=	PROPN
ejpam-5709	44	12	0	0	NUM
ejpam-5709	44	13	,	,	PUNCT
ejpam-5709	44	14	we	we	PRON
ejpam-5709	44	15	say	say	VERB
ejpam-5709	44	16	that	that	SCONJ
ejpam-5709	44	17	α	α	PROPN
ejpam-5709	44	18	is	be	AUX
ejpam-5709	44	19	congruent	congruent	ADJ
ejpam-5709	44	20	to	to	ADP
ejpam-5709	44	21	β	β	PROPN
ejpam-5709	44	22	modulo	modulo	PROPN
ejpam-5709	44	23	γ	γ	X
ejpam-5709	44	24	,	,	PUNCT
ejpam-5709	44	25	denoted	denote	VERB
ejpam-5709	44	26	by	by	ADP
ejpam-5709	44	27	α	α	PROPN
ejpam-5709	44	28	≡	≡	PROPN
ejpam-5709	44	29	β	β	X
ejpam-5709	44	30	(	(	PUNCT
ejpam-5709	44	31	mod	mod	PROPN
ejpam-5709	44	32	γ	γ	PROPN
ejpam-5709	44	33	)	)	PUNCT
ejpam-5709	44	34	,	,	PUNCT
ejpam-5709	44	35	if	if	SCONJ
ejpam-5709	44	36	γ	γ	PROPN
ejpam-5709	44	37	|	|	ADV
ejpam-5709	44	38	(	(	PUNCT
ejpam-5709	44	39	α−	α−	ADP
ejpam-5709	44	40	β	β	NOUN
ejpam-5709	44	41	)	)	PUNCT
ejpam-5709	44	42	.	.	PUNCT
ejpam-5709	45	1	a	a	DET
ejpam-5709	45	2	complete	complete	ADJ
ejpam-5709	45	3	residue	residue	NOUN
ejpam-5709	45	4	system	system	NOUN
ejpam-5709	45	5	modulo	modulo	VERB
ejpam-5709	45	6	β	β	X
ejpam-5709	45	7	in	in	ADP
ejpam-5709	45	8	ok	ok	PROPN
ejpam-5709	45	9	is	be	AUX
ejpam-5709	45	10	defined	define	VERB
ejpam-5709	45	11	as	as	ADP
ejpam-5709	45	12	in	in	ADP
ejpam-5709	45	13	the	the	DET
ejpam-5709	45	14	following	follow	VERB
ejpam-5709	45	15	definition	definition	NOUN
ejpam-5709	45	16	[	[	X
ejpam-5709	45	17	9	9	NUM
ejpam-5709	45	18	]	]	PUNCT
ejpam-5709	45	19	.	.	PUNCT
ejpam-5709	46	1	definition	definition	NOUN
ejpam-5709	46	2	a.	a.	NOUN
ejpam-5709	46	3	a	a	DET
ejpam-5709	46	4	complete	complete	ADJ
ejpam-5709	46	5	residue	residue	NOUN
ejpam-5709	46	6	system	system	NOUN
ejpam-5709	46	7	modulo	modulo	VERB
ejpam-5709	46	8	β	β	X
ejpam-5709	46	9	in	in	ADP
ejpam-5709	46	10	ok	ok	ADJ
ejpam-5709	46	11	,	,	PUNCT
ejpam-5709	46	12	abbreviated	abbreviate	VERB
ejpam-5709	46	13	by	by	ADP
ejpam-5709	46	14	crs(β	crs(β	PROPN
ejpam-5709	46	15	)	)	PUNCT
ejpam-5709	46	16	,	,	PUNCT
ejpam-5709	46	17	means	mean	VERB
ejpam-5709	46	18	a	a	DET
ejpam-5709	46	19	set	set	NOUN
ejpam-5709	46	20	of	of	ADP
ejpam-5709	46	21	|n(β)|	|n(β)|	ADJ
ejpam-5709	46	22	elements	element	NOUN
ejpam-5709	46	23	c	c	NOUN
ejpam-5709	46	24	=	=	SYM
ejpam-5709	46	25	{	{	PUNCT
ejpam-5709	46	26	α1	α1	PROPN
ejpam-5709	46	27	,	,	PUNCT
ejpam-5709	46	28	α2	α2	ADJ
ejpam-5709	46	29	,	,	PUNCT
ejpam-5709	46	30	.	.	PUNCT
ejpam-5709	46	31	.	.	PUNCT
ejpam-5709	47	1	.	.	PUNCT
ejpam-5709	48	1	,	,	PUNCT
ejpam-5709	48	2	α|n(β)|	α|n(β)|	X
ejpam-5709	48	3	}	}	PUNCT
ejpam-5709	48	4	in	in	ADP
ejpam-5709	48	5	ok	ok	ADJ
ejpam-5709	48	6	which	which	PRON
ejpam-5709	48	7	satisfies	satisfy	VERB
ejpam-5709	48	8	the	the	DET
ejpam-5709	48	9	following	following	NOUN
ejpam-5709	48	10	:	:	PUNCT
ejpam-5709	48	11	(	(	PUNCT
ejpam-5709	48	12	i	i	NOUN
ejpam-5709	48	13	)	)	PUNCT
ejpam-5709	48	14	for	for	ADP
ejpam-5709	48	15	each	each	DET
ejpam-5709	48	16	α	α	NOUN
ejpam-5709	48	17	∈	∈	ADJ
ejpam-5709	48	18	ok	ok	INTJ
ejpam-5709	48	19	,	,	PUNCT
ejpam-5709	48	20	there	there	PRON
ejpam-5709	48	21	is	be	VERB
ejpam-5709	48	22	αi	αi	PRON
ejpam-5709	48	23	∈	∈	PROPN
ejpam-5709	48	24	c	c	NOUN
ejpam-5709	48	25	such	such	ADJ
ejpam-5709	48	26	that	that	SCONJ
ejpam-5709	48	27	α	α	PROPN
ejpam-5709	48	28	≡	≡	PROPN
ejpam-5709	48	29	αi	αi	PART
ejpam-5709	48	30	(	(	PUNCT
ejpam-5709	48	31	mod	mod	PROPN
ejpam-5709	48	32	β	β	X
ejpam-5709	48	33	)	)	PUNCT
ejpam-5709	48	34	;	;	PUNCT
ejpam-5709	48	35	p.	p.	PROPN
ejpam-5709	48	36	phetnun	phetnun	PROPN
ejpam-5709	48	37	,	,	PUNCT
ejpam-5709	48	38	n.r	n.r	PROPN
ejpam-5709	48	39	.	.	PROPN
ejpam-5709	48	40	kanasri	kanasri	PROPN
ejpam-5709	48	41	/	/	SYM
ejpam-5709	48	42	eur	eur	PROPN
ejpam-5709	48	43	.	.	PUNCT
ejpam-5709	49	1	j.	j.	PROPN
ejpam-5709	49	2	pure	pure	PROPN
ejpam-5709	49	3	appl	appl	PROPN
ejpam-5709	49	4	.	.	PROPN
ejpam-5709	49	5	math	math	PROPN
ejpam-5709	49	6	,	,	PUNCT
ejpam-5709	49	7	18	18	NUM
ejpam-5709	49	8	(	(	PUNCT
ejpam-5709	49	9	1	1	NUM
ejpam-5709	49	10	)	)	PUNCT
ejpam-5709	49	11	(	(	PUNCT
ejpam-5709	49	12	2025	2025	NUM
ejpam-5709	49	13	)	)	PUNCT
ejpam-5709	49	14	,	,	PUNCT
ejpam-5709	49	15	5709	5709	NUM
ejpam-5709	49	16	3	3	NUM
ejpam-5709	49	17	of	of	ADP
ejpam-5709	49	18	15	15	NUM
ejpam-5709	49	19	(	(	PUNCT
ejpam-5709	49	20	ii	ii	NOUN
ejpam-5709	49	21	)	)	PUNCT
ejpam-5709	49	22	αi	αi	VERB
ejpam-5709	49	23	̸≡	̸≡	PROPN
ejpam-5709	50	1	αj	αj	PROPN
ejpam-5709	50	2	(	(	PUNCT
ejpam-5709	50	3	mod	mod	PROPN
ejpam-5709	50	4	β	β	X
ejpam-5709	50	5	)	)	PUNCT
ejpam-5709	50	6	for	for	ADP
ejpam-5709	50	7	all	all	DET
ejpam-5709	50	8	i	i	PROPN
ejpam-5709	50	9	,	,	PUNCT
ejpam-5709	50	10	j	j	PROPN
ejpam-5709	50	11	∈	∈	PROPN
ejpam-5709	50	12	{	{	PUNCT
ejpam-5709	50	13	1	1	NUM
ejpam-5709	50	14	,	,	PUNCT
ejpam-5709	50	15	2	2	NUM
ejpam-5709	50	16	,	,	PUNCT
ejpam-5709	50	17	.	.	PUNCT
ejpam-5709	50	18	.	.	PUNCT
ejpam-5709	50	19	.	.	PUNCT
ejpam-5709	51	1	,	,	PUNCT
ejpam-5709	51	2	|n(β)|	|n(β)|	VERB
ejpam-5709	51	3	}	}	PUNCT
ejpam-5709	51	4	with	with	ADP
ejpam-5709	51	5	i	i	PROPN
ejpam-5709	51	6	̸=	̸=	PROPN
ejpam-5709	51	7	j.	j.	PROPN
ejpam-5709	51	8	for	for	ADP
ejpam-5709	51	9	example	example	NOUN
ejpam-5709	51	10	,	,	PUNCT
ejpam-5709	51	11	the	the	DET
ejpam-5709	51	12	set	set	NOUN
ejpam-5709	51	13	c	c	NOUN
ejpam-5709	51	14	=	=	PRON
ejpam-5709	51	15	{	{	PUNCT
ejpam-5709	51	16	x+	x+	PROPN
ejpam-5709	51	17	yi	yi	NOUN
ejpam-5709	52	1	|	|	ADV
ejpam-5709	52	2	x	x	PUNCT
ejpam-5709	52	3	=	=	SYM
ejpam-5709	52	4	0	0	NUM
ejpam-5709	52	5	,	,	PUNCT
ejpam-5709	52	6	1	1	NUM
ejpam-5709	52	7	,	,	PUNCT
ejpam-5709	52	8	.	.	PUNCT
ejpam-5709	52	9	.	.	PUNCT
ejpam-5709	53	1	.	.	PUNCT
ejpam-5709	54	1	,	,	PUNCT
ejpam-5709	54	2	a2	a2	PROPN
ejpam-5709	54	3	+	+	CCONJ
ejpam-5709	54	4	b2	b2	NOUN
ejpam-5709	54	5	d	d	NOUN
ejpam-5709	54	6	−	−	PROPN
ejpam-5709	54	7	1	1	NUM
ejpam-5709	54	8	and	and	CCONJ
ejpam-5709	54	9	y	y	PROPN
ejpam-5709	54	10	=	=	SYM
ejpam-5709	54	11	0	0	NUM
ejpam-5709	54	12	,	,	PUNCT
ejpam-5709	54	13	1	1	NUM
ejpam-5709	54	14	,	,	PUNCT
ejpam-5709	54	15	.	.	PUNCT
ejpam-5709	54	16	.	.	PUNCT
ejpam-5709	55	1	.	.	PUNCT
ejpam-5709	56	1	,	,	PUNCT
ejpam-5709	56	2	d−	d−	PROPN
ejpam-5709	56	3	1	1	NUM
ejpam-5709	56	4	}	}	PUNCT
ejpam-5709	56	5	(	(	PUNCT
ejpam-5709	56	6	1	1	X
ejpam-5709	56	7	)	)	PUNCT
ejpam-5709	56	8	is	be	AUX
ejpam-5709	56	9	a	a	DET
ejpam-5709	56	10	crs(β	crs(β	NOUN
ejpam-5709	56	11	)	)	PUNCT
ejpam-5709	56	12	,	,	PUNCT
ejpam-5709	56	13	where	where	SCONJ
ejpam-5709	56	14	β	β	X
ejpam-5709	56	15	=	=	SYM
ejpam-5709	56	16	a+	a+	PUNCT
ejpam-5709	56	17	bi	bi	PROPN
ejpam-5709	56	18	∈	∈	PROPN
ejpam-5709	56	19	z[i	z[i	PROPN
ejpam-5709	56	20	]	]	PUNCT
ejpam-5709	56	21	with	with	ADP
ejpam-5709	56	22	d	d	PROPN
ejpam-5709	56	23	=	=	SYM
ejpam-5709	56	24	gcd	gcd	PROPN
ejpam-5709	56	25	(	(	PUNCT
ejpam-5709	56	26	a	a	DET
ejpam-5709	56	27	,	,	PUNCT
ejpam-5709	56	28	b	b	NOUN
ejpam-5709	56	29	)	)	PUNCT
ejpam-5709	57	1	[	[	X
ejpam-5709	57	2	11	11	NUM
ejpam-5709	57	3	]	]	PUNCT
ejpam-5709	57	4	.	.	PUNCT
ejpam-5709	58	1	it	it	PRON
ejpam-5709	58	2	is	be	AUX
ejpam-5709	58	3	clear	clear	ADJ
ejpam-5709	58	4	that	that	SCONJ
ejpam-5709	58	5	the	the	DET
ejpam-5709	58	6	set	set	NOUN
ejpam-5709	58	7	c′	c′	NOUN
ejpam-5709	58	8	:	:	PUNCT
ejpam-5709	58	9	=	=	SYM
ejpam-5709	58	10	{	{	PUNCT
ejpam-5709	58	11	x+	x+	PROPN
ejpam-5709	58	12	yi	yi	NOUN
ejpam-5709	59	1	|	|	ADV
ejpam-5709	59	2	x	x	PUNCT
ejpam-5709	60	1	=	=	SYM
ejpam-5709	60	2	0	0	NUM
ejpam-5709	60	3	,	,	PUNCT
ejpam-5709	60	4	1	1	NUM
ejpam-5709	60	5	,	,	PUNCT
ejpam-5709	60	6	.	.	PUNCT
ejpam-5709	60	7	.	.	PUNCT
ejpam-5709	61	1	.	.	PUNCT
ejpam-5709	62	1	,	,	PUNCT
ejpam-5709	62	2	max{|a|	max{|a|	X
ejpam-5709	62	3	,	,	PUNCT
ejpam-5709	62	4	|b|	|b|	VERB
ejpam-5709	62	5	}	}	PUNCT
ejpam-5709	62	6	−	−	PROPN
ejpam-5709	62	7	1	1	NUM
ejpam-5709	62	8	and	and	CCONJ
ejpam-5709	62	9	y	y	PROPN
ejpam-5709	62	10	=	=	SYM
ejpam-5709	62	11	0	0	NUM
ejpam-5709	62	12	,	,	PUNCT
ejpam-5709	62	13	1	1	NUM
ejpam-5709	62	14	,	,	PUNCT
ejpam-5709	62	15	.	.	PUNCT
ejpam-5709	62	16	.	.	PUNCT
ejpam-5709	63	1	.	.	PUNCT
ejpam-5709	64	1	,	,	PUNCT
ejpam-5709	64	2	d−	d−	PROPN
ejpam-5709	64	3	1	1	NUM
ejpam-5709	64	4	}	}	SYM
ejpam-5709	64	5	⊆	⊆	NUM
ejpam-5709	64	6	c.	c.	NOUN
ejpam-5709	64	7	in	in	ADP
ejpam-5709	64	8	2017	2017	NUM
ejpam-5709	64	9	,	,	PUNCT
ejpam-5709	64	10	singthongla	singthongla	PROPN
ejpam-5709	64	11	et	et	PROPN
ejpam-5709	64	12	al	al	PROPN
ejpam-5709	64	13	.	.	PUNCT
ejpam-5709	65	1	[	[	X
ejpam-5709	65	2	12	12	NUM
ejpam-5709	65	3	]	]	PUNCT
ejpam-5709	65	4	established	establish	VERB
ejpam-5709	65	5	the	the	DET
ejpam-5709	65	6	irreducibility	irreducibility	NOUN
ejpam-5709	65	7	criterion	criterion	NOUN
ejpam-5709	65	8	for	for	ADP
ejpam-5709	65	9	polynomials	polynomial	NOUN
ejpam-5709	65	10	in	in	ADP
ejpam-5709	65	11	z[i][x	z[i][x	NOUN
ejpam-5709	65	12	]	]	PUNCT
ejpam-5709	65	13	using	use	VERB
ejpam-5709	65	14	the	the	DET
ejpam-5709	65	15	crs(β	crs(β	NOUN
ejpam-5709	65	16	)	)	PUNCT
ejpam-5709	65	17	in	in	ADP
ejpam-5709	65	18	(	(	PUNCT
ejpam-5709	65	19	1	1	NUM
ejpam-5709	65	20	)	)	PUNCT
ejpam-5709	65	21	.	.	PUNCT
ejpam-5709	66	1	the	the	DET
ejpam-5709	66	2	result	result	NOUN
ejpam-5709	66	3	is	be	AUX
ejpam-5709	66	4	as	as	SCONJ
ejpam-5709	66	5	follows	follow	NOUN
ejpam-5709	66	6	.	.	PUNCT
ejpam-5709	67	1	theorem	theorem	ADJ
ejpam-5709	67	2	a.	a.	NOUN
ejpam-5709	67	3	let	let	VERB
ejpam-5709	67	4	β	β	X
ejpam-5709	67	5	∈	∈	PROPN
ejpam-5709	67	6	{	{	PUNCT
ejpam-5709	67	7	2±	2±	NUM
ejpam-5709	67	8	2i	2i	NUM
ejpam-5709	67	9	,	,	PUNCT
ejpam-5709	67	10	1±	1±	NUM
ejpam-5709	67	11	3i	3i	NOUN
ejpam-5709	67	12	,	,	PUNCT
ejpam-5709	67	13	3±	3±	NUM
ejpam-5709	67	14	i	i	NOUN
ejpam-5709	67	15	}	}	PUNCT
ejpam-5709	67	16	or	or	CCONJ
ejpam-5709	67	17	β	β	X
ejpam-5709	67	18	=	=	SYM
ejpam-5709	67	19	a+	a+	PUNCT
ejpam-5709	67	20	bi	bi	PROPN
ejpam-5709	67	21	∈	∈	PROPN
ejpam-5709	67	22	z[i	z[i	PROPN
ejpam-5709	67	23	]	]	X
ejpam-5709	67	24	be	be	VERB
ejpam-5709	67	25	such	such	ADJ
ejpam-5709	67	26	that	that	SCONJ
ejpam-5709	67	27	|β|	|β|	PRON
ejpam-5709	67	28	≥	≥	NOUN
ejpam-5709	67	29	2	2	NUM
ejpam-5709	67	30	+	+	CCONJ
ejpam-5709	67	31	√	√	NUM
ejpam-5709	67	32	2	2	NUM
ejpam-5709	67	33	and	and	CCONJ
ejpam-5709	67	34	a	a	DET
ejpam-5709	67	35	≥	≥	NOUN
ejpam-5709	67	36	1	1	NUM
ejpam-5709	67	37	.	.	PUNCT
ejpam-5709	68	1	for	for	ADP
ejpam-5709	68	2	a	a	DET
ejpam-5709	68	3	gaussian	gaussian	ADJ
ejpam-5709	68	4	prime	prime	NOUN
ejpam-5709	68	5	π	π	PROPN
ejpam-5709	68	6	,	,	PUNCT
ejpam-5709	68	7	if	if	SCONJ
ejpam-5709	68	8	π	π	PROPN
ejpam-5709	68	9	=	=	PUNCT
ejpam-5709	68	10	αnβ	αnβ	PROPN
ejpam-5709	68	11	n	n	PROPN
ejpam-5709	68	12	+	+	CCONJ
ejpam-5709	68	13	αn−1β	αn−1β	PROPN
ejpam-5709	68	14	n−1	n−1	PROPN
ejpam-5709	68	15	+	+	NUM
ejpam-5709	68	16	·	·	PUNCT
ejpam-5709	68	17	·	·	PUNCT
ejpam-5709	68	18	·	·	PUNCT
ejpam-5709	68	19	+	+	NUM
ejpam-5709	68	20	α1β	α1β	X
ejpam-5709	69	1	+	+	CCONJ
ejpam-5709	69	2	α0	α0	ADJ
ejpam-5709	69	3	=	=	NOUN
ejpam-5709	69	4	:	:	PUNCT
ejpam-5709	69	5	f(β	f(β	NUM
ejpam-5709	69	6	)	)	PUNCT
ejpam-5709	69	7	,	,	PUNCT
ejpam-5709	69	8	where	where	SCONJ
ejpam-5709	69	9	n	n	PRON
ejpam-5709	69	10	≥	≥	NOUN
ejpam-5709	69	11	1	1	NUM
ejpam-5709	69	12	,	,	PUNCT
ejpam-5709	69	13	re(αn	re(αn	PROPN
ejpam-5709	69	14	)	)	PUNCT
ejpam-5709	69	15	≥	≥	NOUN
ejpam-5709	69	16	1	1	NUM
ejpam-5709	69	17	,	,	PUNCT
ejpam-5709	69	18	and	and	CCONJ
ejpam-5709	69	19	α0	α0	ADJ
ejpam-5709	69	20	,	,	PUNCT
ejpam-5709	69	21	α1	α1	PROPN
ejpam-5709	69	22	,	,	PUNCT
ejpam-5709	69	23	.	.	PUNCT
ejpam-5709	69	24	.	.	PUNCT
ejpam-5709	69	25	.	.	PUNCT
ejpam-5709	70	1	,	,	PUNCT
ejpam-5709	70	2	αn−1	αn−1	PROPN
ejpam-5709	70	3	∈	∈	PROPN
ejpam-5709	70	4	c′	c′	NOUN
ejpam-5709	70	5	satisfying	satisfy	VERB
ejpam-5709	70	6	re(αn−1	re(αn−1	PRON
ejpam-5709	70	7	)	)	PUNCT
ejpam-5709	70	8	im(αn	im(αn	NOUN
ejpam-5709	70	9	)	)	PUNCT
ejpam-5709	70	10	≥	≥	NOUN
ejpam-5709	70	11	re(αn	re(αn	NOUN
ejpam-5709	70	12	)	)	PUNCT
ejpam-5709	70	13	im(αn−1	im(αn−1	PROPN
ejpam-5709	70	14	)	)	PUNCT
ejpam-5709	70	15	,	,	PUNCT
ejpam-5709	70	16	then	then	ADV
ejpam-5709	70	17	f(x	f(x	PROPN
ejpam-5709	70	18	)	)	PUNCT
ejpam-5709	70	19	is	be	AUX
ejpam-5709	70	20	irreducible	irreducible	ADJ
ejpam-5709	70	21	in	in	ADP
ejpam-5709	70	22	z[i][x	z[i][x	NOUN
ejpam-5709	70	23	]	]	X
ejpam-5709	70	24	.	.	PUNCT
ejpam-5709	71	1	afterward	afterward	ADV
ejpam-5709	71	2	,	,	PUNCT
ejpam-5709	71	3	kanasri	kanasri	PROPN
ejpam-5709	71	4	et	et	PROPN
ejpam-5709	71	5	al	al	PROPN
ejpam-5709	71	6	.	.	PUNCT
ejpam-5709	72	1	[	[	X
ejpam-5709	72	2	4	4	NUM
ejpam-5709	72	3	]	]	X
ejpam-5709	72	4	generalized	generalize	VERB
ejpam-5709	72	5	theorem	theorem	NOUN
ejpam-5709	72	6	a	a	PRON
ejpam-5709	72	7	by	by	ADP
ejpam-5709	72	8	considering	consider	VERB
ejpam-5709	72	9	ωπ	ωπ	PROPN
ejpam-5709	72	10	(	(	PUNCT
ejpam-5709	72	11	ω	ω	PROPN
ejpam-5709	72	12	∈	∈	PROPN
ejpam-5709	72	13	z[i]\{0	z[i]\{0	PROPN
ejpam-5709	72	14	}	}	PUNCT
ejpam-5709	72	15	)	)	PUNCT
ejpam-5709	72	16	instead	instead	ADV
ejpam-5709	72	17	of	of	ADP
ejpam-5709	72	18	π	π	PROPN
ejpam-5709	72	19	as	as	SCONJ
ejpam-5709	72	20	the	the	DET
ejpam-5709	72	21	following	follow	VERB
ejpam-5709	72	22	theorem	theorem	PROPN
ejpam-5709	72	23	.	.	PROPN
ejpam-5709	72	24	theorem	theorem	PROPN
ejpam-5709	72	25	b.	b.	PROPN
ejpam-5709	72	26	let	let	VERB
ejpam-5709	72	27	β	β	VERB
ejpam-5709	72	28	=	=	PUNCT
ejpam-5709	72	29	a	a	DET
ejpam-5709	72	30	+	+	NUM
ejpam-5709	72	31	bi	bi	PROPN
ejpam-5709	72	32	∈	∈	PROPN
ejpam-5709	72	33	z[i	z[i	PROPN
ejpam-5709	72	34	]	]	X
ejpam-5709	72	35	be	be	VERB
ejpam-5709	72	36	such	such	ADJ
ejpam-5709	72	37	that	that	SCONJ
ejpam-5709	72	38	|β|	|β|	PRON
ejpam-5709	72	39	≥	≥	VERB
ejpam-5709	72	40	2|ω|	2|ω|	NUM
ejpam-5709	72	41	+	+	CCONJ
ejpam-5709	72	42	√	√	NUM
ejpam-5709	72	43	2	2	NUM
ejpam-5709	72	44	and	and	CCONJ
ejpam-5709	72	45	a	a	DET
ejpam-5709	72	46	≥	≥	NOUN
ejpam-5709	72	47	|ω|	|ω|	PROPN
ejpam-5709	72	48	.	.	PROPN
ejpam-5709	72	49	for	for	ADP
ejpam-5709	72	50	a	a	DET
ejpam-5709	72	51	gaussian	gaussian	ADJ
ejpam-5709	72	52	prime	prime	NOUN
ejpam-5709	72	53	π	π	PROPN
ejpam-5709	72	54	,	,	PUNCT
ejpam-5709	72	55	if	if	SCONJ
ejpam-5709	72	56	ωπ	ωπ	PRON
ejpam-5709	72	57	=	=	PUNCT
ejpam-5709	72	58	αnβ	αnβ	PROPN
ejpam-5709	72	59	n	n	PROPN
ejpam-5709	72	60	+	+	CCONJ
ejpam-5709	72	61	αn−1β	αn−1β	PROPN
ejpam-5709	72	62	n−1	n−1	PROPN
ejpam-5709	72	63	+	+	NUM
ejpam-5709	72	64	·	·	PUNCT
ejpam-5709	72	65	·	·	PUNCT
ejpam-5709	72	66	·	·	PUNCT
ejpam-5709	73	1	+	+	NUM
ejpam-5709	73	2	α1β	α1β	X
ejpam-5709	73	3	+	+	CCONJ
ejpam-5709	73	4	α0	α0	ADJ
ejpam-5709	73	5	=	=	NOUN
ejpam-5709	73	6	:	:	PUNCT
ejpam-5709	73	7	f(β	f(β	NUM
ejpam-5709	73	8	)	)	PUNCT
ejpam-5709	73	9	,	,	PUNCT
ejpam-5709	73	10	where	where	SCONJ
ejpam-5709	73	11	n	n	PRON
ejpam-5709	73	12	≥	≥	NOUN
ejpam-5709	73	13	1	1	NUM
ejpam-5709	73	14	,	,	PUNCT
ejpam-5709	73	15	re(αn	re(αn	PROPN
ejpam-5709	73	16	)	)	PUNCT
ejpam-5709	73	17	≥	≥	NOUN
ejpam-5709	73	18	1	1	NUM
ejpam-5709	73	19	,	,	PUNCT
ejpam-5709	73	20	and	and	CCONJ
ejpam-5709	73	21	α0	α0	ADJ
ejpam-5709	73	22	,	,	PUNCT
ejpam-5709	73	23	α1	α1	PROPN
ejpam-5709	73	24	,	,	PUNCT
ejpam-5709	73	25	.	.	PUNCT
ejpam-5709	73	26	.	.	PUNCT
ejpam-5709	73	27	.	.	PUNCT
ejpam-5709	74	1	,	,	PUNCT
ejpam-5709	74	2	αn−1	αn−1	PROPN
ejpam-5709	74	3	∈	∈	PROPN
ejpam-5709	74	4	c′	c′	NOUN
ejpam-5709	74	5	satisfying	satisfy	VERB
ejpam-5709	74	6	re(αn−1	re(αn−1	PRON
ejpam-5709	74	7	)	)	PUNCT
ejpam-5709	74	8	im(αn	im(αn	NOUN
ejpam-5709	74	9	)	)	PUNCT
ejpam-5709	74	10	≥	≥	NOUN
ejpam-5709	74	11	re(αn	re(αn	NOUN
ejpam-5709	74	12	)	)	PUNCT
ejpam-5709	74	13	im(αn−1	im(αn−1	PROPN
ejpam-5709	74	14	)	)	PUNCT
ejpam-5709	74	15	,	,	PUNCT
ejpam-5709	74	16	then	then	ADV
ejpam-5709	74	17	f(x	f(x	PROPN
ejpam-5709	74	18	)	)	PUNCT
ejpam-5709	74	19	is	be	AUX
ejpam-5709	74	20	irreducible	irreducible	ADJ
ejpam-5709	74	21	over	over	ADP
ejpam-5709	74	22	q(i	q(i	NOUN
ejpam-5709	74	23	)	)	PUNCT
ejpam-5709	74	24	.	.	PUNCT
ejpam-5709	75	1	for	for	ADP
ejpam-5709	75	2	any	any	DET
ejpam-5709	75	3	quadratic	quadratic	ADJ
ejpam-5709	75	4	field	field	NOUN
ejpam-5709	75	5	k	k	NOUN
ejpam-5709	76	1	=	=	PUNCT
ejpam-5709	76	2	q	q	X
ejpam-5709	76	3	(	(	PUNCT
ejpam-5709	76	4	√	√	NUM
ejpam-5709	76	5	m	m	NOUN
ejpam-5709	76	6	)	)	PUNCT
ejpam-5709	76	7	,	,	PUNCT
ejpam-5709	76	8	tadee	tadee	PROPN
ejpam-5709	76	9	et	et	PROPN
ejpam-5709	76	10	al	al	PROPN
ejpam-5709	76	11	.	.	PUNCT
ejpam-5709	77	1	[	[	X
ejpam-5709	77	2	13	13	NUM
ejpam-5709	77	3	]	]	PUNCT
ejpam-5709	77	4	proved	prove	VERB
ejpam-5709	77	5	for	for	ADP
ejpam-5709	77	6	the	the	DET
ejpam-5709	77	7	case	case	NOUN
ejpam-5709	77	8	m	m	VERB
ejpam-5709	77	9	̸≡	̸≡	NOUN
ejpam-5709	77	10	1	1	NUM
ejpam-5709	77	11	(	(	PUNCT
ejpam-5709	77	12	mod	mod	NOUN
ejpam-5709	77	13	4	4	NUM
ejpam-5709	77	14	)	)	PUNCT
ejpam-5709	77	15	that	that	SCONJ
ejpam-5709	77	16	the	the	DET
ejpam-5709	77	17	set	set	NOUN
ejpam-5709	77	18	c	c	NOUN
ejpam-5709	77	19	=	=	PRON
ejpam-5709	77	20	{	{	PUNCT
ejpam-5709	77	21	x+	x+	ADJ
ejpam-5709	77	22	yσm	yσm	VERB
ejpam-5709	77	23	|	|	ADV
ejpam-5709	77	24	x	x	X
ejpam-5709	78	1	=	=	SYM
ejpam-5709	78	2	0	0	NUM
ejpam-5709	78	3	,	,	PUNCT
ejpam-5709	78	4	1	1	NUM
ejpam-5709	78	5	,	,	PUNCT
ejpam-5709	78	6	.	.	PUNCT
ejpam-5709	78	7	.	.	PUNCT
ejpam-5709	79	1	.	.	PUNCT
ejpam-5709	80	1	,	,	PUNCT
ejpam-5709	80	2	|n(β)|	|n(β)|	NOUN
ejpam-5709	80	3	d	d	NOUN
ejpam-5709	80	4	−	−	PROPN
ejpam-5709	80	5	1	1	NUM
ejpam-5709	80	6	and	and	CCONJ
ejpam-5709	80	7	y	y	PROPN
ejpam-5709	80	8	=	=	SYM
ejpam-5709	80	9	0	0	NUM
ejpam-5709	80	10	,	,	PUNCT
ejpam-5709	80	11	1	1	NUM
ejpam-5709	80	12	,	,	PUNCT
ejpam-5709	80	13	.	.	PUNCT
ejpam-5709	80	14	.	.	PUNCT
ejpam-5709	81	1	.	.	PUNCT
ejpam-5709	82	1	,	,	PUNCT
ejpam-5709	82	2	d−	d−	PROPN
ejpam-5709	82	3	1	1	NUM
ejpam-5709	82	4	}	}	PUNCT
ejpam-5709	82	5	(	(	PUNCT
ejpam-5709	82	6	2	2	X
ejpam-5709	82	7	)	)	PUNCT
ejpam-5709	82	8	is	be	AUX
ejpam-5709	82	9	a	a	DET
ejpam-5709	82	10	crs(β	crs(β	NOUN
ejpam-5709	82	11	)	)	PUNCT
ejpam-5709	82	12	,	,	PUNCT
ejpam-5709	82	13	where	where	SCONJ
ejpam-5709	82	14	β	β	X
ejpam-5709	82	15	=	=	PUNCT
ejpam-5709	82	16	a	a	DET
ejpam-5709	82	17	+	+	NUM
ejpam-5709	82	18	bσm	bσm	NOUN
ejpam-5709	82	19	∈	∈	NOUN
ejpam-5709	82	20	ok	ok	INTJ
ejpam-5709	82	21	with	with	ADP
ejpam-5709	82	22	d	d	PROPN
ejpam-5709	82	23	=	=	SYM
ejpam-5709	82	24	gcd	gcd	PROPN
ejpam-5709	82	25	(	(	PUNCT
ejpam-5709	82	26	a	a	DET
ejpam-5709	82	27	,	,	PUNCT
ejpam-5709	82	28	b	b	NOUN
ejpam-5709	82	29	)	)	PUNCT
ejpam-5709	82	30	.	.	PUNCT
ejpam-5709	83	1	recently	recently	ADV
ejpam-5709	83	2	,	,	PUNCT
ejpam-5709	83	3	phetnun	phetnun	PROPN
ejpam-5709	83	4	et	et	PROPN
ejpam-5709	83	5	al	al	PROPN
ejpam-5709	83	6	.	.	PUNCT
ejpam-5709	84	1	[	[	X
ejpam-5709	84	2	8	8	NUM
ejpam-5709	84	3	]	]	PUNCT
ejpam-5709	84	4	verified	verify	VERB
ejpam-5709	84	5	that	that	SCONJ
ejpam-5709	84	6	(	(	PUNCT
ejpam-5709	84	7	2	2	X
ejpam-5709	84	8	)	)	PUNCT
ejpam-5709	84	9	is	be	AUX
ejpam-5709	84	10	also	also	ADV
ejpam-5709	84	11	a	a	DET
ejpam-5709	84	12	crs(β	crs(β	NOUN
ejpam-5709	84	13	)	)	PUNCT
ejpam-5709	84	14	in	in	ADP
ejpam-5709	84	15	any	any	DET
ejpam-5709	84	16	quadratic	quadratic	ADJ
ejpam-5709	84	17	field	field	NOUN
ejpam-5709	85	1	k	k	NOUN
ejpam-5709	85	2	=	=	PUNCT
ejpam-5709	85	3	q	q	ADJ
ejpam-5709	85	4	(	(	PUNCT
ejpam-5709	85	5	√	√	NUM
ejpam-5709	85	6	m	m	PRON
ejpam-5709	85	7	)	)	PUNCT
ejpam-5709	85	8	with	with	ADP
ejpam-5709	85	9	m	m	PROPN
ejpam-5709	85	10	≡	≡	PROPN
ejpam-5709	85	11	1	1	NUM
ejpam-5709	85	12	(	(	PUNCT
ejpam-5709	85	13	mod	mod	NOUN
ejpam-5709	85	14	4	4	NUM
ejpam-5709	85	15	)	)	PUNCT
ejpam-5709	85	16	.	.	PUNCT
ejpam-5709	86	1	these	these	DET
ejpam-5709	86	2	results	result	NOUN
ejpam-5709	86	3	extend	extend	VERB
ejpam-5709	86	4	the	the	DET
ejpam-5709	86	5	crs(β	crs(β	NOUN
ejpam-5709	86	6	)	)	PUNCT
ejpam-5709	86	7	for	for	ADP
ejpam-5709	86	8	gaussian	gaussian	ADJ
ejpam-5709	86	9	integers	integer	NOUN
ejpam-5709	86	10	in	in	ADP
ejpam-5709	86	11	(	(	PUNCT
ejpam-5709	86	12	1	1	NUM
ejpam-5709	86	13	)	)	PUNCT
ejpam-5709	86	14	to	to	ADP
ejpam-5709	86	15	any	any	DET
ejpam-5709	86	16	quadratic	quadratic	ADJ
ejpam-5709	86	17	field	field	NOUN
ejpam-5709	86	18	.	.	PUNCT
ejpam-5709	87	1	moreover	moreover	ADV
ejpam-5709	87	2	,	,	PUNCT
ejpam-5709	87	3	they	they	PRON
ejpam-5709	87	4	determined	determine	VERB
ejpam-5709	87	5	the	the	DET
ejpam-5709	87	6	so	so	ADV
ejpam-5709	87	7	-	-	PUNCT
ejpam-5709	87	8	called	call	VERB
ejpam-5709	87	9	base	base	NOUN
ejpam-5709	87	10	-	-	PUNCT
ejpam-5709	87	11	β(c	β(c	NUM
ejpam-5709	87	12	)	)	PUNCT
ejpam-5709	87	13	representation	representation	NOUN
ejpam-5709	87	14	for	for	ADP
ejpam-5709	87	15	nonzero	nonzero	ADJ
ejpam-5709	87	16	elements	element	NOUN
ejpam-5709	87	17	of	of	ADP
ejpam-5709	87	18	ok	ok	ADJ
ejpam-5709	87	19	and	and	CCONJ
ejpam-5709	87	20	extended	extended	ADJ
ejpam-5709	87	21	theorem	theorem	VERB
ejpam-5709	87	22	a	a	PRON
ejpam-5709	87	23	to	to	ADP
ejpam-5709	87	24	any	any	DET
ejpam-5709	87	25	imaginary	imaginary	ADJ
ejpam-5709	87	26	quadratic	quadratic	ADJ
ejpam-5709	87	27	field	field	NOUN
ejpam-5709	87	28	using	use	VERB
ejpam-5709	87	29	such	such	DET
ejpam-5709	87	30	a	a	DET
ejpam-5709	87	31	representation	representation	NOUN
ejpam-5709	87	32	.	.	PUNCT
ejpam-5709	88	1	it	it	PRON
ejpam-5709	88	2	was	be	AUX
ejpam-5709	88	3	shown	show	VERB
ejpam-5709	88	4	for	for	ADP
ejpam-5709	88	5	any	any	DET
ejpam-5709	88	6	m	m	NOUN
ejpam-5709	88	7	<	<	X
ejpam-5709	88	8	0	0	PUNCT
ejpam-5709	88	9	in	in	ADP
ejpam-5709	88	10	[	[	X
ejpam-5709	88	11	8	8	NUM
ejpam-5709	88	12	]	]	PUNCT
ejpam-5709	88	13	that	that	SCONJ
ejpam-5709	88	14	the	the	DET
ejpam-5709	88	15	set	set	NOUN
ejpam-5709	88	16	c′	c′	NOUN
ejpam-5709	88	17	:	:	PUNCT
ejpam-5709	88	18	=	=	SYM
ejpam-5709	88	19	{	{	PUNCT
ejpam-5709	88	20	x+	x+	ADJ
ejpam-5709	88	21	yσm	yσm	VERB
ejpam-5709	88	22	|	|	ADV
ejpam-5709	88	23	x	x	X
ejpam-5709	89	1	=	=	SYM
ejpam-5709	89	2	0	0	NUM
ejpam-5709	89	3	,	,	PUNCT
ejpam-5709	89	4	1	1	NUM
ejpam-5709	89	5	,	,	PUNCT
ejpam-5709	89	6	.	.	PUNCT
ejpam-5709	89	7	.	.	PUNCT
ejpam-5709	90	1	.	.	PUNCT
ejpam-5709	91	1	,	,	PUNCT
ejpam-5709	91	2	max{|a|	max{|a|	X
ejpam-5709	91	3	,	,	PUNCT
ejpam-5709	91	4	|b|	|b|	VERB
ejpam-5709	91	5	}	}	PUNCT
ejpam-5709	91	6	−	−	PROPN
ejpam-5709	91	7	1	1	NUM
ejpam-5709	91	8	and	and	CCONJ
ejpam-5709	91	9	y	y	PROPN
ejpam-5709	91	10	=	=	SYM
ejpam-5709	91	11	0	0	NUM
ejpam-5709	91	12	,	,	PUNCT
ejpam-5709	91	13	1	1	NUM
ejpam-5709	91	14	,	,	PUNCT
ejpam-5709	91	15	.	.	PUNCT
ejpam-5709	91	16	.	.	PUNCT
ejpam-5709	92	1	.	.	PUNCT
ejpam-5709	93	1	,	,	PUNCT
ejpam-5709	93	2	d−	d−	PROPN
ejpam-5709	93	3	1	1	NUM
ejpam-5709	93	4	}	}	SYM
ejpam-5709	93	5	⊆	⊆	NUM
ejpam-5709	93	6	c.	c.	NOUN
ejpam-5709	93	7	some	some	PRON
ejpam-5709	93	8	results	result	VERB
ejpam-5709	93	9	in	in	ADP
ejpam-5709	93	10	[	[	X
ejpam-5709	93	11	8	8	NUM
ejpam-5709	93	12	]	]	PUNCT
ejpam-5709	93	13	described	describe	VERB
ejpam-5709	93	14	above	above	ADV
ejpam-5709	93	15	are	be	AUX
ejpam-5709	93	16	as	as	SCONJ
ejpam-5709	93	17	follows	follow	VERB
ejpam-5709	93	18	.	.	PUNCT
ejpam-5709	94	1	p.	p.	NOUN
ejpam-5709	94	2	phetnun	phetnun	PROPN
ejpam-5709	94	3	,	,	PUNCT
ejpam-5709	94	4	n.r	n.r	PROPN
ejpam-5709	94	5	.	.	PROPN
ejpam-5709	94	6	kanasri	kanasri	PROPN
ejpam-5709	94	7	/	/	SYM
ejpam-5709	94	8	eur	eur	PROPN
ejpam-5709	94	9	.	.	PUNCT
ejpam-5709	95	1	j.	j.	PROPN
ejpam-5709	95	2	pure	pure	PROPN
ejpam-5709	95	3	appl	appl	PROPN
ejpam-5709	95	4	.	.	PROPN
ejpam-5709	95	5	math	math	PROPN
ejpam-5709	95	6	,	,	PUNCT
ejpam-5709	95	7	18	18	NUM
ejpam-5709	95	8	(	(	PUNCT
ejpam-5709	95	9	1	1	NUM
ejpam-5709	95	10	)	)	PUNCT
ejpam-5709	95	11	(	(	PUNCT
ejpam-5709	95	12	2025	2025	NUM
ejpam-5709	95	13	)	)	PUNCT
ejpam-5709	95	14	,	,	PUNCT
ejpam-5709	95	15	5709	5709	NUM
ejpam-5709	95	16	4	4	NUM
ejpam-5709	95	17	of	of	ADP
ejpam-5709	95	18	15	15	NUM
ejpam-5709	95	19	definition	definition	NOUN
ejpam-5709	95	20	b.	b.	NOUN
ejpam-5709	95	21	let	let	VERB
ejpam-5709	95	22	k	k	PROPN
ejpam-5709	95	23	=	=	PUNCT
ejpam-5709	95	24	q	q	ADJ
ejpam-5709	95	25	(	(	PUNCT
ejpam-5709	95	26	√	√	NUM
ejpam-5709	95	27	m	m	VERB
ejpam-5709	95	28	)	)	PUNCT
ejpam-5709	95	29	be	be	AUX
ejpam-5709	95	30	an	an	DET
ejpam-5709	95	31	imaginary	imaginary	ADJ
ejpam-5709	95	32	quadratic	quadratic	ADJ
ejpam-5709	95	33	field	field	NOUN
ejpam-5709	95	34	.	.	PUNCT
ejpam-5709	96	1	let	let	VERB
ejpam-5709	96	2	β	β	PRON
ejpam-5709	96	3	∈	∈	VERB
ejpam-5709	96	4	ok\{0	ok\{0	PROPN
ejpam-5709	96	5	}	}	PUNCT
ejpam-5709	96	6	and	and	CCONJ
ejpam-5709	96	7	c	c	PROPN
ejpam-5709	96	8	be	be	AUX
ejpam-5709	96	9	the	the	DET
ejpam-5709	96	10	crs(β	crs(β	NOUN
ejpam-5709	96	11	)	)	PUNCT
ejpam-5709	96	12	as	as	ADP
ejpam-5709	96	13	in	in	ADP
ejpam-5709	96	14	(	(	PUNCT
ejpam-5709	96	15	2	2	NUM
ejpam-5709	96	16	)	)	PUNCT
ejpam-5709	96	17	.	.	PUNCT
ejpam-5709	97	1	we	we	PRON
ejpam-5709	97	2	say	say	VERB
ejpam-5709	97	3	that	that	SCONJ
ejpam-5709	97	4	η	η	PROPN
ejpam-5709	97	5	∈	∈	PROPN
ejpam-5709	97	6	ok\{0	ok\{0	PROPN
ejpam-5709	97	7	}	}	PUNCT
ejpam-5709	97	8	has	have	VERB
ejpam-5709	97	9	a	a	DET
ejpam-5709	97	10	base	base	NOUN
ejpam-5709	97	11	-	-	PUNCT
ejpam-5709	97	12	β(c	β(c	NUM
ejpam-5709	97	13	)	)	PUNCT
ejpam-5709	97	14	representation	representation	NOUN
ejpam-5709	97	15	if	if	SCONJ
ejpam-5709	97	16	η	η	PROPN
ejpam-5709	97	17	=	=	SYM
ejpam-5709	97	18	αnβ	αnβ	PROPN
ejpam-5709	97	19	n	n	PROPN
ejpam-5709	97	20	+	+	CCONJ
ejpam-5709	97	21	αn−1β	αn−1β	PROPN
ejpam-5709	97	22	n−1	n−1	PROPN
ejpam-5709	97	23	+	+	NUM
ejpam-5709	97	24	·	·	PUNCT
ejpam-5709	97	25	·	·	PUNCT
ejpam-5709	97	26	·	·	PUNCT
ejpam-5709	97	27	+	+	NUM
ejpam-5709	97	28	α1β	α1β	X
ejpam-5709	97	29	+	+	CCONJ
ejpam-5709	97	30	α0	α0	ADJ
ejpam-5709	97	31	,	,	PUNCT
ejpam-5709	97	32	(	(	PUNCT
ejpam-5709	97	33	3	3	NUM
ejpam-5709	97	34	)	)	PUNCT
ejpam-5709	97	35	where	where	SCONJ
ejpam-5709	97	36	n	n	PRON
ejpam-5709	97	37	≥	≥	NOUN
ejpam-5709	97	38	1	1	NUM
ejpam-5709	97	39	,	,	PUNCT
ejpam-5709	97	40	αn	αn	NOUN
ejpam-5709	97	41	∈	∈	NOUN
ejpam-5709	97	42	ok\{0	ok\{0	PROPN
ejpam-5709	97	43	}	}	PUNCT
ejpam-5709	97	44	,	,	PUNCT
ejpam-5709	97	45	and	and	CCONJ
ejpam-5709	97	46	αi	αi	NOUN
ejpam-5709	97	47	∈	∈	PROPN
ejpam-5709	98	1	c	c	X
ejpam-5709	98	2	(	(	PUNCT
ejpam-5709	98	3	i	i	NOUN
ejpam-5709	98	4	=	=	NOUN
ejpam-5709	98	5	0	0	NUM
ejpam-5709	98	6	,	,	PUNCT
ejpam-5709	98	7	1	1	NUM
ejpam-5709	98	8	,	,	PUNCT
ejpam-5709	98	9	.	.	PUNCT
ejpam-5709	98	10	.	.	PUNCT
ejpam-5709	98	11	.	.	PUNCT
ejpam-5709	99	1	,	,	PUNCT
ejpam-5709	99	2	n−1	n−1	PROPN
ejpam-5709	99	3	)	)	PUNCT
ejpam-5709	99	4	.	.	PUNCT
ejpam-5709	100	1	if	if	SCONJ
ejpam-5709	100	2	αi	αi	PROPN
ejpam-5709	100	3	∈	∈	PROPN
ejpam-5709	100	4	c′	c′	NOUN
ejpam-5709	100	5	(	(	PUNCT
ejpam-5709	100	6	i	i	NOUN
ejpam-5709	100	7	=	=	NOUN
ejpam-5709	100	8	0	0	NUM
ejpam-5709	100	9	,	,	PUNCT
ejpam-5709	100	10	1	1	NUM
ejpam-5709	100	11	,	,	PUNCT
ejpam-5709	100	12	.	.	PUNCT
ejpam-5709	100	13	.	.	PUNCT
ejpam-5709	100	14	.	.	PUNCT
ejpam-5709	101	1	,	,	PUNCT
ejpam-5709	101	2	n−1	n−1	PROPN
ejpam-5709	101	3	)	)	PUNCT
ejpam-5709	101	4	,	,	PUNCT
ejpam-5709	101	5	then	then	ADV
ejpam-5709	101	6	(	(	PUNCT
ejpam-5709	101	7	3	3	X
ejpam-5709	101	8	)	)	PUNCT
ejpam-5709	101	9	is	be	AUX
ejpam-5709	101	10	called	call	VERB
ejpam-5709	101	11	a	a	DET
ejpam-5709	101	12	base	base	NOUN
ejpam-5709	101	13	-	-	PUNCT
ejpam-5709	101	14	β(c′	β(c′	NUM
ejpam-5709	101	15	)	)	PUNCT
ejpam-5709	101	16	representation	representation	NOUN
ejpam-5709	101	17	of	of	ADP
ejpam-5709	101	18	η	η	PROPN
ejpam-5709	101	19	.	.	PROPN
ejpam-5709	101	20	theorem	theorem	PROPN
ejpam-5709	101	21	c.	c.	PROPN
ejpam-5709	101	22	let	let	VERB
ejpam-5709	101	23	k	k	PROPN
ejpam-5709	101	24	=	=	PUNCT
ejpam-5709	101	25	q	q	ADJ
ejpam-5709	101	26	(	(	PUNCT
ejpam-5709	101	27	√	√	NUM
ejpam-5709	101	28	m	m	VERB
ejpam-5709	101	29	)	)	PUNCT
ejpam-5709	101	30	be	be	AUX
ejpam-5709	101	31	an	an	DET
ejpam-5709	101	32	imaginary	imaginary	ADJ
ejpam-5709	101	33	quadratic	quadratic	ADJ
ejpam-5709	101	34	field	field	NOUN
ejpam-5709	101	35	with	with	ADP
ejpam-5709	101	36	m	m	PROPN
ejpam-5709	101	37	̸≡	̸≡	ADJ
ejpam-5709	101	38	1	1	NUM
ejpam-5709	101	39	(	(	PUNCT
ejpam-5709	101	40	mod	mod	NOUN
ejpam-5709	101	41	4	4	NUM
ejpam-5709	101	42	)	)	PUNCT
ejpam-5709	101	43	.	.	PUNCT
ejpam-5709	102	1	let	let	VERB
ejpam-5709	102	2	β	β	X
ejpam-5709	102	3	=	=	SYM
ejpam-5709	102	4	a+b	a+b	NUM
ejpam-5709	102	5	√	√	NUM
ejpam-5709	102	6	m	m	NUM
ejpam-5709	102	7	∈	∈	NOUN
ejpam-5709	102	8	ok	ok	INTJ
ejpam-5709	102	9	be	be	AUX
ejpam-5709	102	10	such	such	ADJ
ejpam-5709	102	11	that	that	SCONJ
ejpam-5709	102	12	|β|	|β|	PRON
ejpam-5709	102	13	≥	≥	NOUN
ejpam-5709	102	14	2	2	NUM
ejpam-5709	102	15	+	+	NUM
ejpam-5709	102	16	√	√	NUM
ejpam-5709	102	17	1−m	1−m	NUM
ejpam-5709	102	18	and	and	CCONJ
ejpam-5709	102	19	a	a	DET
ejpam-5709	102	20	≥	≥	NOUN
ejpam-5709	102	21	1	1	NUM
ejpam-5709	102	22	+	+	NUM
ejpam-5709	102	23	√	√	NUM
ejpam-5709	102	24	1−m	1−m	NUM
ejpam-5709	102	25	.	.	PUNCT
ejpam-5709	103	1	for	for	ADP
ejpam-5709	103	2	an	an	DET
ejpam-5709	103	3	irreducible	irreducible	ADJ
ejpam-5709	103	4	element	element	NOUN
ejpam-5709	103	5	π	π	NOUN
ejpam-5709	103	6	of	of	ADP
ejpam-5709	103	7	ok	ok	INTJ
ejpam-5709	103	8	with	with	ADP
ejpam-5709	103	9	|π|	|π|	NOUN
ejpam-5709	103	10	≥	≥	NOUN
ejpam-5709	103	11	|β|	|β|	NOUN
ejpam-5709	103	12	,	,	PUNCT
ejpam-5709	103	13	if	if	SCONJ
ejpam-5709	103	14	π	π	PROPN
ejpam-5709	103	15	=	=	PUNCT
ejpam-5709	103	16	αnβ	αnβ	PROPN
ejpam-5709	103	17	n	n	PROPN
ejpam-5709	103	18	+	+	CCONJ
ejpam-5709	103	19	αn−1β	αn−1β	PROPN
ejpam-5709	103	20	n−1	n−1	PROPN
ejpam-5709	103	21	+	+	NUM
ejpam-5709	103	22	·	·	PUNCT
ejpam-5709	103	23	·	·	PUNCT
ejpam-5709	103	24	·	·	PUNCT
ejpam-5709	103	25	+	+	NUM
ejpam-5709	103	26	α1β	α1β	X
ejpam-5709	104	1	+	+	CCONJ
ejpam-5709	104	2	α0	α0	ADJ
ejpam-5709	104	3	=	=	NOUN
ejpam-5709	104	4	:	:	PUNCT
ejpam-5709	104	5	f(β	f(β	NOUN
ejpam-5709	104	6	)	)	PUNCT
ejpam-5709	104	7	is	be	AUX
ejpam-5709	104	8	a	a	DET
ejpam-5709	104	9	base	base	NOUN
ejpam-5709	104	10	-	-	PUNCT
ejpam-5709	104	11	β(c′	β(c′	NUM
ejpam-5709	104	12	)	)	PUNCT
ejpam-5709	104	13	representation	representation	NOUN
ejpam-5709	104	14	with	with	ADP
ejpam-5709	104	15	re(αn	re(αn	PROPN
ejpam-5709	104	16	)	)	PUNCT
ejpam-5709	104	17	≥	≥	NOUN
ejpam-5709	104	18	1	1	NUM
ejpam-5709	104	19	satisfying	satisfy	VERB
ejpam-5709	104	20	re(αn−1	re(αn−1	PRON
ejpam-5709	104	21	)	)	PUNCT
ejpam-5709	104	22	im(αn	im(αn	NOUN
ejpam-5709	104	23	)	)	PUNCT
ejpam-5709	104	24	≥	≥	NOUN
ejpam-5709	104	25	re(αn	re(αn	NOUN
ejpam-5709	104	26	)	)	PUNCT
ejpam-5709	104	27	im(αn−1	im(αn−1	PROPN
ejpam-5709	104	28	)	)	PUNCT
ejpam-5709	104	29	,	,	PUNCT
ejpam-5709	104	30	then	then	ADV
ejpam-5709	104	31	f(x	f(x	PROPN
ejpam-5709	104	32	)	)	PUNCT
ejpam-5709	104	33	is	be	AUX
ejpam-5709	104	34	irreducible	irreducible	ADJ
ejpam-5709	104	35	in	in	ADP
ejpam-5709	104	36	ok	ok	ADJ
ejpam-5709	104	37	[	[	X
ejpam-5709	104	38	x	x	X
ejpam-5709	104	39	]	]	X
ejpam-5709	104	40	.	.	PUNCT
ejpam-5709	105	1	theorem	theorem	PROPN
ejpam-5709	105	2	d.	d.	PROPN
ejpam-5709	105	3	let	let	VERB
ejpam-5709	105	4	k	k	PROPN
ejpam-5709	105	5	=	=	PUNCT
ejpam-5709	105	6	q	q	ADJ
ejpam-5709	105	7	(	(	PUNCT
ejpam-5709	105	8	√	√	NUM
ejpam-5709	105	9	m	m	VERB
ejpam-5709	105	10	)	)	PUNCT
ejpam-5709	105	11	be	be	AUX
ejpam-5709	105	12	an	an	DET
ejpam-5709	105	13	imaginary	imaginary	ADJ
ejpam-5709	105	14	quadratic	quadratic	ADJ
ejpam-5709	105	15	field	field	NOUN
ejpam-5709	105	16	with	with	ADP
ejpam-5709	105	17	m	m	PROPN
ejpam-5709	105	18	≡	≡	PROPN
ejpam-5709	105	19	1	1	NUM
ejpam-5709	105	20	(	(	PUNCT
ejpam-5709	105	21	mod	mod	NOUN
ejpam-5709	105	22	4	4	NUM
ejpam-5709	105	23	)	)	PUNCT
ejpam-5709	105	24	.	.	PUNCT
ejpam-5709	106	1	let	let	VERB
ejpam-5709	106	2	β	β	X
ejpam-5709	106	3	=	=	PRON
ejpam-5709	106	4	a+	a+	PUNCT
ejpam-5709	106	5	bσm	bσm	NOUN
ejpam-5709	106	6	∈	∈	NOUN
ejpam-5709	106	7	ok	ok	INTJ
ejpam-5709	106	8	be	be	AUX
ejpam-5709	106	9	such	such	ADJ
ejpam-5709	106	10	that	that	SCONJ
ejpam-5709	106	11	|β|	|β|	PRON
ejpam-5709	106	12	≥	≥	NUM
ejpam-5709	106	13	2	2	NUM
ejpam-5709	106	14	+	+	CCONJ
ejpam-5709	106	15	√	√	PROPN
ejpam-5709	106	16	(	(	PUNCT
ejpam-5709	106	17	9−m)/4	9−m)/4	NUM
ejpam-5709	106	18	,	,	PUNCT
ejpam-5709	106	19	a	a	DET
ejpam-5709	106	20	≥	≥	NOUN
ejpam-5709	106	21	1	1	NUM
ejpam-5709	106	22	,	,	PUNCT
ejpam-5709	106	23	and	and	CCONJ
ejpam-5709	106	24	a+	a+	X
ejpam-5709	106	25	(	(	PUNCT
ejpam-5709	106	26	b/2	b/2	NUM
ejpam-5709	106	27	)	)	PUNCT
ejpam-5709	106	28	≥	≥	NOUN
ejpam-5709	106	29	1	1	NUM
ejpam-5709	106	30	.	.	X
ejpam-5709	107	1	for	for	ADP
ejpam-5709	107	2	an	an	DET
ejpam-5709	107	3	irreducible	irreducible	ADJ
ejpam-5709	107	4	element	element	NOUN
ejpam-5709	107	5	π	π	NOUN
ejpam-5709	107	6	of	of	ADP
ejpam-5709	107	7	ok	ok	INTJ
ejpam-5709	107	8	with	with	ADP
ejpam-5709	107	9	|π|	|π|	NOUN
ejpam-5709	107	10	>	>	SYM
ejpam-5709	107	11	√	√	PROPN
ejpam-5709	107	12	(	(	PUNCT
ejpam-5709	107	13	9−m)/4	9−m)/4	NUM
ejpam-5709	107	14	(	(	PUNCT
ejpam-5709	107	15	|β|	|β|	NOUN
ejpam-5709	107	16	−	−	PROPN
ejpam-5709	107	17	1	1	NUM
ejpam-5709	107	18	)	)	PUNCT
ejpam-5709	107	19	,	,	PUNCT
ejpam-5709	107	20	if	if	SCONJ
ejpam-5709	107	21	π	π	PROPN
ejpam-5709	107	22	=	=	PUNCT
ejpam-5709	107	23	αnβ	αnβ	PROPN
ejpam-5709	107	24	n	n	PROPN
ejpam-5709	107	25	+	+	CCONJ
ejpam-5709	107	26	αn−1β	αn−1β	PROPN
ejpam-5709	107	27	n−1	n−1	PROPN
ejpam-5709	107	28	+	+	NUM
ejpam-5709	107	29	·	·	PUNCT
ejpam-5709	107	30	·	·	PUNCT
ejpam-5709	107	31	·	·	PUNCT
ejpam-5709	107	32	+	+	NUM
ejpam-5709	107	33	α1β	α1β	X
ejpam-5709	108	1	+	+	CCONJ
ejpam-5709	108	2	α0	α0	ADJ
ejpam-5709	108	3	=	=	NOUN
ejpam-5709	108	4	:	:	PUNCT
ejpam-5709	108	5	f(β	f(β	NOUN
ejpam-5709	108	6	)	)	PUNCT
ejpam-5709	108	7	is	be	AUX
ejpam-5709	108	8	a	a	DET
ejpam-5709	108	9	base	base	NOUN
ejpam-5709	108	10	-	-	PUNCT
ejpam-5709	108	11	β(c′	β(c′	NUM
ejpam-5709	108	12	)	)	PUNCT
ejpam-5709	108	13	representation	representation	NOUN
ejpam-5709	108	14	with	with	ADP
ejpam-5709	108	15	re(αn	re(αn	PROPN
ejpam-5709	108	16	)	)	PUNCT
ejpam-5709	108	17	≥	≥	NOUN
ejpam-5709	108	18	1	1	NUM
ejpam-5709	108	19	satisfying	satisfy	VERB
ejpam-5709	108	20	re(αn−1	re(αn−1	PRON
ejpam-5709	108	21	)	)	PUNCT
ejpam-5709	108	22	im(αn	im(αn	NOUN
ejpam-5709	108	23	)	)	PUNCT
ejpam-5709	108	24	≥	≥	NOUN
ejpam-5709	108	25	re(αn	re(αn	NOUN
ejpam-5709	108	26	)	)	PUNCT
ejpam-5709	108	27	im(αn−1	im(αn−1	PROPN
ejpam-5709	108	28	)	)	PUNCT
ejpam-5709	108	29	,	,	PUNCT
ejpam-5709	108	30	then	then	ADV
ejpam-5709	108	31	f(x	f(x	PROPN
ejpam-5709	108	32	)	)	PUNCT
ejpam-5709	108	33	is	be	AUX
ejpam-5709	108	34	irreducible	irreducible	ADJ
ejpam-5709	108	35	in	in	ADP
ejpam-5709	108	36	ok	ok	ADJ
ejpam-5709	108	37	[	[	X
ejpam-5709	108	38	x	x	X
ejpam-5709	108	39	]	]	X
ejpam-5709	108	40	.	.	PUNCT
ejpam-5709	109	1	recently	recently	ADV
ejpam-5709	109	2	,	,	PUNCT
ejpam-5709	109	3	phetnun	phetnun	NOUN
ejpam-5709	109	4	and	and	CCONJ
ejpam-5709	109	5	kanasri	kanasri	ADJ
ejpam-5709	110	1	[	[	X
ejpam-5709	110	2	7	7	X
ejpam-5709	110	3	]	]	PUNCT
ejpam-5709	110	4	established	establish	VERB
ejpam-5709	110	5	further	further	ADJ
ejpam-5709	110	6	irreducibility	irreducibility	NOUN
ejpam-5709	110	7	criteria	criterion	NOUN
ejpam-5709	110	8	for	for	ADP
ejpam-5709	110	9	polynomials	polynomial	NOUN
ejpam-5709	110	10	in	in	ADP
ejpam-5709	110	11	ok	ok	ADJ
ejpam-5709	110	12	[	[	X
ejpam-5709	110	13	x	x	X
ejpam-5709	110	14	]	]	X
ejpam-5709	110	15	,	,	PUNCT
ejpam-5709	110	16	which	which	PRON
ejpam-5709	110	17	extended	extend	VERB
ejpam-5709	110	18	theorem	theorem	VERB
ejpam-5709	110	19	3.22	3.22	NUM
ejpam-5709	110	20	in	in	ADP
ejpam-5709	110	21	[	[	X
ejpam-5709	110	22	12	12	NUM
ejpam-5709	110	23	]	]	PUNCT
ejpam-5709	110	24	to	to	ADP
ejpam-5709	110	25	any	any	DET
ejpam-5709	110	26	imaginary	imaginary	ADJ
ejpam-5709	110	27	quadratic	quadratic	ADJ
ejpam-5709	110	28	field	field	NOUN
ejpam-5709	110	29	.	.	PUNCT
ejpam-5709	111	1	they	they	PRON
ejpam-5709	111	2	also	also	ADV
ejpam-5709	111	3	provided	provide	VERB
ejpam-5709	111	4	elements	element	NOUN
ejpam-5709	111	5	of	of	ADP
ejpam-5709	111	6	β	β	PRON
ejpam-5709	111	7	that	that	PRON
ejpam-5709	111	8	can	can	AUX
ejpam-5709	111	9	be	be	AUX
ejpam-5709	111	10	applied	apply	VERB
ejpam-5709	111	11	to	to	ADP
ejpam-5709	111	12	these	these	DET
ejpam-5709	111	13	criteria	criterion	NOUN
ejpam-5709	111	14	but	but	CCONJ
ejpam-5709	111	15	not	not	PART
ejpam-5709	111	16	to	to	PART
ejpam-5709	111	17	theorem	theorem	VERB
ejpam-5709	111	18	c	c	PROPN
ejpam-5709	111	19	and	and	CCONJ
ejpam-5709	111	20	theorem	theorem	VERB
ejpam-5709	111	21	d.	d.	PROPN
ejpam-5709	111	22	in	in	ADP
ejpam-5709	111	23	the	the	DET
ejpam-5709	111	24	present	present	ADJ
ejpam-5709	111	25	work	work	NOUN
ejpam-5709	111	26	,	,	PUNCT
ejpam-5709	111	27	we	we	PRON
ejpam-5709	111	28	provide	provide	VERB
ejpam-5709	111	29	the	the	DET
ejpam-5709	111	30	explicit	explicit	ADJ
ejpam-5709	111	31	shapes	shape	NOUN
ejpam-5709	111	32	of	of	ADP
ejpam-5709	111	33	all	all	DET
ejpam-5709	111	34	base	base	NOUN
ejpam-5709	111	35	-	-	PUNCT
ejpam-5709	111	36	β(c	β(c	NUM
ejpam-5709	111	37	)	)	PUNCT
ejpam-5709	111	38	representations	representation	NOUN
ejpam-5709	111	39	for	for	ADP
ejpam-5709	111	40	nonzero	nonzero	ADJ
ejpam-5709	111	41	elements	element	NOUN
ejpam-5709	111	42	of	of	ADP
ejpam-5709	111	43	ok	ok	INTJ
ejpam-5709	111	44	,	,	PUNCT
ejpam-5709	111	45	where	where	SCONJ
ejpam-5709	111	46	k	k	PROPN
ejpam-5709	111	47	is	be	AUX
ejpam-5709	111	48	an	an	DET
ejpam-5709	111	49	imaginary	imaginary	ADJ
ejpam-5709	111	50	quadratic	quadratic	ADJ
ejpam-5709	111	51	field	field	NOUN
ejpam-5709	111	52	.	.	PUNCT
ejpam-5709	112	1	in	in	ADP
ejpam-5709	112	2	addition	addition	NOUN
ejpam-5709	112	3	,	,	PUNCT
ejpam-5709	112	4	we	we	PRON
ejpam-5709	112	5	generalize	generalize	VERB
ejpam-5709	112	6	theorem	theorem	ADJ
ejpam-5709	112	7	c	c	PROPN
ejpam-5709	112	8	and	and	CCONJ
ejpam-5709	112	9	theorem	theorem	VERB
ejpam-5709	112	10	d	d	X
ejpam-5709	112	11	by	by	ADP
ejpam-5709	112	12	considering	consider	VERB
ejpam-5709	112	13	ωπ	ωπ	INTJ
ejpam-5709	112	14	instead	instead	ADV
ejpam-5709	112	15	of	of	ADP
ejpam-5709	112	16	π	π	PROPN
ejpam-5709	112	17	,	,	PUNCT
ejpam-5709	112	18	where	where	SCONJ
ejpam-5709	112	19	ω	ω	PROPN
ejpam-5709	112	20	∈	∈	PROPN
ejpam-5709	112	21	ok\{0	ok\{0	PROPN
ejpam-5709	112	22	}	}	PUNCT
ejpam-5709	112	23	and	and	CCONJ
ejpam-5709	112	24	π	π	PROPN
ejpam-5709	112	25	is	be	AUX
ejpam-5709	112	26	a	a	DET
ejpam-5709	112	27	prime	prime	ADJ
ejpam-5709	112	28	element	element	NOUN
ejpam-5709	112	29	,	,	PUNCT
ejpam-5709	112	30	which	which	PRON
ejpam-5709	112	31	in	in	ADP
ejpam-5709	112	32	turn	turn	NOUN
ejpam-5709	112	33	extend	extend	VERB
ejpam-5709	112	34	theorem	theorem	ADJ
ejpam-5709	112	35	b	b	NOUN
ejpam-5709	112	36	to	to	ADP
ejpam-5709	112	37	any	any	DET
ejpam-5709	112	38	imaginary	imaginary	ADJ
ejpam-5709	112	39	quadratic	quadratic	ADJ
ejpam-5709	112	40	field	field	NOUN
ejpam-5709	112	41	.	.	PUNCT
ejpam-5709	113	1	2	2	X
ejpam-5709	113	2	.	.	X
ejpam-5709	113	3	explicit	explicit	ADJ
ejpam-5709	113	4	shapes	shape	NOUN
ejpam-5709	113	5	of	of	ADP
ejpam-5709	113	6	base	base	NOUN
ejpam-5709	113	7	-	-	PUNCT
ejpam-5709	113	8	β(c	β(c	NUM
ejpam-5709	113	9	)	)	PUNCT
ejpam-5709	113	10	representations	representation	NOUN
ejpam-5709	113	11	let	let	VERB
ejpam-5709	113	12	k	k	PRON
ejpam-5709	113	13	be	be	AUX
ejpam-5709	113	14	an	an	DET
ejpam-5709	113	15	imaginary	imaginary	ADJ
ejpam-5709	113	16	quadratic	quadratic	ADJ
ejpam-5709	113	17	field	field	NOUN
ejpam-5709	113	18	and	and	CCONJ
ejpam-5709	113	19	η	η	PROPN
ejpam-5709	113	20	,	,	PUNCT
ejpam-5709	113	21	β	β	X
ejpam-5709	113	22	=	=	SYM
ejpam-5709	113	23	a+	a+	PUNCT
ejpam-5709	113	24	bσm	bσm	NOUN
ejpam-5709	113	25	be	be	AUX
ejpam-5709	113	26	nonzero	nonzero	ADJ
ejpam-5709	113	27	elements	element	NOUN
ejpam-5709	113	28	of	of	ADP
ejpam-5709	113	29	ok	ok	INTJ
ejpam-5709	113	30	.	.	PUNCT
ejpam-5709	114	1	let	let	VERB
ejpam-5709	114	2	c	c	PRON
ejpam-5709	114	3	be	be	AUX
ejpam-5709	114	4	the	the	DET
ejpam-5709	114	5	base	base	NOUN
ejpam-5709	114	6	-	-	PUNCT
ejpam-5709	114	7	β(c	β(c	NUM
ejpam-5709	114	8	)	)	PUNCT
ejpam-5709	114	9	representations	representation	NOUN
ejpam-5709	114	10	as	as	ADP
ejpam-5709	114	11	in	in	ADP
ejpam-5709	114	12	(	(	PUNCT
ejpam-5709	114	13	2	2	NUM
ejpam-5709	114	14	)	)	PUNCT
ejpam-5709	114	15	.	.	PUNCT
ejpam-5709	115	1	if	if	SCONJ
ejpam-5709	115	2	η	η	PROPN
ejpam-5709	115	3	∈	∈	PROPN
ejpam-5709	115	4	c	c	AUX
ejpam-5709	115	5	,	,	PUNCT
ejpam-5709	115	6	it	it	PRON
ejpam-5709	115	7	is	be	AUX
ejpam-5709	115	8	clear	clear	ADJ
ejpam-5709	115	9	from	from	ADP
ejpam-5709	115	10	definition	definition	NOUN
ejpam-5709	115	11	b	b	PROPN
ejpam-5709	115	12	that	that	SCONJ
ejpam-5709	115	13	η	η	PROPN
ejpam-5709	115	14	can	can	AUX
ejpam-5709	115	15	not	not	PART
ejpam-5709	115	16	be	be	AUX
ejpam-5709	115	17	written	write	VERB
ejpam-5709	115	18	as	as	ADP
ejpam-5709	115	19	a	a	DET
ejpam-5709	115	20	base	base	NOUN
ejpam-5709	115	21	-	-	PUNCT
ejpam-5709	115	22	β(c	β(c	NUM
ejpam-5709	115	23	)	)	PUNCT
ejpam-5709	115	24	representation	representation	NOUN
ejpam-5709	115	25	.	.	PUNCT
ejpam-5709	116	1	assume	assume	VERB
ejpam-5709	116	2	henceforth	henceforth	ADV
ejpam-5709	116	3	that	that	SCONJ
ejpam-5709	116	4	η	η	PROPN
ejpam-5709	116	5	̸∈	̸∈	PROPN
ejpam-5709	116	6	c.	c.	PROPN
ejpam-5709	116	7	by	by	ADP
ejpam-5709	116	8	definition	definition	NOUN
ejpam-5709	116	9	a	a	X
ejpam-5709	116	10	,	,	PUNCT
ejpam-5709	116	11	there	there	PRON
ejpam-5709	116	12	exists	exist	VERB
ejpam-5709	116	13	a	a	DET
ejpam-5709	116	14	unique	unique	ADJ
ejpam-5709	116	15	α0	α0	ADJ
ejpam-5709	116	16	∈	∈	NOUN
ejpam-5709	116	17	c	c	NOUN
ejpam-5709	116	18	such	such	ADJ
ejpam-5709	116	19	that	that	SCONJ
ejpam-5709	116	20	η	η	PROPN
ejpam-5709	116	21	≡	≡	PROPN
ejpam-5709	116	22	α0	α0	PROPN
ejpam-5709	116	23	(	(	PUNCT
ejpam-5709	116	24	mod	mod	PROPN
ejpam-5709	116	25	β	β	PROPN
ejpam-5709	116	26	)	)	PUNCT
ejpam-5709	116	27	,	,	PUNCT
ejpam-5709	117	1	so	so	ADV
ejpam-5709	117	2	η	η	PROPN
ejpam-5709	117	3	=	=	SYM
ejpam-5709	117	4	γ0β	γ0β	PROPN
ejpam-5709	117	5	+	+	CCONJ
ejpam-5709	117	6	α0	α0	ADJ
ejpam-5709	117	7	for	for	ADP
ejpam-5709	117	8	some	some	DET
ejpam-5709	117	9	γ0	γ0	PROPN
ejpam-5709	117	10	∈	∈	PROPN
ejpam-5709	117	11	ok\{0	ok\{0	NOUN
ejpam-5709	117	12	}	}	PUNCT
ejpam-5709	117	13	.	.	PUNCT
ejpam-5709	118	1	if	if	SCONJ
ejpam-5709	118	2	γ0	γ0	NOUN
ejpam-5709	118	3	∈	∈	PROPN
ejpam-5709	118	4	c	c	NOUN
ejpam-5709	118	5	,	,	PUNCT
ejpam-5709	118	6	then	then	ADV
ejpam-5709	118	7	the	the	DET
ejpam-5709	118	8	process	process	NOUN
ejpam-5709	118	9	stops	stop	VERB
ejpam-5709	118	10	,	,	PUNCT
ejpam-5709	118	11	otherwise	otherwise	ADV
ejpam-5709	118	12	,	,	PUNCT
ejpam-5709	118	13	there	there	PRON
ejpam-5709	118	14	exists	exist	VERB
ejpam-5709	118	15	a	a	DET
ejpam-5709	118	16	unique	unique	ADJ
ejpam-5709	118	17	α1	α1	PROPN
ejpam-5709	118	18	∈	∈	NOUN
ejpam-5709	118	19	c	c	NOUN
ejpam-5709	118	20	such	such	ADJ
ejpam-5709	118	21	that	that	SCONJ
ejpam-5709	118	22	γ0	γ0	PROPN
ejpam-5709	118	23	≡	≡	PROPN
ejpam-5709	118	24	α1	α1	PROPN
ejpam-5709	118	25	(	(	PUNCT
ejpam-5709	118	26	mod	mod	PROPN
ejpam-5709	118	27	β	β	X
ejpam-5709	118	28	)	)	PUNCT
ejpam-5709	118	29	,	,	PUNCT
ejpam-5709	118	30	yielding	yield	VERB
ejpam-5709	118	31	γ0	γ0	NOUN
ejpam-5709	118	32	=	=	SYM
ejpam-5709	118	33	γ1β	γ1β	PROPN
ejpam-5709	118	34	+	+	NUM
ejpam-5709	118	35	α1	α1	PROPN
ejpam-5709	118	36	for	for	ADP
ejpam-5709	118	37	some	some	DET
ejpam-5709	118	38	γ1	γ1	PROPN
ejpam-5709	118	39	∈	∈	PROPN
ejpam-5709	118	40	ok\{0	ok\{0	PROPN
ejpam-5709	118	41	}	}	PUNCT
ejpam-5709	118	42	.	.	PUNCT
ejpam-5709	119	1	it	it	PRON
ejpam-5709	119	2	p.	p.	PROPN
ejpam-5709	119	3	phetnun	phetnun	PROPN
ejpam-5709	119	4	,	,	PUNCT
ejpam-5709	119	5	n.r	n.r	PROPN
ejpam-5709	119	6	.	.	PROPN
ejpam-5709	119	7	kanasri	kanasri	PROPN
ejpam-5709	119	8	/	/	SYM
ejpam-5709	119	9	eur	eur	PROPN
ejpam-5709	119	10	.	.	PUNCT
ejpam-5709	120	1	j.	j.	PROPN
ejpam-5709	120	2	pure	pure	PROPN
ejpam-5709	120	3	appl	appl	PROPN
ejpam-5709	120	4	.	.	PROPN
ejpam-5709	120	5	math	math	PROPN
ejpam-5709	120	6	,	,	PUNCT
ejpam-5709	120	7	18	18	NUM
ejpam-5709	120	8	(	(	PUNCT
ejpam-5709	120	9	1	1	NUM
ejpam-5709	120	10	)	)	PUNCT
ejpam-5709	120	11	(	(	PUNCT
ejpam-5709	120	12	2025	2025	NUM
ejpam-5709	120	13	)	)	PUNCT
ejpam-5709	120	14	,	,	PUNCT
ejpam-5709	120	15	5709	5709	NUM
ejpam-5709	120	16	5	5	NUM
ejpam-5709	120	17	of	of	ADP
ejpam-5709	120	18	15	15	NUM
ejpam-5709	120	19	follows	follow	VERB
ejpam-5709	120	20	that	that	SCONJ
ejpam-5709	120	21	η	η	PROPN
ejpam-5709	120	22	=	=	PROPN
ejpam-5709	120	23	γ1β	γ1β	PROPN
ejpam-5709	120	24	2	2	NUM
ejpam-5709	120	25	+	+	CCONJ
ejpam-5709	120	26	α1β	α1β	ADJ
ejpam-5709	120	27	+	+	CCONJ
ejpam-5709	120	28	α0	α0	ADJ
ejpam-5709	120	29	.	.	PUNCT
ejpam-5709	121	1	continuing	continue	VERB
ejpam-5709	121	2	the	the	DET
ejpam-5709	121	3	process	process	NOUN
ejpam-5709	121	4	,	,	PUNCT
ejpam-5709	121	5	we	we	PRON
ejpam-5709	121	6	obtain	obtain	VERB
ejpam-5709	121	7	the	the	DET
ejpam-5709	121	8	sequence	sequence	NOUN
ejpam-5709	121	9	(	(	PUNCT
ejpam-5709	121	10	γi)i≥0	γi)i≥0	NOUN
ejpam-5709	121	11	of	of	ADP
ejpam-5709	121	12	elements	element	NOUN
ejpam-5709	121	13	of	of	ADP
ejpam-5709	121	14	ok	ok	INTJ
ejpam-5709	121	15	such	such	ADJ
ejpam-5709	121	16	that	that	DET
ejpam-5709	121	17	γi−1	γi−1	NOUN
ejpam-5709	121	18	=	=	PUNCT
ejpam-5709	121	19	γiβ	γiβ	PROPN
ejpam-5709	121	20	+	+	CCONJ
ejpam-5709	121	21	αi	αi	VERB
ejpam-5709	121	22	(	(	PUNCT
ejpam-5709	121	23	i	i	PRON
ejpam-5709	121	24	≥	≥	NOUN
ejpam-5709	121	25	0	0	NUM
ejpam-5709	121	26	)	)	PUNCT
ejpam-5709	121	27	(	(	PUNCT
ejpam-5709	121	28	4	4	X
ejpam-5709	121	29	)	)	PUNCT
ejpam-5709	121	30	with	with	ADP
ejpam-5709	121	31	γ−1	γ−1	PROPN
ejpam-5709	121	32	=	=	SYM
ejpam-5709	121	33	η	η	PROPN
ejpam-5709	121	34	.	.	PROPN
ejpam-5709	122	1	if	if	SCONJ
ejpam-5709	122	2	there	there	PRON
ejpam-5709	122	3	exists	exist	VERB
ejpam-5709	122	4	n	n	DET
ejpam-5709	122	5	∈	∈	PROPN
ejpam-5709	122	6	n	n	PART
ejpam-5709	122	7	∪	∪	X
ejpam-5709	122	8	{	{	PUNCT
ejpam-5709	122	9	0	0	NUM
ejpam-5709	122	10	}	}	PUNCT
ejpam-5709	122	11	such	such	ADJ
ejpam-5709	122	12	that	that	PRON
ejpam-5709	122	13	γn	γn	ADP
ejpam-5709	122	14	∈	∈	PROPN
ejpam-5709	122	15	c	c	NOUN
ejpam-5709	122	16	,	,	PUNCT
ejpam-5709	122	17	then	then	ADV
ejpam-5709	122	18	η	η	PROPN
ejpam-5709	122	19	=	=	PROPN
ejpam-5709	122	20	γ0β	γ0β	PROPN
ejpam-5709	122	21	+	+	CCONJ
ejpam-5709	122	22	α0	α0	ADJ
ejpam-5709	122	23	,	,	PUNCT
ejpam-5709	122	24	η	η	PROPN
ejpam-5709	122	25	=	=	PROPN
ejpam-5709	122	26	γ1β	γ1β	PROPN
ejpam-5709	122	27	2	2	NUM
ejpam-5709	123	1	+	+	CCONJ
ejpam-5709	123	2	α1β	α1β	ADJ
ejpam-5709	123	3	+	+	CCONJ
ejpam-5709	123	4	α0	α0	ADJ
ejpam-5709	123	5	,	,	PUNCT
ejpam-5709	123	6	...	...	PUNCT
ejpam-5709	123	7	η	η	PROPN
ejpam-5709	123	8	=	=	PROPN
ejpam-5709	123	9	γnβ	γnβ	NOUN
ejpam-5709	123	10	n+1	n+1	PROPN
ejpam-5709	123	11	+	+	NUM
ejpam-5709	123	12	αnβ	αnβ	PROPN
ejpam-5709	123	13	n	n	PROPN
ejpam-5709	123	14	+	+	CCONJ
ejpam-5709	123	15	·	·	PUNCT
ejpam-5709	123	16	·	·	PUNCT
ejpam-5709	123	17	·	·	PUNCT
ejpam-5709	124	1	+	+	NUM
ejpam-5709	124	2	α1β	α1β	X
ejpam-5709	124	3	+	+	CCONJ
ejpam-5709	124	4	α0	α0	ADJ
ejpam-5709	124	5	(	(	PUNCT
ejpam-5709	124	6	5	5	NUM
ejpam-5709	124	7	)	)	PUNCT
ejpam-5709	124	8	are	be	AUX
ejpam-5709	124	9	n+	n+	ADP
ejpam-5709	124	10	1	1	NUM
ejpam-5709	124	11	base	base	NOUN
ejpam-5709	124	12	-	-	PUNCT
ejpam-5709	124	13	β(c	β(c	NUM
ejpam-5709	124	14	)	)	PUNCT
ejpam-5709	124	15	representations	representation	NOUN
ejpam-5709	124	16	of	of	ADP
ejpam-5709	124	17	η	η	PROPN
ejpam-5709	124	18	.	.	PUNCT
ejpam-5709	124	19	the	the	DET
ejpam-5709	124	20	following	follow	VERB
ejpam-5709	124	21	theorem	theorem	NOUN
ejpam-5709	124	22	shows	show	VERB
ejpam-5709	124	23	that	that	SCONJ
ejpam-5709	124	24	the	the	DET
ejpam-5709	124	25	equations	equation	NOUN
ejpam-5709	124	26	in	in	ADP
ejpam-5709	124	27	(	(	PUNCT
ejpam-5709	124	28	5	5	NUM
ejpam-5709	124	29	)	)	PUNCT
ejpam-5709	124	30	are	be	AUX
ejpam-5709	124	31	all	all	PRON
ejpam-5709	124	32	base	base	NOUN
ejpam-5709	124	33	-	-	PUNCT
ejpam-5709	124	34	β(c	β(c	NUM
ejpam-5709	124	35	)	)	PUNCT
ejpam-5709	124	36	representations	representation	NOUN
ejpam-5709	124	37	of	of	ADP
ejpam-5709	124	38	η	η	PROPN
ejpam-5709	124	39	for	for	ADP
ejpam-5709	124	40	the	the	DET
ejpam-5709	124	41	case	case	NOUN
ejpam-5709	124	42	γn	γn	ADP
ejpam-5709	124	43	∈	∈	PROPN
ejpam-5709	124	44	c	c	NOUN
ejpam-5709	124	45	for	for	ADP
ejpam-5709	124	46	some	some	DET
ejpam-5709	124	47	n	n	PRON
ejpam-5709	124	48	∈	∈	PROPN
ejpam-5709	124	49	n	n	NOUN
ejpam-5709	124	50	∪	∪	X
ejpam-5709	124	51	{	{	PUNCT
ejpam-5709	124	52	0	0	NUM
ejpam-5709	124	53	}	}	PUNCT
ejpam-5709	124	54	.	.	PUNCT
ejpam-5709	125	1	theorem	theorem	NOUN
ejpam-5709	125	2	1	1	NUM
ejpam-5709	125	3	.	.	PUNCT
ejpam-5709	126	1	let	let	VERB
ejpam-5709	126	2	k	k	NOUN
ejpam-5709	126	3	=	=	PUNCT
ejpam-5709	126	4	q	q	ADJ
ejpam-5709	126	5	(	(	PUNCT
ejpam-5709	126	6	√	√	NUM
ejpam-5709	126	7	m	m	VERB
ejpam-5709	126	8	)	)	PUNCT
ejpam-5709	126	9	be	be	AUX
ejpam-5709	126	10	an	an	DET
ejpam-5709	126	11	imaginary	imaginary	ADJ
ejpam-5709	126	12	quadratic	quadratic	ADJ
ejpam-5709	126	13	field	field	NOUN
ejpam-5709	126	14	and	and	CCONJ
ejpam-5709	126	15	let	let	VERB
ejpam-5709	126	16	β	β	X
ejpam-5709	126	17	,	,	PUNCT
ejpam-5709	126	18	η	η	PROPN
ejpam-5709	126	19	∈	∈	PROPN
ejpam-5709	126	20	ok\{0	ok\{0	PROPN
ejpam-5709	126	21	}	}	PUNCT
ejpam-5709	126	22	with	with	ADP
ejpam-5709	126	23	η	η	PROPN
ejpam-5709	126	24	̸∈	̸∈	PROPN
ejpam-5709	126	25	c.	c.	PROPN
ejpam-5709	126	26	let	let	VERB
ejpam-5709	126	27	(	(	PUNCT
ejpam-5709	126	28	γi)i≥0	γi)i≥0	NOUN
ejpam-5709	126	29	be	be	AUX
ejpam-5709	126	30	the	the	DET
ejpam-5709	126	31	sequence	sequence	NOUN
ejpam-5709	126	32	described	describe	VERB
ejpam-5709	126	33	in	in	ADP
ejpam-5709	126	34	(	(	PUNCT
ejpam-5709	126	35	4	4	NUM
ejpam-5709	126	36	)	)	PUNCT
ejpam-5709	126	37	.	.	PUNCT
ejpam-5709	127	1	if	if	SCONJ
ejpam-5709	127	2	γn	γn	ADP
ejpam-5709	127	3	∈	∈	PROPN
ejpam-5709	127	4	c	c	NOUN
ejpam-5709	127	5	for	for	ADP
ejpam-5709	127	6	some	some	DET
ejpam-5709	127	7	n	n	PRON
ejpam-5709	127	8	∈	∈	PROPN
ejpam-5709	127	9	n	n	NOUN
ejpam-5709	127	10	∪	∪	X
ejpam-5709	127	11	{	{	PUNCT
ejpam-5709	127	12	0	0	NUM
ejpam-5709	127	13	}	}	PUNCT
ejpam-5709	127	14	,	,	PUNCT
ejpam-5709	127	15	then	then	ADV
ejpam-5709	127	16	all	all	DET
ejpam-5709	127	17	equations	equation	NOUN
ejpam-5709	127	18	in	in	ADP
ejpam-5709	127	19	(	(	PUNCT
ejpam-5709	127	20	5	5	NUM
ejpam-5709	127	21	)	)	PUNCT
ejpam-5709	127	22	are	be	AUX
ejpam-5709	127	23	the	the	DET
ejpam-5709	127	24	base	base	NOUN
ejpam-5709	127	25	-	-	PUNCT
ejpam-5709	127	26	β(c	β(c	NUM
ejpam-5709	127	27	)	)	PUNCT
ejpam-5709	127	28	representations	representation	NOUN
ejpam-5709	127	29	of	of	ADP
ejpam-5709	127	30	η	η	PROPN
ejpam-5709	127	31	.	.	PROPN
ejpam-5709	127	32	proof	proof	NOUN
ejpam-5709	127	33	.	.	PUNCT
ejpam-5709	128	1	by	by	ADP
ejpam-5709	128	2	the	the	DET
ejpam-5709	128	3	description	description	NOUN
ejpam-5709	128	4	mentioned	mention	VERB
ejpam-5709	128	5	above	above	ADV
ejpam-5709	128	6	,	,	PUNCT
ejpam-5709	128	7	all	all	DET
ejpam-5709	128	8	n	n	DET
ejpam-5709	128	9	+	+	CCONJ
ejpam-5709	128	10	1	1	NUM
ejpam-5709	128	11	equations	equation	NOUN
ejpam-5709	128	12	in	in	ADP
ejpam-5709	128	13	(	(	PUNCT
ejpam-5709	128	14	5	5	NUM
ejpam-5709	128	15	)	)	PUNCT
ejpam-5709	128	16	are	be	AUX
ejpam-5709	128	17	base	base	NOUN
ejpam-5709	128	18	-	-	PUNCT
ejpam-5709	128	19	β(c	β(c	NUM
ejpam-5709	128	20	)	)	PUNCT
ejpam-5709	128	21	representations	representation	NOUN
ejpam-5709	128	22	of	of	ADP
ejpam-5709	128	23	η	η	PROPN
ejpam-5709	128	24	.	.	PROPN
ejpam-5709	128	25	next	next	ADV
ejpam-5709	128	26	,	,	PUNCT
ejpam-5709	128	27	let	let	VERB
ejpam-5709	128	28	η	η	PROPN
ejpam-5709	128	29	=	=	PROPN
ejpam-5709	128	30	δkβ	δkβ	PROPN
ejpam-5709	128	31	k	k	NOUN
ejpam-5709	128	32	+	+	CCONJ
ejpam-5709	128	33	δk−1β	δk−1β	VERB
ejpam-5709	128	34	k−1	k−1	PROPN
ejpam-5709	128	35	+	+	CCONJ
ejpam-5709	128	36	·	·	PUNCT
ejpam-5709	128	37	·	·	PUNCT
ejpam-5709	128	38	·	·	PUNCT
ejpam-5709	129	1	+	+	NUM
ejpam-5709	129	2	δ1β	δ1β	X
ejpam-5709	129	3	+	+	NUM
ejpam-5709	129	4	δ0	δ0	NOUN
ejpam-5709	129	5	(	(	PUNCT
ejpam-5709	129	6	6	6	NUM
ejpam-5709	129	7	)	)	PUNCT
ejpam-5709	129	8	be	be	AUX
ejpam-5709	129	9	any	any	DET
ejpam-5709	129	10	base	base	NOUN
ejpam-5709	129	11	-	-	PUNCT
ejpam-5709	129	12	β(c	β(c	NUM
ejpam-5709	129	13	)	)	PUNCT
ejpam-5709	129	14	representation	representation	NOUN
ejpam-5709	129	15	of	of	ADP
ejpam-5709	129	16	η	η	PROPN
ejpam-5709	129	17	.	.	PROPN
ejpam-5709	130	1	then	then	ADV
ejpam-5709	130	2	k	k	PROPN
ejpam-5709	130	3	∈	∈	PROPN
ejpam-5709	130	4	n	n	CCONJ
ejpam-5709	130	5	,	,	PUNCT
ejpam-5709	130	6	δi	δi	PROPN
ejpam-5709	130	7	∈	∈	PROPN
ejpam-5709	130	8	c	c	X
ejpam-5709	130	9	(	(	PUNCT
ejpam-5709	130	10	0	0	NUM
ejpam-5709	130	11	≤	≤	NUM
ejpam-5709	130	12	i	i	PRON
ejpam-5709	130	13	≤	≤	PROPN
ejpam-5709	130	14	k−1	k−1	PROPN
ejpam-5709	130	15	)	)	PUNCT
ejpam-5709	130	16	,	,	PUNCT
ejpam-5709	130	17	and	and	CCONJ
ejpam-5709	130	18	δk	δk	ADP
ejpam-5709	130	19	∈	∈	PROPN
ejpam-5709	130	20	ok\{0	ok\{0	NOUN
ejpam-5709	130	21	}	}	PUNCT
ejpam-5709	130	22	.	.	PUNCT
ejpam-5709	131	1	suppose	suppose	VERB
ejpam-5709	131	2	that	that	SCONJ
ejpam-5709	131	3	k	k	PROPN
ejpam-5709	131	4	≥	≥	PROPN
ejpam-5709	131	5	n	n	PROPN
ejpam-5709	131	6	+	+	NUM
ejpam-5709	131	7	2	2	NUM
ejpam-5709	131	8	.	.	PUNCT
ejpam-5709	131	9	by	by	ADP
ejpam-5709	131	10	the	the	DET
ejpam-5709	131	11	last	last	ADJ
ejpam-5709	131	12	equation	equation	NOUN
ejpam-5709	131	13	in	in	ADP
ejpam-5709	131	14	(	(	PUNCT
ejpam-5709	131	15	5	5	NUM
ejpam-5709	131	16	)	)	PUNCT
ejpam-5709	131	17	together	together	ADV
ejpam-5709	131	18	with	with	ADP
ejpam-5709	131	19	(	(	PUNCT
ejpam-5709	131	20	6	6	NUM
ejpam-5709	131	21	)	)	PUNCT
ejpam-5709	131	22	,	,	PUNCT
ejpam-5709	131	23	we	we	PRON
ejpam-5709	131	24	obtain	obtain	VERB
ejpam-5709	131	25	that	that	SCONJ
ejpam-5709	131	26	η	η	PROPN
ejpam-5709	131	27	≡	≡	PROPN
ejpam-5709	131	28	α0	α0	PROPN
ejpam-5709	131	29	(	(	PUNCT
ejpam-5709	131	30	mod	mod	PROPN
ejpam-5709	131	31	β	β	X
ejpam-5709	131	32	)	)	PUNCT
ejpam-5709	131	33	and	and	CCONJ
ejpam-5709	131	34	η	η	PROPN
ejpam-5709	131	35	≡	≡	PROPN
ejpam-5709	131	36	δ0	δ0	NOUN
ejpam-5709	131	37	(	(	PUNCT
ejpam-5709	131	38	mod	mod	PROPN
ejpam-5709	131	39	β	β	X
ejpam-5709	131	40	)	)	PUNCT
ejpam-5709	131	41	.	.	PUNCT
ejpam-5709	132	1	it	it	PRON
ejpam-5709	132	2	follows	follow	VERB
ejpam-5709	132	3	from	from	ADP
ejpam-5709	132	4	definition	definition	NOUN
ejpam-5709	132	5	a(ii	a(ii	NUM
ejpam-5709	132	6	)	)	PUNCT
ejpam-5709	132	7	that	that	SCONJ
ejpam-5709	132	8	α0	α0	ADJ
ejpam-5709	132	9	=	=	SYM
ejpam-5709	132	10	δ0	δ0	NOUN
ejpam-5709	132	11	.	.	PUNCT
ejpam-5709	133	1	then	then	ADV
ejpam-5709	133	2	η1	η1	NOUN
ejpam-5709	133	3	:	:	PUNCT
ejpam-5709	133	4	=	=	NOUN
ejpam-5709	133	5	γnβ	γnβ	NOUN
ejpam-5709	133	6	n	n	PROPN
ejpam-5709	133	7	+	+	CCONJ
ejpam-5709	133	8	αnβ	αnβ	PROPN
ejpam-5709	133	9	n−1	n−1	PROPN
ejpam-5709	133	10	+	+	CCONJ
ejpam-5709	133	11	·	·	PUNCT
ejpam-5709	133	12	·	·	PUNCT
ejpam-5709	133	13	·	·	PUNCT
ejpam-5709	134	1	+	+	NUM
ejpam-5709	134	2	α2β	α2β	PROPN
ejpam-5709	134	3	+	+	CCONJ
ejpam-5709	134	4	α1	α1	PROPN
ejpam-5709	134	5	=	=	SYM
ejpam-5709	134	6	δkβ	δkβ	NOUN
ejpam-5709	134	7	k−1	k−1	PROPN
ejpam-5709	134	8	+	+	CCONJ
ejpam-5709	134	9	δk−1β	δk−1β	VERB
ejpam-5709	134	10	k−2	k−2	PROPN
ejpam-5709	134	11	+	+	CCONJ
ejpam-5709	134	12	·	·	PUNCT
ejpam-5709	134	13	·	·	PUNCT
ejpam-5709	134	14	·	·	PUNCT
ejpam-5709	134	15	+	+	PUNCT
ejpam-5709	134	16	δ2β	δ2β	SYM
ejpam-5709	134	17	+	+	NUM
ejpam-5709	134	18	δ1	δ1	NOUN
ejpam-5709	134	19	,	,	PUNCT
ejpam-5709	134	20	so	so	SCONJ
ejpam-5709	134	21	α1	α1	PROPN
ejpam-5709	134	22	≡	≡	PROPN
ejpam-5709	134	23	δ1	δ1	NOUN
ejpam-5709	134	24	(	(	PUNCT
ejpam-5709	134	25	mod	mod	PROPN
ejpam-5709	134	26	β	β	NOUN
ejpam-5709	134	27	)	)	PUNCT
ejpam-5709	134	28	.	.	PUNCT
ejpam-5709	135	1	again	again	ADV
ejpam-5709	135	2	,	,	PUNCT
ejpam-5709	135	3	by	by	ADP
ejpam-5709	135	4	definition	definition	NOUN
ejpam-5709	135	5	a(ii	a(ii	VERB
ejpam-5709	135	6	)	)	PUNCT
ejpam-5709	135	7	,	,	PUNCT
ejpam-5709	135	8	we	we	PRON
ejpam-5709	135	9	obtain	obtain	VERB
ejpam-5709	135	10	α1	α1	NOUN
ejpam-5709	135	11	=	=	SYM
ejpam-5709	135	12	δ1	δ1	NOUN
ejpam-5709	135	13	and	and	CCONJ
ejpam-5709	135	14	so	so	ADV
ejpam-5709	135	15	η2	η2	PROPN
ejpam-5709	135	16	:	:	PUNCT
ejpam-5709	135	17	=	=	SYM
ejpam-5709	135	18	γnβ	γnβ	NOUN
ejpam-5709	135	19	n−1	n−1	PROPN
ejpam-5709	135	20	+	+	CCONJ
ejpam-5709	135	21	αnβ	αnβ	PROPN
ejpam-5709	135	22	n−2	n−2	PROPN
ejpam-5709	135	23	+	+	PROPN
ejpam-5709	135	24	·	·	PUNCT
ejpam-5709	135	25	·	·	PUNCT
ejpam-5709	135	26	·	·	PUNCT
ejpam-5709	135	27	+	+	NUM
ejpam-5709	135	28	α2	α2	ADJ
ejpam-5709	135	29	=	=	SYM
ejpam-5709	135	30	δkβ	δkβ	NOUN
ejpam-5709	135	31	k−2	k−2	NOUN
ejpam-5709	135	32	+	+	CCONJ
ejpam-5709	135	33	δk−1β	δk−1β	VERB
ejpam-5709	135	34	k−3	k−3	PROPN
ejpam-5709	135	35	+	+	CCONJ
ejpam-5709	135	36	·	·	PUNCT
ejpam-5709	135	37	·	·	PUNCT
ejpam-5709	135	38	·	·	PUNCT
ejpam-5709	136	1	+	+	CCONJ
ejpam-5709	136	2	δ2	δ2	VERB
ejpam-5709	136	3	.	.	PUNCT
ejpam-5709	137	1	proceeding	proceed	VERB
ejpam-5709	137	2	in	in	ADP
ejpam-5709	137	3	the	the	DET
ejpam-5709	137	4	same	same	ADJ
ejpam-5709	137	5	manner	manner	NOUN
ejpam-5709	137	6	,	,	PUNCT
ejpam-5709	137	7	we	we	PRON
ejpam-5709	137	8	deduce	deduce	VERB
ejpam-5709	137	9	that	that	SCONJ
ejpam-5709	137	10	αi	αi	VERB
ejpam-5709	137	11	=	=	NOUN
ejpam-5709	137	12	δi	δi	PROPN
ejpam-5709	137	13	(	(	PUNCT
ejpam-5709	137	14	0	0	NUM
ejpam-5709	137	15	≤	≤	NUM
ejpam-5709	137	16	i	i	NOUN
ejpam-5709	137	17	≤	≤	NOUN
ejpam-5709	137	18	n	n	CCONJ
ejpam-5709	137	19	)	)	PUNCT
ejpam-5709	137	20	and	and	CCONJ
ejpam-5709	137	21	γn	γn	NOUN
ejpam-5709	137	22	=	=	SYM
ejpam-5709	137	23	δkβ	δkβ	NOUN
ejpam-5709	137	24	k−n−1	k−n−1	PROPN
ejpam-5709	137	25	+	+	NOUN
ejpam-5709	137	26	·	·	PUNCT
ejpam-5709	137	27	·	·	PUNCT
ejpam-5709	137	28	·	·	PUNCT
ejpam-5709	138	1	+	+	NUM
ejpam-5709	138	2	δn+2β	δn+2β	NOUN
ejpam-5709	138	3	+	+	CCONJ
ejpam-5709	138	4	δn+1	δn+1	ADJ
ejpam-5709	138	5	,	,	PUNCT
ejpam-5709	138	6	implying	imply	VERB
ejpam-5709	138	7	γn	γn	ADP
ejpam-5709	138	8	≡	≡	PROPN
ejpam-5709	138	9	δn+1	δn+1	PROPN
ejpam-5709	138	10	(	(	PUNCT
ejpam-5709	138	11	mod	mod	PROPN
ejpam-5709	138	12	β	β	X
ejpam-5709	138	13	)	)	PUNCT
ejpam-5709	138	14	.	.	PUNCT
ejpam-5709	139	1	since	since	SCONJ
ejpam-5709	139	2	γn	γn	NUM
ejpam-5709	139	3	,	,	PUNCT
ejpam-5709	139	4	δn+1	δn+1	PROPN
ejpam-5709	139	5	∈	∈	PROPN
ejpam-5709	139	6	c	c	X
ejpam-5709	139	7	,	,	PUNCT
ejpam-5709	139	8	it	it	PRON
ejpam-5709	139	9	follows	follow	VERB
ejpam-5709	139	10	that	that	SCONJ
ejpam-5709	139	11	γn	γn	AUX
ejpam-5709	139	12	=	=	SYM
ejpam-5709	139	13	δn+1	δn+1	PROPN
ejpam-5709	139	14	.	.	PUNCT
ejpam-5709	140	1	thus	thus	ADV
ejpam-5709	140	2	we	we	PRON
ejpam-5709	140	3	have	have	VERB
ejpam-5709	140	4	δkβ	δkβ	NOUN
ejpam-5709	140	5	k−n−2	k−n−2	NOUN
ejpam-5709	140	6	+	+	CCONJ
ejpam-5709	140	7	·	·	PUNCT
ejpam-5709	140	8	·	·	PUNCT
ejpam-5709	140	9	·	·	PUNCT
ejpam-5709	141	1	+	+	CCONJ
ejpam-5709	141	2	δn+3β	δn+3β	VERB
ejpam-5709	141	3	+	+	CCONJ
ejpam-5709	141	4	δn+2	δn+2	PROPN
ejpam-5709	141	5	=	=	SYM
ejpam-5709	141	6	0	0	NUM
ejpam-5709	141	7	,	,	PUNCT
ejpam-5709	141	8	(	(	PUNCT
ejpam-5709	141	9	7	7	NUM
ejpam-5709	141	10	)	)	PUNCT
ejpam-5709	141	11	so	so	ADV
ejpam-5709	141	12	δn+2	δn+2	ADV
ejpam-5709	141	13	≡	≡	PROPN
ejpam-5709	141	14	0	0	PUNCT
ejpam-5709	142	1	(	(	PUNCT
ejpam-5709	142	2	mod	mod	PROPN
ejpam-5709	142	3	β	β	NOUN
ejpam-5709	142	4	)	)	PUNCT
ejpam-5709	142	5	.	.	PUNCT
ejpam-5709	143	1	if	if	SCONJ
ejpam-5709	143	2	k	k	PROPN
ejpam-5709	143	3	=	=	PUNCT
ejpam-5709	143	4	n+	n+	ADP
ejpam-5709	143	5	2	2	NUM
ejpam-5709	143	6	,	,	PUNCT
ejpam-5709	143	7	then	then	ADV
ejpam-5709	143	8	(	(	PUNCT
ejpam-5709	143	9	7	7	X
ejpam-5709	143	10	)	)	PUNCT
ejpam-5709	143	11	implies	imply	VERB
ejpam-5709	143	12	δk	δk	ADP
ejpam-5709	143	13	=	=	SYM
ejpam-5709	143	14	0	0	NUM
ejpam-5709	143	15	,	,	PUNCT
ejpam-5709	143	16	a	a	DET
ejpam-5709	143	17	contradiction	contradiction	NOUN
ejpam-5709	143	18	.	.	PUNCT
ejpam-5709	144	1	if	if	SCONJ
ejpam-5709	144	2	k	k	PROPN
ejpam-5709	144	3	>	>	X
ejpam-5709	144	4	n+	n+	NUM
ejpam-5709	144	5	2	2	NUM
ejpam-5709	144	6	,	,	PUNCT
ejpam-5709	144	7	then	then	ADV
ejpam-5709	144	8	δn+2	δn+2	CCONJ
ejpam-5709	144	9	∈	∈	PROPN
ejpam-5709	144	10	c	c	NOUN
ejpam-5709	144	11	and	and	CCONJ
ejpam-5709	144	12	thus	thus	ADV
ejpam-5709	144	13	δn+2	δn+2	ADV
ejpam-5709	144	14	=	=	SYM
ejpam-5709	144	15	0	0	NUM
ejpam-5709	144	16	by	by	ADP
ejpam-5709	144	17	definition	definition	NOUN
ejpam-5709	144	18	a(ii	a(ii	NUM
ejpam-5709	144	19	)	)	PUNCT
ejpam-5709	144	20	.	.	PUNCT
ejpam-5709	145	1	continuing	continue	VERB
ejpam-5709	145	2	similarly	similarly	ADV
ejpam-5709	145	3	,	,	PUNCT
ejpam-5709	145	4	we	we	PRON
ejpam-5709	145	5	finally	finally	ADV
ejpam-5709	145	6	obtain	obtain	VERB
ejpam-5709	145	7	δk	δk	ADP
ejpam-5709	145	8	=	=	NOUN
ejpam-5709	145	9	0	0	NUM
ejpam-5709	145	10	,	,	PUNCT
ejpam-5709	145	11	which	which	PRON
ejpam-5709	145	12	is	be	AUX
ejpam-5709	145	13	a	a	DET
ejpam-5709	145	14	contradiction	contradiction	NOUN
ejpam-5709	145	15	.	.	PUNCT
ejpam-5709	146	1	hence	hence	ADV
ejpam-5709	146	2	k	k	PROPN
ejpam-5709	146	3	≤	≤	PROPN
ejpam-5709	146	4	n+	n+	PUNCT
ejpam-5709	146	5	1	1	NUM
ejpam-5709	146	6	,	,	PUNCT
ejpam-5709	146	7	implying	imply	VERB
ejpam-5709	146	8	the	the	DET
ejpam-5709	146	9	base	base	NOUN
ejpam-5709	146	10	-	-	PUNCT
ejpam-5709	146	11	β(c	β(c	NUM
ejpam-5709	146	12	)	)	PUNCT
ejpam-5709	146	13	representation	representation	NOUN
ejpam-5709	146	14	of	of	ADP
ejpam-5709	146	15	η	η	PROPN
ejpam-5709	146	16	in	in	ADP
ejpam-5709	146	17	(	(	PUNCT
ejpam-5709	146	18	6	6	NUM
ejpam-5709	146	19	)	)	PUNCT
ejpam-5709	146	20	is	be	AUX
ejpam-5709	146	21	one	one	NUM
ejpam-5709	146	22	of	of	ADP
ejpam-5709	146	23	that	that	PRON
ejpam-5709	146	24	in	in	ADP
ejpam-5709	146	25	(	(	PUNCT
ejpam-5709	146	26	5	5	NUM
ejpam-5709	146	27	)	)	PUNCT
ejpam-5709	146	28	as	as	SCONJ
ejpam-5709	146	29	desired	desire	VERB
ejpam-5709	146	30	.	.	PUNCT
ejpam-5709	147	1	the	the	DET
ejpam-5709	147	2	base	base	NOUN
ejpam-5709	147	3	-	-	PUNCT
ejpam-5709	147	4	β(c	β(c	NUM
ejpam-5709	147	5	)	)	PUNCT
ejpam-5709	147	6	representations	representation	NOUN
ejpam-5709	147	7	for	for	ADP
ejpam-5709	147	8	the	the	DET
ejpam-5709	147	9	case	case	NOUN
ejpam-5709	147	10	γn	γn	ADP
ejpam-5709	147	11	∈	∈	PROPN
ejpam-5709	147	12	c	c	NOUN
ejpam-5709	147	13	for	for	ADP
ejpam-5709	147	14	some	some	DET
ejpam-5709	147	15	n	n	PRON
ejpam-5709	147	16	∈	∈	PROPN
ejpam-5709	147	17	n	n	NOUN
ejpam-5709	147	18	∪	∪	X
ejpam-5709	147	19	{	{	PUNCT
ejpam-5709	147	20	0	0	NUM
ejpam-5709	147	21	}	}	PUNCT
ejpam-5709	147	22	are	be	AUX
ejpam-5709	147	23	shown	show	VERB
ejpam-5709	147	24	in	in	ADP
ejpam-5709	147	25	the	the	DET
ejpam-5709	147	26	following	follow	VERB
ejpam-5709	147	27	example	example	NOUN
ejpam-5709	147	28	.	.	PUNCT
ejpam-5709	148	1	p.	p.	NOUN
ejpam-5709	148	2	phetnun	phetnun	PROPN
ejpam-5709	148	3	,	,	PUNCT
ejpam-5709	148	4	n.r	n.r	PROPN
ejpam-5709	148	5	.	.	PROPN
ejpam-5709	148	6	kanasri	kanasri	PROPN
ejpam-5709	148	7	/	/	SYM
ejpam-5709	148	8	eur	eur	PROPN
ejpam-5709	148	9	.	.	PUNCT
ejpam-5709	149	1	j.	j.	PROPN
ejpam-5709	149	2	pure	pure	PROPN
ejpam-5709	149	3	appl	appl	PROPN
ejpam-5709	149	4	.	.	PROPN
ejpam-5709	149	5	math	math	PROPN
ejpam-5709	149	6	,	,	PUNCT
ejpam-5709	149	7	18	18	NUM
ejpam-5709	149	8	(	(	PUNCT
ejpam-5709	149	9	1	1	NUM
ejpam-5709	149	10	)	)	PUNCT
ejpam-5709	149	11	(	(	PUNCT
ejpam-5709	149	12	2025	2025	NUM
ejpam-5709	149	13	)	)	PUNCT
ejpam-5709	149	14	,	,	PUNCT
ejpam-5709	149	15	5709	5709	NUM
ejpam-5709	149	16	6	6	NUM
ejpam-5709	149	17	of	of	ADP
ejpam-5709	149	18	15	15	NUM
ejpam-5709	149	19	example	example	NOUN
ejpam-5709	149	20	1	1	NUM
ejpam-5709	149	21	.	.	PUNCT
ejpam-5709	150	1	let	let	VERB
ejpam-5709	150	2	k	k	NOUN
ejpam-5709	150	3	=	=	PUNCT
ejpam-5709	150	4	q	q	PROPN
ejpam-5709	150	5	(	(	PUNCT
ejpam-5709	150	6	√	√	NUM
ejpam-5709	150	7	−1	−1	NOUN
ejpam-5709	150	8	)	)	PUNCT
ejpam-5709	150	9	,	,	PUNCT
ejpam-5709	150	10	β	β	X
ejpam-5709	150	11	=	=	PUNCT
ejpam-5709	150	12	−3	−3	PROPN
ejpam-5709	151	1	+	+	NUM
ejpam-5709	151	2	2i	2i	NUM
ejpam-5709	151	3	,	,	PUNCT
ejpam-5709	151	4	and	and	CCONJ
ejpam-5709	151	5	η	η	PROPN
ejpam-5709	151	6	=	=	PROPN
ejpam-5709	151	7	−439	−439	PROPN
ejpam-5709	151	8	−	−	PROPN
ejpam-5709	151	9	258i	258i	PROPN
ejpam-5709	151	10	.	.	PUNCT
ejpam-5709	152	1	then	then	ADV
ejpam-5709	152	2	d	d	X
ejpam-5709	152	3	=	=	SYM
ejpam-5709	152	4	1	1	NUM
ejpam-5709	152	5	,	,	PUNCT
ejpam-5709	152	6	n(β	n(β	NUM
ejpam-5709	152	7	)	)	PUNCT
ejpam-5709	152	8	=	=	SYM
ejpam-5709	152	9	13	13	NUM
ejpam-5709	152	10	,	,	PUNCT
ejpam-5709	152	11	and	and	CCONJ
ejpam-5709	152	12	thus	thus	ADV
ejpam-5709	152	13	c	c	X
ejpam-5709	152	14	=	=	SYM
ejpam-5709	152	15	{	{	PUNCT
ejpam-5709	152	16	0	0	NUM
ejpam-5709	152	17	,	,	PUNCT
ejpam-5709	152	18	1	1	NUM
ejpam-5709	152	19	,	,	PUNCT
ejpam-5709	152	20	.	.	PUNCT
ejpam-5709	152	21	.	.	PUNCT
ejpam-5709	152	22	.	.	PUNCT
ejpam-5709	153	1	,	,	PUNCT
ejpam-5709	153	2	12	12	NUM
ejpam-5709	153	3	}	}	PUNCT
ejpam-5709	153	4	is	be	AUX
ejpam-5709	153	5	a	a	DET
ejpam-5709	153	6	crs(β	crs(β	NOUN
ejpam-5709	153	7	)	)	PUNCT
ejpam-5709	153	8	.	.	PUNCT
ejpam-5709	154	1	one	one	PRON
ejpam-5709	154	2	can	can	AUX
ejpam-5709	154	3	compute	compute	VERB
ejpam-5709	154	4	that	that	PRON
ejpam-5709	154	5	η	η	PROPN
ejpam-5709	154	6	=	=	PROPN
ejpam-5709	154	7	(	(	PUNCT
ejpam-5709	154	8	63	63	NUM
ejpam-5709	154	9	+	+	SYM
ejpam-5709	154	10	128i)β	128i)β	NUM
ejpam-5709	154	11	+	+	CCONJ
ejpam-5709	154	12	6	6	NUM
ejpam-5709	154	13	,	,	PUNCT
ejpam-5709	154	14	63	63	NUM
ejpam-5709	154	15	+	+	SYM
ejpam-5709	154	16	128i	128i	NOUN
ejpam-5709	154	17	=	=	SYM
ejpam-5709	154	18	(	(	PUNCT
ejpam-5709	154	19	7−	7−	NUM
ejpam-5709	154	20	38i)β	38i)β	NUM
ejpam-5709	154	21	+	+	CCONJ
ejpam-5709	154	22	8	8	NUM
ejpam-5709	154	23	,	,	PUNCT
ejpam-5709	154	24	7−	7−	NUM
ejpam-5709	154	25	38i	38i	NUM
ejpam-5709	154	26	=	=	SYM
ejpam-5709	154	27	(	(	PUNCT
ejpam-5709	154	28	−7	−7	PROPN
ejpam-5709	154	29	+	+	NOUN
ejpam-5709	154	30	8i)β	8i)β	NUM
ejpam-5709	154	31	+	+	CCONJ
ejpam-5709	154	32	2	2	NUM
ejpam-5709	154	33	,	,	PUNCT
ejpam-5709	154	34	−7	−7	NOUN
ejpam-5709	154	35	+	+	NOUN
ejpam-5709	154	36	8i	8i	NUM
ejpam-5709	154	37	=	=	NOUN
ejpam-5709	154	38	4β	4β	NOUN
ejpam-5709	154	39	+	+	NOUN
ejpam-5709	154	40	5	5	X
ejpam-5709	154	41	.	.	PUNCT
ejpam-5709	154	42	thus	thus	ADV
ejpam-5709	154	43	,	,	PUNCT
ejpam-5709	154	44	γ0	γ0	NOUN
ejpam-5709	154	45	=	=	SYM
ejpam-5709	154	46	63	63	NUM
ejpam-5709	154	47	+	+	NUM
ejpam-5709	154	48	128i	128i	NOUN
ejpam-5709	154	49	,	,	PUNCT
ejpam-5709	154	50	γ1	γ1	NOUN
ejpam-5709	154	51	=	=	PROPN
ejpam-5709	154	52	7−	7−	NUM
ejpam-5709	154	53	38i	38i	NUM
ejpam-5709	154	54	,	,	PUNCT
ejpam-5709	154	55	γ2	γ2	NOUN
ejpam-5709	154	56	=	=	SYM
ejpam-5709	154	57	−7	−7	ADP
ejpam-5709	154	58	+	+	NOUN
ejpam-5709	154	59	8i	8i	NUM
ejpam-5709	154	60	,	,	PUNCT
ejpam-5709	154	61	γ3	γ3	NOUN
ejpam-5709	154	62	=	=	PUNCT
ejpam-5709	154	63	4	4	NUM
ejpam-5709	154	64	∈	∈	NOUN
ejpam-5709	154	65	c	c	NOUN
ejpam-5709	154	66	and	and	CCONJ
ejpam-5709	154	67	α0	α0	ADJ
ejpam-5709	154	68	=	=	SYM
ejpam-5709	154	69	6	6	NUM
ejpam-5709	154	70	,	,	PUNCT
ejpam-5709	154	71	α1	α1	PROPN
ejpam-5709	154	72	=	=	SYM
ejpam-5709	154	73	8	8	NUM
ejpam-5709	154	74	,	,	PUNCT
ejpam-5709	154	75	α2	α2	NOUN
ejpam-5709	154	76	=	=	SYM
ejpam-5709	154	77	2	2	NUM
ejpam-5709	154	78	,	,	PUNCT
ejpam-5709	154	79	α3	α3	NOUN
ejpam-5709	154	80	=	=	SYM
ejpam-5709	154	81	5	5	X
ejpam-5709	154	82	.	.	PUNCT
ejpam-5709	154	83	by	by	ADP
ejpam-5709	154	84	theorem	theorem	NOUN
ejpam-5709	154	85	1	1	NUM
ejpam-5709	154	86	,	,	PUNCT
ejpam-5709	154	87	we	we	PRON
ejpam-5709	154	88	conclude	conclude	VERB
ejpam-5709	154	89	that	that	SCONJ
ejpam-5709	154	90	η	η	PROPN
ejpam-5709	154	91	=	=	PROPN
ejpam-5709	154	92	γkβ	γkβ	PROPN
ejpam-5709	155	1	k+1	k+1	X
ejpam-5709	156	1	+	+	CCONJ
ejpam-5709	157	1	αkβ	αkβ	NOUN
ejpam-5709	157	2	k	k	X
ejpam-5709	157	3	+	+	CCONJ
ejpam-5709	157	4	·	·	PUNCT
ejpam-5709	157	5	·	·	PUNCT
ejpam-5709	157	6	·	·	PUNCT
ejpam-5709	157	7	+	+	NUM
ejpam-5709	157	8	α1β	α1β	X
ejpam-5709	157	9	+	+	CCONJ
ejpam-5709	157	10	α0	α0	ADJ
ejpam-5709	157	11	(	(	PUNCT
ejpam-5709	157	12	0	0	NUM
ejpam-5709	157	13	≤	≤	NUM
ejpam-5709	158	1	k	k	X
ejpam-5709	158	2	≤	≤	NUM
ejpam-5709	158	3	3	3	NUM
ejpam-5709	158	4	)	)	PUNCT
ejpam-5709	158	5	are	be	AUX
ejpam-5709	158	6	the	the	DET
ejpam-5709	158	7	base	base	NOUN
ejpam-5709	158	8	-	-	PUNCT
ejpam-5709	158	9	β(c	β(c	NUM
ejpam-5709	158	10	)	)	PUNCT
ejpam-5709	158	11	representations	representation	NOUN
ejpam-5709	158	12	of	of	ADP
ejpam-5709	158	13	η	η	PROPN
ejpam-5709	158	14	.	.	PROPN
ejpam-5709	158	15	to	to	PART
ejpam-5709	158	16	provide	provide	VERB
ejpam-5709	158	17	the	the	DET
ejpam-5709	158	18	explicit	explicit	ADJ
ejpam-5709	158	19	shapes	shape	NOUN
ejpam-5709	158	20	of	of	ADP
ejpam-5709	158	21	the	the	DET
ejpam-5709	158	22	base	base	NOUN
ejpam-5709	158	23	-	-	PUNCT
ejpam-5709	158	24	β(c	β(c	NUM
ejpam-5709	158	25	)	)	PUNCT
ejpam-5709	158	26	representations	representation	NOUN
ejpam-5709	158	27	of	of	ADP
ejpam-5709	158	28	η	η	PROPN
ejpam-5709	158	29	for	for	ADP
ejpam-5709	158	30	the	the	DET
ejpam-5709	158	31	case	case	NOUN
ejpam-5709	158	32	γi	γi	X
ejpam-5709	158	33	̸∈	̸∈	PROPN
ejpam-5709	158	34	c	c	PROPN
ejpam-5709	158	35	for	for	ADP
ejpam-5709	158	36	all	all	DET
ejpam-5709	158	37	i	i	PRON
ejpam-5709	158	38	∈	∈	VERB
ejpam-5709	158	39	n	n	PART
ejpam-5709	158	40	∪	∪	X
ejpam-5709	158	41	{	{	PUNCT
ejpam-5709	158	42	0	0	NUM
ejpam-5709	158	43	}	}	PUNCT
ejpam-5709	158	44	,	,	PUNCT
ejpam-5709	158	45	we	we	PRON
ejpam-5709	158	46	require	require	VERB
ejpam-5709	158	47	the	the	DET
ejpam-5709	158	48	following	follow	VERB
ejpam-5709	158	49	lemma	lemma	PROPN
ejpam-5709	158	50	.	.	PUNCT
ejpam-5709	159	1	lemma	lemma	PROPN
ejpam-5709	159	2	1	1	X
ejpam-5709	159	3	.	.	PUNCT
ejpam-5709	160	1	let	let	VERB
ejpam-5709	160	2	k	k	NOUN
ejpam-5709	160	3	=	=	PUNCT
ejpam-5709	160	4	q	q	ADJ
ejpam-5709	160	5	(	(	PUNCT
ejpam-5709	160	6	√	√	NUM
ejpam-5709	160	7	m	m	VERB
ejpam-5709	160	8	)	)	PUNCT
ejpam-5709	160	9	be	be	AUX
ejpam-5709	160	10	an	an	DET
ejpam-5709	160	11	imaginary	imaginary	ADJ
ejpam-5709	160	12	quadratic	quadratic	ADJ
ejpam-5709	160	13	field	field	NOUN
ejpam-5709	160	14	and	and	CCONJ
ejpam-5709	160	15	let	let	VERB
ejpam-5709	160	16	β	β	VERB
ejpam-5709	160	17	=	=	PUNCT
ejpam-5709	160	18	a	a	DET
ejpam-5709	160	19	+	+	NUM
ejpam-5709	160	20	bσm	bσm	NOUN
ejpam-5709	160	21	∈	∈	NOUN
ejpam-5709	160	22	ok	ok	INTJ
ejpam-5709	160	23	such	such	ADJ
ejpam-5709	160	24	that	that	SCONJ
ejpam-5709	160	25	|β|	|β|	PRON
ejpam-5709	160	26	≥	≥	NUM
ejpam-5709	160	27	2	2	NUM
ejpam-5709	160	28	and	and	CCONJ
ejpam-5709	160	29	d	d	NOUN
ejpam-5709	160	30	=	=	SYM
ejpam-5709	160	31	gcd(a	gcd(a	PROPN
ejpam-5709	160	32	,	,	PUNCT
ejpam-5709	160	33	b	b	NOUN
ejpam-5709	160	34	)	)	PUNCT
ejpam-5709	160	35	.	.	PUNCT
ejpam-5709	161	1	let	let	VERB
ejpam-5709	161	2	s	s	PRON
ejpam-5709	161	3	=	=	PUNCT
ejpam-5709	161	4			PUNCT
ejpam-5709	161	5	√	√	NOUN
ejpam-5709	161	6	(	(	PUNCT
ejpam-5709	161	7	|n(β)|	|n(β)|	PROPN
ejpam-5709	161	8	d	d	NOUN
ejpam-5709	161	9	−	−	PROPN
ejpam-5709	161	10	1	1	NUM
ejpam-5709	161	11	)	)	SYM
ejpam-5709	161	12	2	2	NUM
ejpam-5709	161	13	−m(d−	−m(d−	PROPN
ejpam-5709	161	14	1)2	1)2	NUM
ejpam-5709	161	15	if	if	SCONJ
ejpam-5709	161	16	m	m	VERB
ejpam-5709	161	17	̸≡	̸≡	VERB
ejpam-5709	161	18	1	1	NUM
ejpam-5709	161	19	(	(	PUNCT
ejpam-5709	161	20	mod	mod	PROPN
ejpam-5709	161	21	4),√	4),√	PROPN
ejpam-5709	161	22	(	(	PUNCT
ejpam-5709	161	23	|n(β)|	|n(β)|	PROPN
ejpam-5709	161	24	d	d	NOUN
ejpam-5709	161	25	−	−	PROPN
ejpam-5709	161	26	1	1	NUM
ejpam-5709	161	27	)	)	SYM
ejpam-5709	161	28	2	2	NUM
ejpam-5709	161	29	+	+	CCONJ
ejpam-5709	161	30	(	(	PUNCT
ejpam-5709	161	31	|n(β)|	|n(β)|	ADJ
ejpam-5709	161	32	d	d	NOUN
ejpam-5709	161	33	−	−	PROPN
ejpam-5709	161	34	1	1	NUM
ejpam-5709	161	35	)	)	PUNCT
ejpam-5709	161	36	(	(	PUNCT
ejpam-5709	161	37	d−	d−	PROPN
ejpam-5709	161	38	1	1	NUM
ejpam-5709	161	39	)	)	PUNCT
ejpam-5709	161	40	+	+	CCONJ
ejpam-5709	162	1	(	(	PUNCT
ejpam-5709	162	2	d−	d−	PROPN
ejpam-5709	162	3	1)2	1)2	NUM
ejpam-5709	162	4	(	(	PUNCT
ejpam-5709	162	5	1−m	1−m	NUM
ejpam-5709	162	6	4	4	NUM
ejpam-5709	162	7	)	)	PUNCT
ejpam-5709	162	8	if	if	SCONJ
ejpam-5709	162	9	m	m	VERB
ejpam-5709	162	10	≡	≡	PROPN
ejpam-5709	162	11	1	1	NUM
ejpam-5709	162	12	(	(	PUNCT
ejpam-5709	162	13	mod	mod	NOUN
ejpam-5709	162	14	4	4	NUM
ejpam-5709	162	15	)	)	PUNCT
ejpam-5709	162	16	.	.	PUNCT
ejpam-5709	163	1	(	(	PUNCT
ejpam-5709	163	2	8)	8)	NUM
ejpam-5709	163	3	for	for	ADP
ejpam-5709	163	4	η	η	PROPN
ejpam-5709	163	5	∈	∈	PROPN
ejpam-5709	163	6	ok	ok	ADJ
ejpam-5709	163	7	with	with	ADP
ejpam-5709	163	8	|η|	|η|	PROPN
ejpam-5709	163	9	≥	≥	NOUN
ejpam-5709	163	10	s	s	NOUN
ejpam-5709	163	11	,	,	PUNCT
ejpam-5709	163	12	if	if	SCONJ
ejpam-5709	163	13	η	η	PROPN
ejpam-5709	163	14	=	=	SYM
ejpam-5709	163	15	γ0β+α0	γ0β+α0	PROPN
ejpam-5709	163	16	,	,	PUNCT
ejpam-5709	163	17	where	where	SCONJ
ejpam-5709	163	18	γ0	γ0	PROPN
ejpam-5709	163	19	∈	∈	PROPN
ejpam-5709	163	20	ok\{0	ok\{0	PROPN
ejpam-5709	163	21	}	}	PUNCT
ejpam-5709	163	22	and	and	CCONJ
ejpam-5709	163	23	α0	α0	PROPN
ejpam-5709	163	24	∈	∈	PROPN
ejpam-5709	163	25	c	c	NOUN
ejpam-5709	163	26	,	,	PUNCT
ejpam-5709	163	27	then	then	ADV
ejpam-5709	163	28	|η|	|η|	PROPN
ejpam-5709	163	29	≥	≥	NUM
ejpam-5709	163	30	|γ0|	|γ0|	NOUN
ejpam-5709	163	31	.	.	PUNCT
ejpam-5709	164	1	proof	proof	NOUN
ejpam-5709	164	2	.	.	PUNCT
ejpam-5709	165	1	suppose	suppose	VERB
ejpam-5709	165	2	to	to	ADP
ejpam-5709	165	3	the	the	DET
ejpam-5709	165	4	contrary	contrary	NOUN
ejpam-5709	165	5	that	that	DET
ejpam-5709	165	6	|γ0|	|γ0|	NOUN
ejpam-5709	165	7	>	>	X
ejpam-5709	165	8	|η|	|η|	PROPN
ejpam-5709	165	9	.	.	PROPN
ejpam-5709	166	1	then	then	ADV
ejpam-5709	166	2	|γ0|	|γ0|	VERB
ejpam-5709	166	3	>	>	X
ejpam-5709	166	4	|γ0β	|γ0β	PROPN
ejpam-5709	166	5	+	+	CCONJ
ejpam-5709	166	6	α0|	α0|	NUM
ejpam-5709	166	7	≥	≥	NOUN
ejpam-5709	166	8	|γ0||β|	|γ0||β|	PROPN
ejpam-5709	166	9	−	−	NOUN
ejpam-5709	166	10	|α0|	|α0|	NOUN
ejpam-5709	166	11	and	and	CCONJ
ejpam-5709	166	12	thus	thus	ADV
ejpam-5709	166	13	|α0|	|α0|	X
ejpam-5709	166	14	>	>	X
ejpam-5709	166	15	|γ0|(|β|	|γ0|(|β|	ADV
ejpam-5709	166	16	−	−	PROPN
ejpam-5709	166	17	1	1	NUM
ejpam-5709	166	18	)	)	PUNCT
ejpam-5709	166	19	.	.	PUNCT
ejpam-5709	167	1	since	since	SCONJ
ejpam-5709	167	2	α0	α0	PROPN
ejpam-5709	167	3	∈	∈	PROPN
ejpam-5709	167	4	c	c	X
ejpam-5709	167	5	,	,	PUNCT
ejpam-5709	167	6	we	we	PRON
ejpam-5709	167	7	have	have	VERB
ejpam-5709	167	8	|α0|	|α0|	NOUN
ejpam-5709	167	9	≤	≤	NOUN
ejpam-5709	167	10	s	s	NOUN
ejpam-5709	167	11	and	and	CCONJ
ejpam-5709	167	12	hence	hence	ADV
ejpam-5709	167	13	|γ0|	|γ0|	NOUN
ejpam-5709	167	14	>	>	X
ejpam-5709	168	1	|η|	|η|	PROPN
ejpam-5709	168	2	≥	≥	NOUN
ejpam-5709	168	3	s	s	PART
ejpam-5709	168	4	≥	≥	NOUN
ejpam-5709	168	5	|α0|	|α0|	NOUN
ejpam-5709	168	6	>	>	X
ejpam-5709	168	7	|γ0|(|β|	|γ0|(|β|	ADV
ejpam-5709	168	8	−	−	PROPN
ejpam-5709	168	9	1	1	NUM
ejpam-5709	168	10	)	)	PUNCT
ejpam-5709	168	11	≥	≥	NOUN
ejpam-5709	168	12	|γ0|	|γ0|	NOUN
ejpam-5709	168	13	,	,	PUNCT
ejpam-5709	168	14	which	which	PRON
ejpam-5709	168	15	is	be	AUX
ejpam-5709	168	16	a	a	DET
ejpam-5709	168	17	contradiction	contradiction	NOUN
ejpam-5709	168	18	.	.	PUNCT
ejpam-5709	169	1	theorem	theorem	NOUN
ejpam-5709	169	2	2	2	NUM
ejpam-5709	169	3	.	.	PUNCT
ejpam-5709	170	1	let	let	VERB
ejpam-5709	170	2	k	k	NOUN
ejpam-5709	170	3	=	=	PUNCT
ejpam-5709	170	4	q	q	ADJ
ejpam-5709	170	5	(	(	PUNCT
ejpam-5709	170	6	√	√	NUM
ejpam-5709	170	7	m	m	VERB
ejpam-5709	170	8	)	)	PUNCT
ejpam-5709	170	9	be	be	AUX
ejpam-5709	170	10	an	an	DET
ejpam-5709	170	11	imaginary	imaginary	ADJ
ejpam-5709	170	12	quadratic	quadratic	ADJ
ejpam-5709	170	13	field	field	NOUN
ejpam-5709	170	14	and	and	CCONJ
ejpam-5709	170	15	let	let	VERB
ejpam-5709	170	16	β	β	X
ejpam-5709	170	17	=	=	SYM
ejpam-5709	170	18	a+bσm	a+bσm	PROPN
ejpam-5709	170	19	,	,	PUNCT
ejpam-5709	170	20	η	η	PROPN
ejpam-5709	170	21	∈	∈	PROPN
ejpam-5709	170	22	ok	ok	INTJ
ejpam-5709	170	23	be	be	AUX
ejpam-5709	170	24	such	such	ADJ
ejpam-5709	170	25	that	that	SCONJ
ejpam-5709	170	26	|β|	|β|	PRON
ejpam-5709	170	27	≥	≥	NUM
ejpam-5709	170	28	2	2	NUM
ejpam-5709	170	29	,	,	PUNCT
ejpam-5709	170	30	η	η	PROPN
ejpam-5709	170	31	̸∈	̸∈	PROPN
ejpam-5709	170	32	c	c	PROPN
ejpam-5709	170	33	,	,	PUNCT
ejpam-5709	170	34	and	and	CCONJ
ejpam-5709	170	35	|η|	|η|	PROPN
ejpam-5709	170	36	≥	≥	NUM
ejpam-5709	170	37	s	s	PROPN
ejpam-5709	170	38	,	,	PUNCT
ejpam-5709	170	39	where	where	SCONJ
ejpam-5709	170	40	s	s	NOUN
ejpam-5709	170	41	is	be	AUX
ejpam-5709	170	42	defined	define	VERB
ejpam-5709	170	43	as	as	ADP
ejpam-5709	170	44	in	in	ADP
ejpam-5709	170	45	(	(	PUNCT
ejpam-5709	170	46	8)	8)	NUM
ejpam-5709	170	47	.	.	PUNCT
ejpam-5709	171	1	let	let	ADJ
ejpam-5709	171	2	(	(	PUNCT
ejpam-5709	171	3	γi)i≥0	γi)i≥0	NOUN
ejpam-5709	171	4	be	be	AUX
ejpam-5709	171	5	the	the	DET
ejpam-5709	171	6	sequence	sequence	NOUN
ejpam-5709	171	7	described	describe	VERB
ejpam-5709	171	8	in	in	ADP
ejpam-5709	171	9	(	(	PUNCT
ejpam-5709	171	10	4	4	NUM
ejpam-5709	171	11	)	)	PUNCT
ejpam-5709	171	12	.	.	PUNCT
ejpam-5709	172	1	if	if	SCONJ
ejpam-5709	172	2	γi	γi	PROPN
ejpam-5709	172	3	̸∈	̸∈	PROPN
ejpam-5709	172	4	c	c	PROPN
ejpam-5709	172	5	for	for	ADP
ejpam-5709	172	6	all	all	DET
ejpam-5709	172	7	i	i	PRON
ejpam-5709	172	8	∈	∈	VERB
ejpam-5709	172	9	n	n	PART
ejpam-5709	172	10	∪	∪	X
ejpam-5709	172	11	{	{	PUNCT
ejpam-5709	172	12	0	0	NUM
ejpam-5709	172	13	}	}	PUNCT
ejpam-5709	172	14	,	,	PUNCT
ejpam-5709	172	15	then	then	ADV
ejpam-5709	172	16	there	there	PRON
ejpam-5709	172	17	exist	exist	VERB
ejpam-5709	172	18	nonnegative	nonnegative	ADJ
ejpam-5709	172	19	integers	integer	NOUN
ejpam-5709	172	20	u	u	NOUN
ejpam-5709	172	21	and	and	CCONJ
ejpam-5709	172	22	v	v	NOUN
ejpam-5709	172	23	with	with	ADP
ejpam-5709	172	24	u	u	NOUN
ejpam-5709	172	25	<	<	X
ejpam-5709	172	26	v	v	ADP
ejpam-5709	172	27	such	such	ADJ
ejpam-5709	172	28	that	that	DET
ejpam-5709	172	29	γu	γu	NOUN
ejpam-5709	172	30	=	=	PRON
ejpam-5709	172	31	γv	γv	NOUN
ejpam-5709	172	32	and	and	CCONJ
ejpam-5709	172	33	the	the	DET
ejpam-5709	172	34	base	base	NOUN
ejpam-5709	172	35	-	-	PUNCT
ejpam-5709	172	36	β(c	β(c	NUM
ejpam-5709	172	37	)	)	PUNCT
ejpam-5709	172	38	representations	representation	NOUN
ejpam-5709	172	39	of	of	ADP
ejpam-5709	172	40	η	η	PROPN
ejpam-5709	172	41	are	be	AUX
ejpam-5709	172	42	of	of	ADP
ejpam-5709	172	43	the	the	DET
ejpam-5709	172	44	form	form	NOUN
ejpam-5709	172	45	η	η	PROPN
ejpam-5709	172	46	=	=	PROPN
ejpam-5709	172	47	γkβ	γkβ	NOUN
ejpam-5709	172	48	k+1	k+1	X
ejpam-5709	172	49	+	+	CCONJ
ejpam-5709	172	50	∑	∑	PROPN
ejpam-5709	172	51	0≤i≤k	0≤i≤k	NUM
ejpam-5709	172	52	αiβ	αiβ	NOUN
ejpam-5709	172	53	i	i	PRON
ejpam-5709	172	54	(	(	PUNCT
ejpam-5709	172	55	0	0	NUM
ejpam-5709	172	56	≤	≤	NUM
ejpam-5709	172	57	k	k	X
ejpam-5709	172	58	≤	≤	NUM
ejpam-5709	172	59	u	u	NOUN
ejpam-5709	172	60	)	)	PUNCT
ejpam-5709	172	61	,	,	PUNCT
ejpam-5709	172	62	(	(	PUNCT
ejpam-5709	172	63	9	9	X
ejpam-5709	172	64	)	)	PUNCT
ejpam-5709	172	65	η	η	PROPN
ejpam-5709	172	66	=	=	PROPN
ejpam-5709	172	67	γkβ	γkβ	NOUN
ejpam-5709	172	68	k+1	k+1	X
ejpam-5709	173	1	+	+	CCONJ
ejpam-5709	173	2	∑	∑	PROPN
ejpam-5709	173	3	u	u	NOUN
ejpam-5709	173	4	<	<	X
ejpam-5709	173	5	j≤k	j≤k	X
ejpam-5709	173	6	αjβ	αjβ	PROPN
ejpam-5709	173	7	j	j	PROPN
ejpam-5709	174	1	+	+	CCONJ
ejpam-5709	174	2	∑	∑	PROPN
ejpam-5709	174	3	0≤i≤u	0≤i≤u	NUM
ejpam-5709	174	4	αiβ	αiβ	NOUN
ejpam-5709	174	5	i	i	PRON
ejpam-5709	174	6	(	(	PUNCT
ejpam-5709	174	7	u	u	X
ejpam-5709	174	8	<	<	X
ejpam-5709	174	9	k	k	X
ejpam-5709	174	10	<	<	X
ejpam-5709	174	11	v	v	NOUN
ejpam-5709	174	12	)	)	PUNCT
ejpam-5709	174	13	,	,	PUNCT
ejpam-5709	174	14	(	(	PUNCT
ejpam-5709	174	15	10	10	NUM
ejpam-5709	174	16	)	)	PUNCT
ejpam-5709	174	17	η	η	PROPN
ejpam-5709	174	18	=	=	SYM
ejpam-5709	174	19	γuβ	γuβ	PROPN
ejpam-5709	174	20	(	(	PUNCT
ejpam-5709	174	21	v−u)n+u+1	v−u)n+u+1	PROPN
ejpam-5709	174	22	+	+	CCONJ
ejpam-5709	174	23	αvβ	αvβ	ADJ
ejpam-5709	174	24	(	(	PUNCT
ejpam-5709	174	25	v−u)n+u	v−u)n+u	NOUN
ejpam-5709	174	26	+	+	CCONJ
ejpam-5709	174	27	αv−1β	αv−1β	NUM
ejpam-5709	174	28	(	(	PUNCT
ejpam-5709	174	29	v−u)n+u−1	v−u)n+u−1	PROPN
ejpam-5709	174	30	+	+	CCONJ
ejpam-5709	174	31	·	·	PUNCT
ejpam-5709	174	32	·	·	PUNCT
ejpam-5709	174	33	·	·	PUNCT
ejpam-5709	175	1	+	+	NUM
ejpam-5709	175	2	αu+1β	αu+1β	NUM
ejpam-5709	175	3	(	(	PUNCT
ejpam-5709	175	4	v−u)n+2u−v+1	v−u)n+2u−v+1	NOUN
ejpam-5709	175	5	+	+	X
ejpam-5709	175	6	·	·	PUNCT
ejpam-5709	175	7	·	·	PUNCT
ejpam-5709	175	8	·	·	PUNCT
ejpam-5709	175	9	+	+	CCONJ
ejpam-5709	175	10	∑	∑	PUNCT
ejpam-5709	175	11	u	u	NOUN
ejpam-5709	175	12	<	<	X
ejpam-5709	175	13	j≤v	j≤v	PROPN
ejpam-5709	175	14	αjβ	αjβ	PROPN
ejpam-5709	175	15	j	j	PROPN
ejpam-5709	175	16	+	+	CCONJ
ejpam-5709	175	17	∑	∑	PROPN
ejpam-5709	175	18	0≤i≤u	0≤i≤u	NUM
ejpam-5709	175	19	αiβ	αiβ	PROPN
ejpam-5709	175	20	i	i	PRON
ejpam-5709	175	21	(	(	PUNCT
ejpam-5709	175	22	n	n	NOUN
ejpam-5709	175	23	∈	∈	PROPN
ejpam-5709	175	24	n	n	CCONJ
ejpam-5709	175	25	)	)	PUNCT
ejpam-5709	175	26	,	,	PUNCT
ejpam-5709	175	27	(	(	PUNCT
ejpam-5709	175	28	11	11	X
ejpam-5709	175	29	)	)	PUNCT
ejpam-5709	175	30	p.	p.	NOUN
ejpam-5709	175	31	phetnun	phetnun	PROPN
ejpam-5709	175	32	,	,	PUNCT
ejpam-5709	175	33	n.r	n.r	PROPN
ejpam-5709	175	34	.	.	PROPN
ejpam-5709	175	35	kanasri	kanasri	PROPN
ejpam-5709	175	36	/	/	SYM
ejpam-5709	175	37	eur	eur	PROPN
ejpam-5709	175	38	.	.	PUNCT
ejpam-5709	176	1	j.	j.	PROPN
ejpam-5709	176	2	pure	pure	PROPN
ejpam-5709	176	3	appl	appl	PROPN
ejpam-5709	176	4	.	.	PROPN
ejpam-5709	176	5	math	math	PROPN
ejpam-5709	176	6	,	,	PUNCT
ejpam-5709	176	7	18	18	NUM
ejpam-5709	176	8	(	(	PUNCT
ejpam-5709	176	9	1	1	NUM
ejpam-5709	176	10	)	)	PUNCT
ejpam-5709	176	11	(	(	PUNCT
ejpam-5709	176	12	2025	2025	NUM
ejpam-5709	176	13	)	)	PUNCT
ejpam-5709	176	14	,	,	PUNCT
ejpam-5709	176	15	5709	5709	NUM
ejpam-5709	176	16	7	7	NUM
ejpam-5709	176	17	of	of	ADP
ejpam-5709	176	18	15	15	NUM
ejpam-5709	176	19	η	η	NOUN
ejpam-5709	176	20	=	=	PROPN
ejpam-5709	176	21	γu+1β	γu+1β	PROPN
ejpam-5709	176	22	(	(	PUNCT
ejpam-5709	176	23	v−u)n+u+2	v−u)n+u+2	PROPN
ejpam-5709	176	24	+	+	CCONJ
ejpam-5709	176	25	αu+1β	αu+1β	PROPN
ejpam-5709	176	26	(	(	PUNCT
ejpam-5709	176	27	v−u)n+u+1	v−u)n+u+1	PROPN
ejpam-5709	176	28	+	+	CCONJ
ejpam-5709	176	29	αvβ	αvβ	ADJ
ejpam-5709	176	30	(	(	PUNCT
ejpam-5709	176	31	v−u)n+u	v−u)n+u	NOUN
ejpam-5709	176	32	+	+	CCONJ
ejpam-5709	176	33	αv−1β	αv−1β	NUM
ejpam-5709	176	34	(	(	PUNCT
ejpam-5709	176	35	v−u)n+u−1	v−u)n+u−1	PROPN
ejpam-5709	176	36	+	+	CCONJ
ejpam-5709	176	37	·	·	PUNCT
ejpam-5709	176	38	·	·	PUNCT
ejpam-5709	176	39	·	·	PUNCT
ejpam-5709	177	1	+	+	NUM
ejpam-5709	177	2	αu+1β	αu+1β	NUM
ejpam-5709	177	3	(	(	PUNCT
ejpam-5709	177	4	v−u)n+2u−v+1	v−u)n+2u−v+1	NOUN
ejpam-5709	177	5	+	+	X
ejpam-5709	177	6	·	·	PUNCT
ejpam-5709	177	7	·	·	PUNCT
ejpam-5709	177	8	·	·	PUNCT
ejpam-5709	177	9	+	+	CCONJ
ejpam-5709	177	10	∑	∑	PUNCT
ejpam-5709	177	11	u	u	NOUN
ejpam-5709	177	12	<	<	X
ejpam-5709	177	13	j≤v	j≤v	PROPN
ejpam-5709	177	14	αjβ	αjβ	PROPN
ejpam-5709	177	15	j	j	PROPN
ejpam-5709	177	16	+	+	CCONJ
ejpam-5709	177	17	∑	∑	PROPN
ejpam-5709	177	18	0≤i≤u	0≤i≤u	NUM
ejpam-5709	177	19	αiβ	αiβ	PROPN
ejpam-5709	177	20	i	i	PRON
ejpam-5709	177	21	(	(	PUNCT
ejpam-5709	177	22	n	n	NOUN
ejpam-5709	177	23	∈	∈	PROPN
ejpam-5709	177	24	n	n	CCONJ
ejpam-5709	177	25	)	)	PUNCT
ejpam-5709	177	26	,	,	PUNCT
ejpam-5709	177	27	η	η	PROPN
ejpam-5709	177	28	=	=	PROPN
ejpam-5709	177	29	γu+2β	γu+2β	NOUN
ejpam-5709	177	30	(	(	PUNCT
ejpam-5709	177	31	v−u)n+u+3	v−u)n+u+3	NOUN
ejpam-5709	177	32	+	+	CCONJ
ejpam-5709	177	33	αu+2β	αu+2β	NOUN
ejpam-5709	177	34	(	(	PUNCT
ejpam-5709	177	35	v−u)n+u+2	v−u)n+u+2	PROPN
ejpam-5709	177	36	+	+	CCONJ
ejpam-5709	177	37	αu+1β	αu+1β	PROPN
ejpam-5709	177	38	(	(	PUNCT
ejpam-5709	177	39	v−u)n+u+1	v−u)n+u+1	PROPN
ejpam-5709	177	40	+	+	CCONJ
ejpam-5709	177	41	αvβ	αvβ	ADJ
ejpam-5709	177	42	(	(	PUNCT
ejpam-5709	177	43	v−u)n+u	v−u)n+u	NOUN
ejpam-5709	177	44	+	+	CCONJ
ejpam-5709	177	45	αv−1β	αv−1β	NUM
ejpam-5709	177	46	(	(	PUNCT
ejpam-5709	177	47	v−u)n+u−1	v−u)n+u−1	PROPN
ejpam-5709	177	48	+	+	CCONJ
ejpam-5709	177	49	·	·	PUNCT
ejpam-5709	177	50	·	·	PUNCT
ejpam-5709	177	51	·	·	PUNCT
ejpam-5709	177	52	+	+	NUM
ejpam-5709	177	53	αu+1β	αu+1β	NUM
ejpam-5709	177	54	(	(	PUNCT
ejpam-5709	177	55	v−u)n+2u−v+1	v−u)n+2u−v+1	NOUN
ejpam-5709	177	56	+	+	X
ejpam-5709	177	57	·	·	PUNCT
ejpam-5709	177	58	·	·	PUNCT
ejpam-5709	177	59	·	·	PUNCT
ejpam-5709	177	60	+	+	CCONJ
ejpam-5709	177	61	∑	∑	PUNCT
ejpam-5709	177	62	u	u	NOUN
ejpam-5709	177	63	<	<	X
ejpam-5709	177	64	j≤v	j≤v	PROPN
ejpam-5709	177	65	αjβ	αjβ	PROPN
ejpam-5709	177	66	j	j	PROPN
ejpam-5709	177	67	+	+	CCONJ
ejpam-5709	177	68	∑	∑	PROPN
ejpam-5709	177	69	0≤i≤u	0≤i≤u	NUM
ejpam-5709	177	70	αiβ	αiβ	PROPN
ejpam-5709	177	71	i	i	PRON
ejpam-5709	177	72	(	(	PUNCT
ejpam-5709	177	73	n	n	NOUN
ejpam-5709	177	74	∈	∈	PROPN
ejpam-5709	177	75	n	n	CCONJ
ejpam-5709	177	76	)	)	PUNCT
ejpam-5709	177	77	,	,	PUNCT
ejpam-5709	177	78	...	...	PUNCT
ejpam-5709	178	1	η	η	X
ejpam-5709	178	2	=	=	PRON
ejpam-5709	178	3	γv−1β	γv−1β	PROPN
ejpam-5709	178	4	(	(	PUNCT
ejpam-5709	178	5	v−u)n+v	v−u)n+v	NOUN
ejpam-5709	178	6	+	+	CCONJ
ejpam-5709	178	7	αv−1β	αv−1β	NUM
ejpam-5709	178	8	(	(	PUNCT
ejpam-5709	178	9	v−u)n+v−1	v−u)n+v−1	PROPN
ejpam-5709	178	10	+	+	CCONJ
ejpam-5709	178	11	αv−2β	αv−2β	PROPN
ejpam-5709	178	12	(	(	PUNCT
ejpam-5709	178	13	v−u)n+v−2	v−u)n+v−2	NOUN
ejpam-5709	178	14	+	+	CCONJ
ejpam-5709	178	15	·	·	PUNCT
ejpam-5709	178	16	·	·	PUNCT
ejpam-5709	178	17	·	·	PUNCT
ejpam-5709	178	18	+	+	NUM
ejpam-5709	178	19	αu+1β	αu+1β	NUM
ejpam-5709	178	20	(	(	PUNCT
ejpam-5709	178	21	v−u)n+u+1	v−u)n+u+1	PROPN
ejpam-5709	178	22	+	+	CCONJ
ejpam-5709	178	23	αvβ	αvβ	ADJ
ejpam-5709	178	24	(	(	PUNCT
ejpam-5709	178	25	v−u)n+u	v−u)n+u	NOUN
ejpam-5709	178	26	+	+	CCONJ
ejpam-5709	178	27	αv−1β	αv−1β	NUM
ejpam-5709	178	28	(	(	PUNCT
ejpam-5709	178	29	v−u)n+u−1	v−u)n+u−1	PROPN
ejpam-5709	178	30	+	+	CCONJ
ejpam-5709	178	31	·	·	PUNCT
ejpam-5709	178	32	·	·	PUNCT
ejpam-5709	178	33	·	·	PUNCT
ejpam-5709	178	34	+	+	NUM
ejpam-5709	178	35	αu+1β	αu+1β	NUM
ejpam-5709	178	36	(	(	PUNCT
ejpam-5709	178	37	v−u)n+2u−v+1	v−u)n+2u−v+1	NOUN
ejpam-5709	178	38	+	+	X
ejpam-5709	178	39	·	·	PUNCT
ejpam-5709	178	40	·	·	PUNCT
ejpam-5709	178	41	·	·	PUNCT
ejpam-5709	178	42	+	+	CCONJ
ejpam-5709	178	43	∑	∑	PUNCT
ejpam-5709	178	44	u	u	NOUN
ejpam-5709	178	45	<	<	X
ejpam-5709	178	46	j≤v	j≤v	PROPN
ejpam-5709	178	47	αjβ	αjβ	PROPN
ejpam-5709	178	48	j	j	PROPN
ejpam-5709	178	49	+	+	CCONJ
ejpam-5709	178	50	∑	∑	PROPN
ejpam-5709	178	51	0≤i≤u	0≤i≤u	NUM
ejpam-5709	178	52	αiβ	αiβ	PROPN
ejpam-5709	178	53	i	i	PRON
ejpam-5709	178	54	(	(	PUNCT
ejpam-5709	178	55	n	n	NOUN
ejpam-5709	178	56	∈	∈	PROPN
ejpam-5709	178	57	n	n	CCONJ
ejpam-5709	178	58	)	)	PUNCT
ejpam-5709	178	59	.	.	PUNCT
ejpam-5709	179	1	(	(	PUNCT
ejpam-5709	179	2	12	12	NUM
ejpam-5709	179	3	)	)	PUNCT
ejpam-5709	179	4	proof	proof	NOUN
ejpam-5709	179	5	.	.	PUNCT
ejpam-5709	180	1	we	we	PRON
ejpam-5709	180	2	first	first	ADV
ejpam-5709	180	3	verify	verify	VERB
ejpam-5709	180	4	two	two	NUM
ejpam-5709	180	5	important	important	ADJ
ejpam-5709	180	6	claims	claim	NOUN
ejpam-5709	180	7	:	:	PUNCT
ejpam-5709	180	8	claim	claim	NOUN
ejpam-5709	180	9	1	1	NUM
ejpam-5709	180	10	.	.	PUNCT
ejpam-5709	180	11	|γi|	|γi|	NOUN
ejpam-5709	180	12	≤	≤	NUM
ejpam-5709	180	13	max{2s	max{2s	PROPN
ejpam-5709	180	14	,	,	PUNCT
ejpam-5709	180	15	|η|	|η|	PROPN
ejpam-5709	180	16	}	}	PUNCT
ejpam-5709	180	17	=	=	NOUN
ejpam-5709	180	18	:	:	PUNCT
ejpam-5709	180	19	n	n	PROPN
ejpam-5709	180	20	for	for	ADP
ejpam-5709	180	21	all	all	PRON
ejpam-5709	180	22	i	i	PRON
ejpam-5709	180	23	∈	∈	VERB
ejpam-5709	180	24	n	n	PART
ejpam-5709	180	25	∪	∪	X
ejpam-5709	180	26	{	{	PUNCT
ejpam-5709	180	27	0	0	NUM
ejpam-5709	180	28	}	}	PUNCT
ejpam-5709	180	29	.	.	PUNCT
ejpam-5709	181	1	proof	proof	NOUN
ejpam-5709	181	2	of	of	ADP
ejpam-5709	181	3	claim	claim	NOUN
ejpam-5709	181	4	1	1	NUM
ejpam-5709	181	5	.	.	X
ejpam-5709	182	1	for	for	ADP
ejpam-5709	182	2	i	i	PRON
ejpam-5709	182	3	=	=	NOUN
ejpam-5709	182	4	0	0	NUM
ejpam-5709	182	5	,	,	PUNCT
ejpam-5709	182	6	we	we	PRON
ejpam-5709	182	7	have	have	VERB
ejpam-5709	182	8	η	η	NOUN
ejpam-5709	182	9	=	=	NOUN
ejpam-5709	182	10	γ0β	γ0β	PROPN
ejpam-5709	182	11	+	+	CCONJ
ejpam-5709	182	12	α0	α0	ADJ
ejpam-5709	182	13	.	.	PUNCT
ejpam-5709	183	1	since	since	SCONJ
ejpam-5709	183	2	|η|	|η|	PROPN
ejpam-5709	183	3	≥	≥	NOUN
ejpam-5709	183	4	s	s	PART
ejpam-5709	183	5	,	,	PUNCT
ejpam-5709	183	6	it	it	PRON
ejpam-5709	183	7	follows	follow	VERB
ejpam-5709	183	8	from	from	ADP
ejpam-5709	183	9	lemma	lemma	PROPN
ejpam-5709	183	10	1	1	NUM
ejpam-5709	183	11	that	that	PRON
ejpam-5709	183	12	|γ0|	|γ0|	NOUN
ejpam-5709	183	13	≤	≤	PUNCT
ejpam-5709	183	14	|η|	|η|	PROPN
ejpam-5709	183	15	≤	≤	NOUN
ejpam-5709	183	16	n	n	ADV
ejpam-5709	183	17	.	.	PUNCT
ejpam-5709	184	1	for	for	ADP
ejpam-5709	184	2	i	i	PRON
ejpam-5709	184	3	=	=	NOUN
ejpam-5709	184	4	1	1	NUM
ejpam-5709	184	5	,	,	PUNCT
ejpam-5709	184	6	we	we	PRON
ejpam-5709	184	7	have	have	VERB
ejpam-5709	184	8	γ0	γ0	NOUN
ejpam-5709	184	9	=	=	SYM
ejpam-5709	184	10	γ1β	γ1β	PROPN
ejpam-5709	184	11	+	+	NUM
ejpam-5709	184	12	α1	α1	PROPN
ejpam-5709	184	13	,	,	PUNCT
ejpam-5709	184	14	where	where	SCONJ
ejpam-5709	184	15	γ1	γ1	PROPN
ejpam-5709	184	16	∈	∈	PROPN
ejpam-5709	184	17	ok\{0	ok\{0	PROPN
ejpam-5709	184	18	}	}	PUNCT
ejpam-5709	184	19	and	and	CCONJ
ejpam-5709	184	20	α1	α1	PROPN
ejpam-5709	184	21	∈	∈	PROPN
ejpam-5709	184	22	c	c	NOUN
ejpam-5709	184	23	,	,	PUNCT
ejpam-5709	184	24	so	so	ADV
ejpam-5709	184	25	|α1|	|α1|	VERB
ejpam-5709	184	26	≤	≤	ADJ
ejpam-5709	184	27	s.	s.	PROPN
ejpam-5709	184	28	if	if	SCONJ
ejpam-5709	184	29	|γ0|	|γ0|	PROPN
ejpam-5709	184	30	≥	≥	NOUN
ejpam-5709	184	31	s	s	PROPN
ejpam-5709	184	32	,	,	PUNCT
ejpam-5709	184	33	then	then	ADV
ejpam-5709	184	34	it	it	PRON
ejpam-5709	184	35	follows	follow	VERB
ejpam-5709	184	36	from	from	ADP
ejpam-5709	184	37	lemma	lemma	PROPN
ejpam-5709	184	38	1	1	NUM
ejpam-5709	184	39	again	again	ADV
ejpam-5709	184	40	that	that	SCONJ
ejpam-5709	184	41	|γ1|	|γ1|	PROPN
ejpam-5709	184	42	≤	≤	NUM
ejpam-5709	184	43	|γ0|	|γ0|	NOUN
ejpam-5709	184	44	≤	≤	NOUN
ejpam-5709	184	45	n	n	ADV
ejpam-5709	184	46	.	.	PUNCT
ejpam-5709	185	1	if	if	SCONJ
ejpam-5709	185	2	|γ0|	|γ0|	NOUN
ejpam-5709	185	3	<	<	X
ejpam-5709	185	4	s	s	PROPN
ejpam-5709	185	5	,	,	PUNCT
ejpam-5709	185	6	then	then	ADV
ejpam-5709	185	7	s	s	AUX
ejpam-5709	185	8	>	>	X
ejpam-5709	185	9	|γ1β	|γ1β	PROPN
ejpam-5709	186	1	+	+	CCONJ
ejpam-5709	186	2	α1|	α1|	NOUN
ejpam-5709	186	3	≥	≥	NOUN
ejpam-5709	186	4	|γ1||β|	|γ1||β|	NUM
ejpam-5709	186	5	−	−	PROPN
ejpam-5709	186	6	|α1|	|α1|	NOUN
ejpam-5709	186	7	and	and	CCONJ
ejpam-5709	186	8	thus	thus	ADV
ejpam-5709	186	9	|γ1|	|γ1|	X
ejpam-5709	186	10	<	<	X
ejpam-5709	186	11	2|γ1|	2|γ1|	PROPN
ejpam-5709	186	12	≤	≤	NOUN
ejpam-5709	186	13	|γ1||β|	|γ1||β|	ADP
ejpam-5709	186	14	<	<	X
ejpam-5709	186	15	s	s	X
ejpam-5709	187	1	+	+	X
ejpam-5709	187	2	|α1|	|α1|	VERB
ejpam-5709	187	3	≤	≤	NOUN
ejpam-5709	187	4	s	s	PART
ejpam-5709	187	5	+	+	NUM
ejpam-5709	187	6	s	s	NOUN
ejpam-5709	187	7	=	=	SYM
ejpam-5709	187	8	2s	2s	PROPN
ejpam-5709	187	9	≤	≤	PROPN
ejpam-5709	187	10	n.	n.	NOUN
ejpam-5709	187	11	proceeding	proceeding	NOUN
ejpam-5709	187	12	in	in	ADP
ejpam-5709	187	13	the	the	DET
ejpam-5709	187	14	same	same	ADJ
ejpam-5709	187	15	manner	manner	NOUN
ejpam-5709	187	16	,	,	PUNCT
ejpam-5709	187	17	we	we	PRON
ejpam-5709	187	18	obtain	obtain	VERB
ejpam-5709	187	19	|γi|	|γi|	NOUN
ejpam-5709	187	20	≤	≤	NUM
ejpam-5709	187	21	n	n	NOUN
ejpam-5709	187	22	for	for	ADP
ejpam-5709	187	23	all	all	PRON
ejpam-5709	187	24	i	i	PRON
ejpam-5709	187	25	∈	∈	VERB
ejpam-5709	187	26	n	n	PART
ejpam-5709	187	27	∪	∪	X
ejpam-5709	187	28	{	{	PUNCT
ejpam-5709	187	29	0	0	NUM
ejpam-5709	187	30	}	}	PUNCT
ejpam-5709	187	31	.	.	PUNCT
ejpam-5709	188	1	claim	claim	NOUN
ejpam-5709	188	2	2	2	NUM
ejpam-5709	188	3	.	.	X
ejpam-5709	188	4	there	there	PRON
ejpam-5709	188	5	exist	exist	VERB
ejpam-5709	188	6	nonnegative	nonnegative	ADJ
ejpam-5709	188	7	integers	integer	NOUN
ejpam-5709	188	8	u	u	NOUN
ejpam-5709	188	9	and	and	CCONJ
ejpam-5709	188	10	v	v	NOUN
ejpam-5709	188	11	with	with	ADP
ejpam-5709	188	12	u	u	NOUN
ejpam-5709	188	13	<	<	X
ejpam-5709	188	14	v	v	ADP
ejpam-5709	188	15	such	such	ADJ
ejpam-5709	188	16	that	that	DET
ejpam-5709	188	17	γu	γu	NOUN
ejpam-5709	188	18	=	=	NOUN
ejpam-5709	188	19	γv	γv	NOUN
ejpam-5709	188	20	.	.	PUNCT
ejpam-5709	189	1	proof	proof	NOUN
ejpam-5709	189	2	of	of	ADP
ejpam-5709	189	3	claim	claim	NOUN
ejpam-5709	189	4	2	2	X
ejpam-5709	189	5	.	.	PUNCT
ejpam-5709	189	6	by	by	ADP
ejpam-5709	189	7	claim	claim	NOUN
ejpam-5709	189	8	1	1	NUM
ejpam-5709	189	9	,	,	PUNCT
ejpam-5709	189	10	we	we	PRON
ejpam-5709	189	11	have	have	VERB
ejpam-5709	189	12	|γi|	|γi|	NOUN
ejpam-5709	189	13	≤	≤	NUM
ejpam-5709	189	14	n	n	NOUN
ejpam-5709	189	15	for	for	ADP
ejpam-5709	189	16	all	all	DET
ejpam-5709	189	17	i	i	PRON
ejpam-5709	189	18	∈	∈	VERB
ejpam-5709	189	19	n	n	PART
ejpam-5709	189	20	∪	∪	X
ejpam-5709	189	21	{	{	PUNCT
ejpam-5709	189	22	0	0	NUM
ejpam-5709	189	23	}	}	PUNCT
ejpam-5709	189	24	.	.	PUNCT
ejpam-5709	190	1	for	for	ADP
ejpam-5709	190	2	each	each	DET
ejpam-5709	190	3	i	i	PRON
ejpam-5709	190	4	∈	∈	PROPN
ejpam-5709	190	5	n	n	PART
ejpam-5709	190	6	∪	∪	VERB
ejpam-5709	190	7	{	{	PUNCT
ejpam-5709	190	8	0	0	NUM
ejpam-5709	190	9	}	}	PUNCT
ejpam-5709	190	10	,	,	PUNCT
ejpam-5709	190	11	we	we	PRON
ejpam-5709	190	12	denote	denote	VERB
ejpam-5709	190	13	γi	γi	INTJ
ejpam-5709	190	14	:	:	PUNCT
ejpam-5709	190	15	=	=	VERB
ejpam-5709	190	16	ai	ai	VERB
ejpam-5709	190	17	+	+	X
ejpam-5709	190	18	biσm	biσm	ADJ
ejpam-5709	190	19	and	and	CCONJ
ejpam-5709	190	20	a	a	PRON
ejpam-5709	190	21	:	:	PUNCT
ejpam-5709	190	22	=	=	SYM
ejpam-5709	190	23	{	{	PUNCT
ejpam-5709	190	24	(	(	PUNCT
ejpam-5709	190	25	a0	a0	NOUN
ejpam-5709	190	26	,	,	PUNCT
ejpam-5709	190	27	b0	b0	NOUN
ejpam-5709	190	28	)	)	PUNCT
ejpam-5709	190	29	,	,	PUNCT
ejpam-5709	190	30	(	(	PUNCT
ejpam-5709	190	31	a1	a1	NOUN
ejpam-5709	190	32	,	,	PUNCT
ejpam-5709	190	33	b1	b1	NOUN
ejpam-5709	190	34	)	)	PUNCT
ejpam-5709	190	35	,	,	PUNCT
ejpam-5709	190	36	(	(	PUNCT
ejpam-5709	190	37	a2	a2	PROPN
ejpam-5709	190	38	,	,	PUNCT
ejpam-5709	190	39	b2	b2	NOUN
ejpam-5709	190	40	)	)	PUNCT
ejpam-5709	190	41	,	,	PUNCT
ejpam-5709	190	42	.	.	PUNCT
ejpam-5709	190	43	.	.	PUNCT
ejpam-5709	191	1	.	.	PUNCT
ejpam-5709	192	1	}	}	PUNCT
ejpam-5709	192	2	,	,	PUNCT
ejpam-5709	192	3	where	where	SCONJ
ejpam-5709	192	4	ai	ai	VERB
ejpam-5709	192	5	,	,	PUNCT
ejpam-5709	192	6	bi	bi	PROPN
ejpam-5709	192	7	∈	∈	PROPN
ejpam-5709	192	8	z	z	PROPN
ejpam-5709	192	9	for	for	ADP
ejpam-5709	192	10	all	all	PRON
ejpam-5709	192	11	i	i	PRON
ejpam-5709	192	12	∈	∈	VERB
ejpam-5709	192	13	n	n	PART
ejpam-5709	192	14	∪	∪	X
ejpam-5709	192	15	{	{	PUNCT
ejpam-5709	192	16	0	0	NUM
ejpam-5709	192	17	}	}	PUNCT
ejpam-5709	192	18	.	.	PUNCT
ejpam-5709	193	1	we	we	PRON
ejpam-5709	193	2	first	first	ADV
ejpam-5709	193	3	show	show	VERB
ejpam-5709	193	4	that	that	SCONJ
ejpam-5709	193	5	a	a	PRON
ejpam-5709	193	6	is	be	AUX
ejpam-5709	193	7	a	a	DET
ejpam-5709	193	8	finite	finite	NOUN
ejpam-5709	193	9	set	set	VERB
ejpam-5709	193	10	by	by	ADP
ejpam-5709	193	11	considering	consider	VERB
ejpam-5709	193	12	the	the	DET
ejpam-5709	193	13	following	follow	VERB
ejpam-5709	193	14	two	two	NUM
ejpam-5709	193	15	cases	case	NOUN
ejpam-5709	193	16	:	:	PUNCT
ejpam-5709	193	17	case	case	NOUN
ejpam-5709	193	18	1	1	NUM
ejpam-5709	193	19	.	.	PUNCT
ejpam-5709	194	1	m	m	VERB
ejpam-5709	194	2	̸≡	̸≡	PROPN
ejpam-5709	194	3	1	1	NUM
ejpam-5709	194	4	(	(	PUNCT
ejpam-5709	194	5	mod	mod	NOUN
ejpam-5709	194	6	4	4	NUM
ejpam-5709	194	7	)	)	PUNCT
ejpam-5709	194	8	.	.	PUNCT
ejpam-5709	195	1	then	then	ADV
ejpam-5709	195	2	a2i	a2i	X
ejpam-5709	195	3	−mb2i	−mb2i	X
ejpam-5709	195	4	=	=	SYM
ejpam-5709	195	5	n	n	PROPN
ejpam-5709	195	6	(	(	PUNCT
ejpam-5709	195	7	γi	γi	NOUN
ejpam-5709	195	8	)	)	PUNCT
ejpam-5709	195	9	=	=	SYM
ejpam-5709	195	10	|γi|2	|γi|2	PROPN
ejpam-5709	195	11	≤	≤	NOUN
ejpam-5709	195	12	n2	n2	NOUN
ejpam-5709	195	13	for	for	ADP
ejpam-5709	195	14	all	all	DET
ejpam-5709	195	15	i	i	PRON
ejpam-5709	195	16	∈	∈	VERB
ejpam-5709	195	17	n	n	PART
ejpam-5709	195	18	∪	∪	X
ejpam-5709	195	19	{	{	PUNCT
ejpam-5709	195	20	0	0	NUM
ejpam-5709	195	21	}	}	PUNCT
ejpam-5709	195	22	.	.	PUNCT
ejpam-5709	196	1	thus	thus	ADV
ejpam-5709	196	2	a2i	a2i	X
ejpam-5709	196	3	≤	≤	NUM
ejpam-5709	196	4	n2	n2	NOUN
ejpam-5709	196	5	and	and	CCONJ
ejpam-5709	196	6	b2i	b2i	NOUN
ejpam-5709	196	7	≤	≤	ADJ
ejpam-5709	196	8	n2	n2	NOUN
ejpam-5709	196	9	and	and	CCONJ
ejpam-5709	196	10	so	so	ADV
ejpam-5709	196	11	|ai|	|ai|	ADJ
ejpam-5709	196	12	≤	≤	ADJ
ejpam-5709	196	13	n	n	ADP
ejpam-5709	196	14	and	and	CCONJ
ejpam-5709	196	15	|bi|	|bi|	ADJ
ejpam-5709	196	16	≤	≤	NOUN
ejpam-5709	196	17	n	n	NOUN
ejpam-5709	196	18	.	.	PUNCT
ejpam-5709	197	1	it	it	PRON
ejpam-5709	197	2	follows	follow	VERB
ejpam-5709	197	3	that	that	SCONJ
ejpam-5709	197	4	the	the	DET
ejpam-5709	197	5	set	set	NOUN
ejpam-5709	197	6	a	a	PRON
ejpam-5709	197	7	is	be	AUX
ejpam-5709	197	8	finite	finite	ADJ
ejpam-5709	197	9	.	.	PUNCT
ejpam-5709	197	10	case	case	NOUN
ejpam-5709	197	11	2	2	NUM
ejpam-5709	197	12	.	.	PUNCT
ejpam-5709	197	13	m	m	VERB
ejpam-5709	197	14	≡	≡	PROPN
ejpam-5709	197	15	1	1	NUM
ejpam-5709	197	16	(	(	PUNCT
ejpam-5709	197	17	mod	mod	NOUN
ejpam-5709	197	18	4	4	NUM
ejpam-5709	197	19	)	)	PUNCT
ejpam-5709	197	20	.	.	PUNCT
ejpam-5709	198	1	then	then	ADV
ejpam-5709	198	2	a2i	a2i	VERB
ejpam-5709	198	3	+	+	CCONJ
ejpam-5709	198	4	aibi	aibi	NOUN
ejpam-5709	198	5	+	+	CCONJ
ejpam-5709	198	6	b2i	b2i	CCONJ
ejpam-5709	198	7	(	(	PUNCT
ejpam-5709	198	8	1	1	NUM
ejpam-5709	198	9	−	−	PRON
ejpam-5709	198	10	m)/4	m)/4	PROPN
ejpam-5709	198	11	=	=	SYM
ejpam-5709	199	1	n	n	CCONJ
ejpam-5709	199	2	(	(	PUNCT
ejpam-5709	199	3	γi	γi	NOUN
ejpam-5709	199	4	)	)	PUNCT
ejpam-5709	199	5	=	=	SYM
ejpam-5709	199	6	|γi|2	|γi|2	PROPN
ejpam-5709	199	7	≤	≤	NOUN
ejpam-5709	199	8	n2	n2	NOUN
ejpam-5709	199	9	for	for	ADP
ejpam-5709	199	10	all	all	DET
ejpam-5709	199	11	i	i	PRON
ejpam-5709	199	12	∈	∈	PROPN
ejpam-5709	199	13	n∪{0	n∪{0	NOUN
ejpam-5709	199	14	}	}	PUNCT
ejpam-5709	199	15	,	,	PUNCT
ejpam-5709	199	16	so	so	CCONJ
ejpam-5709	199	17	(	(	PUNCT
ejpam-5709	199	18	2ai	2ai	ADJ
ejpam-5709	199	19	+	+	CCONJ
ejpam-5709	199	20	bi	bi	NOUN
ejpam-5709	199	21	)	)	PUNCT
ejpam-5709	199	22	2	2	NUM
ejpam-5709	199	23	−mb2i	−mb2i	NOUN
ejpam-5709	199	24	=	=	SYM
ejpam-5709	199	25	4a2i	4a2i	NOUN
ejpam-5709	200	1	+4aibi	+4aibi	PROPN
ejpam-5709	200	2	+	+	CCONJ
ejpam-5709	200	3	b2i	b2i	X
ejpam-5709	200	4	−mb2i	−mb2i	X
ejpam-5709	201	1	≤	≤	ADJ
ejpam-5709	201	2	4n2	4n2	NUM
ejpam-5709	201	3	.	.	PUNCT
ejpam-5709	202	1	it	it	PRON
ejpam-5709	202	2	follows	follow	VERB
ejpam-5709	202	3	that	that	SCONJ
ejpam-5709	202	4	|bi|	|bi|	ADJ
ejpam-5709	202	5	≤	≤	NUM
ejpam-5709	202	6	2n	2n	NUM
ejpam-5709	202	7	and	and	CCONJ
ejpam-5709	202	8	|2ai	|2ai	NOUN
ejpam-5709	202	9	+	+	CCONJ
ejpam-5709	202	10	bi|	bi|	NOUN
ejpam-5709	202	11	≤	≤	NOUN
ejpam-5709	202	12	2n	2n	NUM
ejpam-5709	202	13	and	and	CCONJ
ejpam-5709	202	14	thus	thus	ADV
ejpam-5709	202	15	|ai|	|ai|	ADJ
ejpam-5709	202	16	≤	≤	ADJ
ejpam-5709	202	17	|2ai|	|2ai|	ADJ
ejpam-5709	202	18	≤	≤	NOUN
ejpam-5709	202	19	|2ai	|2ai	NOUN
ejpam-5709	202	20	+	+	CCONJ
ejpam-5709	203	1	bi|	bi|	NOUN
ejpam-5709	203	2	+	+	CCONJ
ejpam-5709	203	3	|bi|	|bi|	ADJ
ejpam-5709	203	4	≤	≤	NUM
ejpam-5709	203	5	2n	2n	NUM
ejpam-5709	204	1	+	+	CCONJ
ejpam-5709	204	2	2n	2n	NUM
ejpam-5709	204	3	=	=	SYM
ejpam-5709	204	4	4n	4n	NOUN
ejpam-5709	204	5	,	,	PUNCT
ejpam-5709	204	6	implying	imply	VERB
ejpam-5709	204	7	the	the	DET
ejpam-5709	204	8	set	set	NOUN
ejpam-5709	204	9	a	a	PRON
ejpam-5709	204	10	is	be	AUX
ejpam-5709	204	11	finite	finite	ADJ
ejpam-5709	204	12	.	.	PUNCT
ejpam-5709	205	1	consequently	consequently	ADV
ejpam-5709	205	2	,	,	PUNCT
ejpam-5709	205	3	(	(	PUNCT
ejpam-5709	205	4	au	au	ADP
ejpam-5709	205	5	,	,	PUNCT
ejpam-5709	205	6	bu	bu	ADJ
ejpam-5709	205	7	)	)	PUNCT
ejpam-5709	205	8	=	=	SYM
ejpam-5709	205	9	(	(	PUNCT
ejpam-5709	205	10	av	av	PROPN
ejpam-5709	205	11	,	,	PUNCT
ejpam-5709	205	12	bv	bv	PROPN
ejpam-5709	205	13	)	)	PUNCT
ejpam-5709	205	14	for	for	ADP
ejpam-5709	205	15	some	some	DET
ejpam-5709	205	16	nonnegative	nonnegative	ADJ
ejpam-5709	205	17	integers	integer	NOUN
ejpam-5709	205	18	u	u	NOUN
ejpam-5709	205	19	and	and	CCONJ
ejpam-5709	205	20	v	v	NOUN
ejpam-5709	205	21	with	with	ADP
ejpam-5709	205	22	u	u	PRON
ejpam-5709	205	23	<	<	X
ejpam-5709	205	24	v.	v.	ADP
ejpam-5709	205	25	this	this	PRON
ejpam-5709	205	26	implies	imply	VERB
ejpam-5709	205	27	that	that	PRON
ejpam-5709	205	28	γu	γu	ADV
ejpam-5709	205	29	=	=	SYM
ejpam-5709	205	30	au	au	PROPN
ejpam-5709	205	31	+	+	PUNCT
ejpam-5709	205	32	buσm	buσm	NOUN
ejpam-5709	205	33	=	=	SYM
ejpam-5709	205	34	av	av	PROPN
ejpam-5709	205	35	+	+	CCONJ
ejpam-5709	205	36	bvσm	bvσm	NOUN
ejpam-5709	205	37	=	=	SYM
ejpam-5709	205	38	γv	γv	NOUN
ejpam-5709	205	39	.	.	PUNCT
ejpam-5709	206	1	next	next	ADV
ejpam-5709	206	2	,	,	PUNCT
ejpam-5709	206	3	we	we	PRON
ejpam-5709	206	4	show	show	VERB
ejpam-5709	206	5	that	that	SCONJ
ejpam-5709	206	6	the	the	DET
ejpam-5709	206	7	base	base	NOUN
ejpam-5709	206	8	-	-	PUNCT
ejpam-5709	206	9	β(c	β(c	NUM
ejpam-5709	206	10	)	)	PUNCT
ejpam-5709	206	11	representations	representation	NOUN
ejpam-5709	206	12	of	of	ADP
ejpam-5709	206	13	η	η	PROPN
ejpam-5709	206	14	are	be	AUX
ejpam-5709	206	15	the	the	DET
ejpam-5709	206	16	equations	equation	NOUN
ejpam-5709	206	17	in	in	ADP
ejpam-5709	206	18	(	(	PUNCT
ejpam-5709	206	19	9	9	NUM
ejpam-5709	206	20	)	)	PUNCT
ejpam-5709	206	21	,	,	PUNCT
ejpam-5709	206	22	(	(	PUNCT
ejpam-5709	206	23	10	10	NUM
ejpam-5709	206	24	)	)	PUNCT
ejpam-5709	206	25	,	,	PUNCT
ejpam-5709	206	26	(	(	PUNCT
ejpam-5709	206	27	11	11	NUM
ejpam-5709	206	28	)	)	PUNCT
ejpam-5709	206	29	,	,	PUNCT
ejpam-5709	206	30	and	and	CCONJ
ejpam-5709	206	31	(	(	PUNCT
ejpam-5709	206	32	12	12	NUM
ejpam-5709	206	33	)	)	PUNCT
ejpam-5709	206	34	.	.	PUNCT
ejpam-5709	207	1	recall	recall	VERB
ejpam-5709	207	2	that	that	PRON
ejpam-5709	207	3	η	η	NOUN
ejpam-5709	207	4	=	=	PROPN
ejpam-5709	207	5	γ0β	γ0β	PROPN
ejpam-5709	207	6	+	+	CCONJ
ejpam-5709	207	7	α0	α0	ADJ
ejpam-5709	207	8	and	and	CCONJ
ejpam-5709	207	9	γi−1	γi−1	PROPN
ejpam-5709	207	10	=	=	PUNCT
ejpam-5709	207	11	γiβ	γiβ	PROPN
ejpam-5709	208	1	+	+	CCONJ
ejpam-5709	208	2	αi	αi	VERB
ejpam-5709	208	3	(	(	PUNCT
ejpam-5709	208	4	i	i	PRON
ejpam-5709	208	5	≥	≥	VERB
ejpam-5709	208	6	1	1	NUM
ejpam-5709	208	7	)	)	PUNCT
ejpam-5709	208	8	,	,	PUNCT
ejpam-5709	208	9	(	(	PUNCT
ejpam-5709	208	10	13	13	NUM
ejpam-5709	208	11	)	)	PUNCT
ejpam-5709	208	12	where	where	SCONJ
ejpam-5709	208	13	γi	γi	X
ejpam-5709	208	14	∈	∈	PROPN
ejpam-5709	208	15	ok\{0	ok\{0	PROPN
ejpam-5709	208	16	}	}	PUNCT
ejpam-5709	208	17	and	and	CCONJ
ejpam-5709	208	18	αi	αi	PRON
ejpam-5709	208	19	∈	∈	PROPN
ejpam-5709	208	20	c	c	PROPN
ejpam-5709	208	21	for	for	ADP
ejpam-5709	208	22	all	all	DET
ejpam-5709	208	23	i	i	PRON
ejpam-5709	208	24	≥	≥	VERB
ejpam-5709	208	25	0	0	NUM
ejpam-5709	208	26	.	.	PUNCT
ejpam-5709	209	1	then	then	ADV
ejpam-5709	209	2	we	we	PRON
ejpam-5709	209	3	obtain	obtain	VERB
ejpam-5709	209	4	η	η	NOUN
ejpam-5709	209	5	=	=	PROPN
ejpam-5709	209	6	γkβ	γkβ	NOUN
ejpam-5709	209	7	k+1	k+1	X
ejpam-5709	210	1	+	+	CCONJ
ejpam-5709	210	2	αkβ	αkβ	NOUN
ejpam-5709	210	3	k	k	X
ejpam-5709	211	1	+	+	CCONJ
ejpam-5709	211	2	αk−1β	αk−1β	PROPN
ejpam-5709	211	3	k−1	k−1	PROPN
ejpam-5709	211	4	+	+	CCONJ
ejpam-5709	211	5	·	·	PUNCT
ejpam-5709	211	6	·	·	PUNCT
ejpam-5709	211	7	·	·	PUNCT
ejpam-5709	211	8	+	+	NUM
ejpam-5709	211	9	α1β	α1β	X
ejpam-5709	211	10	+	+	CCONJ
ejpam-5709	211	11	α0	α0	ADJ
ejpam-5709	211	12	(	(	PUNCT
ejpam-5709	211	13	0	0	NUM
ejpam-5709	211	14	≤	≤	NUM
ejpam-5709	211	15	k	k	X
ejpam-5709	211	16	≤	≤	NUM
ejpam-5709	211	17	u	u	NOUN
ejpam-5709	211	18	)	)	PUNCT
ejpam-5709	211	19	,	,	PUNCT
ejpam-5709	211	20	(	(	PUNCT
ejpam-5709	211	21	14	14	NUM
ejpam-5709	211	22	)	)	PUNCT
ejpam-5709	211	23	p.	p.	NOUN
ejpam-5709	211	24	phetnun	phetnun	PROPN
ejpam-5709	211	25	,	,	PUNCT
ejpam-5709	211	26	n.r	n.r	PROPN
ejpam-5709	211	27	.	.	PROPN
ejpam-5709	211	28	kanasri	kanasri	PROPN
ejpam-5709	211	29	/	/	SYM
ejpam-5709	211	30	eur	eur	PROPN
ejpam-5709	211	31	.	.	PUNCT
ejpam-5709	212	1	j.	j.	PROPN
ejpam-5709	212	2	pure	pure	PROPN
ejpam-5709	212	3	appl	appl	PROPN
ejpam-5709	212	4	.	.	PROPN
ejpam-5709	212	5	math	math	PROPN
ejpam-5709	212	6	,	,	PUNCT
ejpam-5709	212	7	18	18	NUM
ejpam-5709	212	8	(	(	PUNCT
ejpam-5709	212	9	1	1	NUM
ejpam-5709	212	10	)	)	PUNCT
ejpam-5709	212	11	(	(	PUNCT
ejpam-5709	212	12	2025	2025	NUM
ejpam-5709	212	13	)	)	PUNCT
ejpam-5709	212	14	,	,	PUNCT
ejpam-5709	212	15	5709	5709	NUM
ejpam-5709	212	16	8	8	NUM
ejpam-5709	212	17	of	of	ADP
ejpam-5709	212	18	15	15	NUM
ejpam-5709	212	19	which	which	PRON
ejpam-5709	212	20	are	be	AUX
ejpam-5709	212	21	base	base	NOUN
ejpam-5709	212	22	-	-	PUNCT
ejpam-5709	212	23	β(c	β(c	NUM
ejpam-5709	212	24	)	)	PUNCT
ejpam-5709	212	25	representations	representation	NOUN
ejpam-5709	212	26	in	in	ADP
ejpam-5709	212	27	(	(	PUNCT
ejpam-5709	212	28	9	9	NUM
ejpam-5709	212	29	)	)	PUNCT
ejpam-5709	212	30	.	.	PUNCT
ejpam-5709	213	1	again	again	ADV
ejpam-5709	213	2	,	,	PUNCT
ejpam-5709	213	3	using	use	VERB
ejpam-5709	213	4	(	(	PUNCT
ejpam-5709	213	5	13	13	NUM
ejpam-5709	213	6	)	)	PUNCT
ejpam-5709	213	7	repeatedly	repeatedly	ADV
ejpam-5709	213	8	,	,	PUNCT
ejpam-5709	213	9	we	we	PRON
ejpam-5709	213	10	get	get	VERB
ejpam-5709	213	11	η	η	NOUN
ejpam-5709	213	12	=	=	PROPN
ejpam-5709	213	13	γkβ	γkβ	NOUN
ejpam-5709	214	1	k+1	k+1	X
ejpam-5709	215	1	+	+	CCONJ
ejpam-5709	216	1	αkβ	αkβ	NOUN
ejpam-5709	216	2	k	k	X
ejpam-5709	216	3	+	+	CCONJ
ejpam-5709	216	4	·	·	PUNCT
ejpam-5709	216	5	·	·	PUNCT
ejpam-5709	216	6	·	·	PUNCT
ejpam-5709	216	7	+	+	NUM
ejpam-5709	216	8	αu+1β	αu+1β	NUM
ejpam-5709	216	9	u+1	u+1	ADJ
ejpam-5709	217	1	+	+	CCONJ
ejpam-5709	217	2	αuβ	αuβ	NOUN
ejpam-5709	217	3	u	u	NOUN
ejpam-5709	217	4	+	+	PROPN
ejpam-5709	217	5	·	·	PUNCT
ejpam-5709	217	6	·	·	PUNCT
ejpam-5709	217	7	·	·	PUNCT
ejpam-5709	217	8	+	+	NUM
ejpam-5709	217	9	α1β	α1β	X
ejpam-5709	218	1	+	+	CCONJ
ejpam-5709	218	2	α0	α0	ADJ
ejpam-5709	218	3	(	(	PUNCT
ejpam-5709	218	4	u	u	NOUN
ejpam-5709	218	5	<	<	X
ejpam-5709	218	6	k	k	X
ejpam-5709	218	7	<	<	X
ejpam-5709	218	8	v	v	NOUN
ejpam-5709	218	9	)	)	PUNCT
ejpam-5709	218	10	,	,	PUNCT
ejpam-5709	218	11	which	which	PRON
ejpam-5709	218	12	are	be	AUX
ejpam-5709	218	13	base	base	NOUN
ejpam-5709	218	14	-	-	PUNCT
ejpam-5709	218	15	β(c	β(c	NUM
ejpam-5709	218	16	)	)	PUNCT
ejpam-5709	218	17	representations	representation	NOUN
ejpam-5709	218	18	in	in	ADP
ejpam-5709	218	19	(	(	PUNCT
ejpam-5709	218	20	10	10	NUM
ejpam-5709	218	21	)	)	PUNCT
ejpam-5709	218	22	.	.	PUNCT
ejpam-5709	219	1	since	since	SCONJ
ejpam-5709	219	2	γu	γu	NOUN
ejpam-5709	219	3	=	=	SYM
ejpam-5709	219	4	γv	γv	PROPN
ejpam-5709	219	5	,	,	PUNCT
ejpam-5709	219	6	we	we	PRON
ejpam-5709	219	7	obtain	obtain	VERB
ejpam-5709	219	8	from	from	ADP
ejpam-5709	219	9	(	(	PUNCT
ejpam-5709	219	10	13	13	NUM
ejpam-5709	219	11	)	)	PUNCT
ejpam-5709	219	12	that	that	PRON
ejpam-5709	219	13	γu	γu	NOUN
ejpam-5709	219	14	=	=	PRON
ejpam-5709	219	15	γu+1β	γu+1β	PROPN
ejpam-5709	219	16	+	+	CCONJ
ejpam-5709	219	17	αu+1	αu+1	ADJ
ejpam-5709	219	18	=	=	SYM
ejpam-5709	219	19	γu+2β	γu+2β	NOUN
ejpam-5709	219	20	2	2	NUM
ejpam-5709	219	21	+	+	NUM
ejpam-5709	219	22	αu+2β	αu+2β	NOUN
ejpam-5709	220	1	+	+	CCONJ
ejpam-5709	220	2	αu+1	αu+1	NOUN
ejpam-5709	220	3	...	...	PUNCT
ejpam-5709	221	1	=	=	SYM
ejpam-5709	221	2	γuβ	γuβ	NOUN
ejpam-5709	221	3	v−u	v−u	NOUN
ejpam-5709	221	4	+	+	CCONJ
ejpam-5709	221	5	αvβ	αvβ	ADJ
ejpam-5709	222	1	v−u−1	v−u−1	PROPN
ejpam-5709	222	2	+	+	CCONJ
ejpam-5709	222	3	αv−1β	αv−1β	NUM
ejpam-5709	222	4	v−u−2	v−u−2	NOUN
ejpam-5709	222	5	+	+	X
ejpam-5709	222	6	·	·	PUNCT
ejpam-5709	222	7	·	·	PUNCT
ejpam-5709	222	8	·	·	PUNCT
ejpam-5709	222	9	+	+	NUM
ejpam-5709	222	10	αu+2β	αu+2β	NOUN
ejpam-5709	222	11	+	+	CCONJ
ejpam-5709	222	12	αu+1	αu+1	X
ejpam-5709	222	13	.	.	PUNCT
ejpam-5709	222	14	(	(	PUNCT
ejpam-5709	222	15	15	15	NUM
ejpam-5709	222	16	)	)	PUNCT
ejpam-5709	222	17	substituting	substituting	NOUN
ejpam-5709	222	18	(	(	PUNCT
ejpam-5709	222	19	15	15	NUM
ejpam-5709	222	20	)	)	PUNCT
ejpam-5709	222	21	into	into	ADP
ejpam-5709	222	22	(	(	PUNCT
ejpam-5709	222	23	14	14	NUM
ejpam-5709	222	24	)	)	PUNCT
ejpam-5709	222	25	with	with	ADP
ejpam-5709	222	26	k	k	PROPN
ejpam-5709	222	27	=	=	PUNCT
ejpam-5709	222	28	u	u	PROPN
ejpam-5709	222	29	leads	lead	VERB
ejpam-5709	222	30	to	to	ADP
ejpam-5709	222	31	η	η	PROPN
ejpam-5709	222	32	=	=	PROPN
ejpam-5709	222	33	γuβ	γuβ	PROPN
ejpam-5709	222	34	v+1+αvβ	v+1+αvβ	X
ejpam-5709	222	35	v+αv−1β	v+αv−1β	PROPN
ejpam-5709	223	1	v−1	v−1	PROPN
ejpam-5709	223	2	+	+	PROPN
ejpam-5709	223	3	·	·	PUNCT
ejpam-5709	223	4	·	·	PUNCT
ejpam-5709	223	5	·	·	PUNCT
ejpam-5709	223	6	+	+	NOUN
ejpam-5709	223	7	αu+1β	αu+1β	NUM
ejpam-5709	223	8	u+1+αuβ	u+1+αuβ	ADP
ejpam-5709	223	9	u+αu−1β	u+αu−1β	PUNCT
ejpam-5709	224	1	u−1	u−1	PROPN
ejpam-5709	224	2	+	+	PROPN
ejpam-5709	224	3	·	·	PUNCT
ejpam-5709	224	4	·	·	PUNCT
ejpam-5709	224	5	·	·	PUNCT
ejpam-5709	224	6	+	+	NOUN
ejpam-5709	224	7	α1β+α0	α1β+α0	NOUN
ejpam-5709	224	8	.	.	PUNCT
ejpam-5709	225	1	(	(	PUNCT
ejpam-5709	225	2	16	16	NUM
ejpam-5709	225	3	)	)	PUNCT
ejpam-5709	225	4	again	again	ADV
ejpam-5709	225	5	,	,	PUNCT
ejpam-5709	225	6	substituting	substitute	VERB
ejpam-5709	225	7	(	(	PUNCT
ejpam-5709	225	8	15	15	NUM
ejpam-5709	225	9	)	)	PUNCT
ejpam-5709	225	10	into	into	ADP
ejpam-5709	225	11	(	(	PUNCT
ejpam-5709	225	12	16	16	NUM
ejpam-5709	225	13	)	)	PUNCT
ejpam-5709	225	14	yields	yield	VERB
ejpam-5709	225	15	η	η	PROPN
ejpam-5709	225	16	=	=	PROPN
ejpam-5709	225	17	γuβ	γuβ	PROPN
ejpam-5709	225	18	2v−u+1	2v−u+1	NUM
ejpam-5709	225	19	+	+	CCONJ
ejpam-5709	225	20	αvβ	αvβ	ADJ
ejpam-5709	225	21	2v−u	2v−u	NOUN
ejpam-5709	225	22	+	+	CCONJ
ejpam-5709	226	1	αv−1β	αv−1β	NUM
ejpam-5709	226	2	2v−u−1	2v−u−1	NUM
ejpam-5709	226	3	+	+	CCONJ
ejpam-5709	226	4	·	·	PUNCT
ejpam-5709	226	5	·	·	PUNCT
ejpam-5709	226	6	·	·	PUNCT
ejpam-5709	226	7	+	+	NUM
ejpam-5709	226	8	αu+1β	αu+1β	NUM
ejpam-5709	226	9	v+1	v+1	PROPN
ejpam-5709	226	10	+	+	CCONJ
ejpam-5709	226	11	αvβ	αvβ	ADJ
ejpam-5709	226	12	v	v	ADP
ejpam-5709	226	13	+	+	NOUN
ejpam-5709	226	14	αv−1β	αv−1β	NUM
ejpam-5709	226	15	v−1	v−1	PROPN
ejpam-5709	226	16	+	+	CCONJ
ejpam-5709	226	17	·	·	PUNCT
ejpam-5709	226	18	·	·	PUNCT
ejpam-5709	226	19	·	·	PUNCT
ejpam-5709	227	1	+	+	NUM
ejpam-5709	227	2	αu+1β	αu+1β	NUM
ejpam-5709	227	3	u+1	u+1	ADJ
ejpam-5709	227	4	+	+	CCONJ
ejpam-5709	227	5	αuβ	αuβ	NOUN
ejpam-5709	227	6	u	u	NOUN
ejpam-5709	227	7	+	+	CCONJ
ejpam-5709	227	8	αu−1β	αu−1β	NUM
ejpam-5709	227	9	u−1	u−1	PROPN
ejpam-5709	227	10	+	+	CCONJ
ejpam-5709	227	11	·	·	PUNCT
ejpam-5709	227	12	·	·	PUNCT
ejpam-5709	227	13	·	·	PUNCT
ejpam-5709	227	14	+	+	NUM
ejpam-5709	227	15	α1β	α1β	X
ejpam-5709	227	16	+	+	CCONJ
ejpam-5709	227	17	α0	α0	ADJ
ejpam-5709	227	18	.	.	PUNCT
ejpam-5709	228	1	(	(	PUNCT
ejpam-5709	228	2	17	17	NUM
ejpam-5709	228	3	)	)	PUNCT
ejpam-5709	228	4	continuing	continue	VERB
ejpam-5709	228	5	this	this	DET
ejpam-5709	228	6	process	process	NOUN
ejpam-5709	228	7	,	,	PUNCT
ejpam-5709	228	8	in	in	ADP
ejpam-5709	228	9	general	general	ADJ
ejpam-5709	228	10	,	,	PUNCT
ejpam-5709	228	11	we	we	PRON
ejpam-5709	228	12	have	have	VERB
ejpam-5709	228	13	η	η	NOUN
ejpam-5709	228	14	=	=	PROPN
ejpam-5709	228	15	γuβ	γuβ	PROPN
ejpam-5709	228	16	(	(	PUNCT
ejpam-5709	228	17	v−u)n+u+1	v−u)n+u+1	PROPN
ejpam-5709	228	18	+	+	CCONJ
ejpam-5709	228	19	αvβ	αvβ	ADJ
ejpam-5709	228	20	(	(	PUNCT
ejpam-5709	228	21	v−u)n+u	v−u)n+u	NOUN
ejpam-5709	228	22	+	+	CCONJ
ejpam-5709	228	23	αv−1β	αv−1β	NUM
ejpam-5709	228	24	(	(	PUNCT
ejpam-5709	228	25	v−u)n+u−1	v−u)n+u−1	PROPN
ejpam-5709	228	26	+	+	CCONJ
ejpam-5709	228	27	·	·	PUNCT
ejpam-5709	228	28	·	·	PUNCT
ejpam-5709	228	29	·	·	PUNCT
ejpam-5709	229	1	+	+	NUM
ejpam-5709	229	2	αu+1β	αu+1β	NUM
ejpam-5709	229	3	(	(	PUNCT
ejpam-5709	229	4	v−u)n+2u−v+1	v−u)n+2u−v+1	NOUN
ejpam-5709	229	5	+	+	X
ejpam-5709	229	6	·	·	PUNCT
ejpam-5709	229	7	·	·	PUNCT
ejpam-5709	229	8	·	·	PUNCT
ejpam-5709	229	9	+	+	CCONJ
ejpam-5709	229	10	∑	∑	PUNCT
ejpam-5709	229	11	u	u	NOUN
ejpam-5709	229	12	<	<	X
ejpam-5709	229	13	j≤v	j≤v	PROPN
ejpam-5709	229	14	αjβ	αjβ	PROPN
ejpam-5709	229	15	j	j	PROPN
ejpam-5709	229	16	+	+	CCONJ
ejpam-5709	229	17	∑	∑	PROPN
ejpam-5709	229	18	0≤i≤u	0≤i≤u	NUM
ejpam-5709	229	19	αiβ	αiβ	PROPN
ejpam-5709	229	20	i	i	PRON
ejpam-5709	229	21	(	(	PUNCT
ejpam-5709	229	22	n	n	NOUN
ejpam-5709	229	23	∈	∈	PROPN
ejpam-5709	229	24	n	n	CCONJ
ejpam-5709	229	25	)	)	PUNCT
ejpam-5709	229	26	,	,	PUNCT
ejpam-5709	229	27	(	(	PUNCT
ejpam-5709	229	28	18	18	X
ejpam-5709	229	29	)	)	PUNCT
ejpam-5709	229	30	yielding	yield	VERB
ejpam-5709	229	31	base	base	NOUN
ejpam-5709	229	32	-	-	PUNCT
ejpam-5709	229	33	β(c	β(c	NUM
ejpam-5709	229	34	)	)	PUNCT
ejpam-5709	229	35	representations	representation	NOUN
ejpam-5709	229	36	in	in	ADP
ejpam-5709	229	37	(	(	PUNCT
ejpam-5709	229	38	11	11	NUM
ejpam-5709	229	39	)	)	PUNCT
ejpam-5709	229	40	.	.	PUNCT
ejpam-5709	230	1	using	use	VERB
ejpam-5709	230	2	(	(	PUNCT
ejpam-5709	230	3	13	13	NUM
ejpam-5709	230	4	)	)	PUNCT
ejpam-5709	230	5	successively	successively	ADV
ejpam-5709	230	6	and	and	CCONJ
ejpam-5709	230	7	(	(	PUNCT
ejpam-5709	230	8	18	18	NUM
ejpam-5709	230	9	)	)	PUNCT
ejpam-5709	230	10	,	,	PUNCT
ejpam-5709	230	11	we	we	PRON
ejpam-5709	230	12	obtain	obtain	VERB
ejpam-5709	230	13	the	the	DET
ejpam-5709	230	14	base	base	NOUN
ejpam-5709	230	15	-	-	PUNCT
ejpam-5709	230	16	β(c	β(c	NUM
ejpam-5709	230	17	)	)	PUNCT
ejpam-5709	230	18	representations	representation	NOUN
ejpam-5709	230	19	in	in	ADP
ejpam-5709	230	20	(	(	PUNCT
ejpam-5709	230	21	12	12	NUM
ejpam-5709	230	22	)	)	PUNCT
ejpam-5709	230	23	.	.	PUNCT
ejpam-5709	231	1	finally	finally	ADV
ejpam-5709	231	2	,	,	PUNCT
ejpam-5709	231	3	we	we	PRON
ejpam-5709	231	4	let	let	VERB
ejpam-5709	231	5	η	η	PROPN
ejpam-5709	231	6	=	=	PROPN
ejpam-5709	231	7	δlβ	δlβ	PROPN
ejpam-5709	231	8	l	l	PROPN
ejpam-5709	231	9	+	+	X
ejpam-5709	231	10	·	·	PUNCT
ejpam-5709	231	11	·	·	PUNCT
ejpam-5709	231	12	·	·	PUNCT
ejpam-5709	232	1	+	+	CCONJ
ejpam-5709	232	2	δ1β	δ1β	X
ejpam-5709	232	3	+	+	PUNCT
ejpam-5709	232	4	δ0	δ0	NOUN
ejpam-5709	232	5	be	be	AUX
ejpam-5709	232	6	any	any	DET
ejpam-5709	232	7	base	base	NOUN
ejpam-5709	232	8	-	-	PUNCT
ejpam-5709	232	9	β(c	β(c	NUM
ejpam-5709	232	10	)	)	PUNCT
ejpam-5709	232	11	representation	representation	NOUN
ejpam-5709	232	12	of	of	ADP
ejpam-5709	232	13	η	η	PROPN
ejpam-5709	232	14	,	,	PUNCT
ejpam-5709	232	15	where	where	SCONJ
ejpam-5709	232	16	l	l	PROPN
ejpam-5709	232	17	∈	∈	PROPN
ejpam-5709	232	18	n	n	CCONJ
ejpam-5709	232	19	,	,	PUNCT
ejpam-5709	232	20	δl	δl	PROPN
ejpam-5709	232	21	∈	∈	PROPN
ejpam-5709	232	22	ok\{0	ok\{0	PROPN
ejpam-5709	232	23	}	}	PUNCT
ejpam-5709	232	24	,	,	PUNCT
ejpam-5709	232	25	and	and	CCONJ
ejpam-5709	232	26	δi	δi	ADP
ejpam-5709	232	27	∈	∈	PROPN
ejpam-5709	232	28	c	c	NOUN
ejpam-5709	232	29	for	for	ADP
ejpam-5709	232	30	all	all	PRON
ejpam-5709	232	31	i	i	PRON
ejpam-5709	232	32	∈	∈	PROPN
ejpam-5709	232	33	{	{	PUNCT
ejpam-5709	232	34	0	0	NUM
ejpam-5709	232	35	,	,	PUNCT
ejpam-5709	232	36	1	1	NUM
ejpam-5709	232	37	,	,	PUNCT
ejpam-5709	232	38	.	.	PUNCT
ejpam-5709	232	39	.	.	PUNCT
ejpam-5709	232	40	.	.	PUNCT
ejpam-5709	233	1	,	,	PUNCT
ejpam-5709	234	1	l	l	NOUN
ejpam-5709	234	2	−	−	NOUN
ejpam-5709	234	3	1	1	NUM
ejpam-5709	234	4	}	}	PUNCT
ejpam-5709	234	5	.	.	PUNCT
ejpam-5709	235	1	then	then	ADV
ejpam-5709	235	2	η	η	X
ejpam-5709	235	3	=	=	SYM
ejpam-5709	235	4	γl−1β	γl−1β	PROPN
ejpam-5709	235	5	l	l	NOUN
ejpam-5709	235	6	+	+	CCONJ
ejpam-5709	235	7	αl−1β	αl−1β	NUM
ejpam-5709	235	8	l−1	l−1	PROPN
ejpam-5709	235	9	+	+	CCONJ
ejpam-5709	235	10	·	·	PUNCT
ejpam-5709	235	11	·	·	PUNCT
ejpam-5709	235	12	·	·	PUNCT
ejpam-5709	235	13	+	+	NUM
ejpam-5709	235	14	α1β	α1β	X
ejpam-5709	236	1	+	+	CCONJ
ejpam-5709	236	2	α0	α0	ADJ
ejpam-5709	236	3	is	be	AUX
ejpam-5709	236	4	one	one	NUM
ejpam-5709	236	5	of	of	ADP
ejpam-5709	236	6	the	the	DET
ejpam-5709	236	7	equations	equation	NOUN
ejpam-5709	236	8	in	in	ADP
ejpam-5709	236	9	(	(	PUNCT
ejpam-5709	236	10	9	9	NUM
ejpam-5709	236	11	)	)	PUNCT
ejpam-5709	236	12	,	,	PUNCT
ejpam-5709	236	13	(	(	PUNCT
ejpam-5709	236	14	10	10	NUM
ejpam-5709	236	15	)	)	PUNCT
ejpam-5709	236	16	,	,	PUNCT
ejpam-5709	236	17	(	(	PUNCT
ejpam-5709	236	18	11	11	NUM
ejpam-5709	236	19	)	)	PUNCT
ejpam-5709	236	20	,	,	PUNCT
ejpam-5709	236	21	or	or	CCONJ
ejpam-5709	236	22	(	(	PUNCT
ejpam-5709	236	23	12	12	NUM
ejpam-5709	236	24	)	)	PUNCT
ejpam-5709	236	25	.	.	PUNCT
ejpam-5709	237	1	by	by	ADP
ejpam-5709	237	2	the	the	DET
ejpam-5709	237	3	same	same	ADJ
ejpam-5709	237	4	proof	proof	NOUN
ejpam-5709	237	5	as	as	ADP
ejpam-5709	237	6	in	in	ADP
ejpam-5709	237	7	theorem	theorem	NOUN
ejpam-5709	237	8	1	1	NUM
ejpam-5709	237	9	,	,	PUNCT
ejpam-5709	237	10	we	we	PRON
ejpam-5709	237	11	can	can	AUX
ejpam-5709	237	12	conclude	conclude	VERB
ejpam-5709	237	13	that	that	PRON
ejpam-5709	237	14	αi	αi	ADV
ejpam-5709	237	15	=	=	PUNCT
ejpam-5709	237	16	δi	δi	PROPN
ejpam-5709	237	17	(	(	PUNCT
ejpam-5709	237	18	0	0	NUM
ejpam-5709	237	19	≤	≤	NUM
ejpam-5709	237	20	i	i	NOUN
ejpam-5709	237	21	≤	≤	NOUN
ejpam-5709	237	22	l	l	NOUN
ejpam-5709	237	23	−	−	NOUN
ejpam-5709	237	24	1	1	NUM
ejpam-5709	237	25	)	)	PUNCT
ejpam-5709	237	26	and	and	CCONJ
ejpam-5709	237	27	γl−1	γl−1	PROPN
ejpam-5709	237	28	=	=	PUNCT
ejpam-5709	238	1	δl	δl	PROPN
ejpam-5709	238	2	.	.	PUNCT
ejpam-5709	239	1	this	this	PRON
ejpam-5709	239	2	completes	complete	VERB
ejpam-5709	239	3	the	the	DET
ejpam-5709	239	4	proof	proof	NOUN
ejpam-5709	239	5	.	.	PUNCT
ejpam-5709	240	1	note	note	VERB
ejpam-5709	240	2	that	that	SCONJ
ejpam-5709	240	3	the	the	DET
ejpam-5709	240	4	base	base	NOUN
ejpam-5709	240	5	-	-	PUNCT
ejpam-5709	240	6	β(c	β(c	NUM
ejpam-5709	240	7	)	)	PUNCT
ejpam-5709	240	8	representations	representation	NOUN
ejpam-5709	240	9	of	of	ADP
ejpam-5709	240	10	η	η	PROPN
ejpam-5709	240	11	in	in	ADP
ejpam-5709	240	12	(	(	PUNCT
ejpam-5709	240	13	11	11	NUM
ejpam-5709	240	14	)	)	PUNCT
ejpam-5709	240	15	and	and	CCONJ
ejpam-5709	240	16	(	(	PUNCT
ejpam-5709	240	17	12	12	NUM
ejpam-5709	240	18	)	)	PUNCT
ejpam-5709	240	19	are	be	AUX
ejpam-5709	240	20	periodic	periodic	ADJ
ejpam-5709	240	21	with	with	ADP
ejpam-5709	240	22	period	period	NOUN
ejpam-5709	240	23	v	v	ADP
ejpam-5709	240	24	−	−	NOUN
ejpam-5709	240	25	u.	u.	NOUN
ejpam-5709	240	26	the	the	DET
ejpam-5709	240	27	following	follow	VERB
ejpam-5709	240	28	examples	example	NOUN
ejpam-5709	240	29	show	show	VERB
ejpam-5709	240	30	the	the	DET
ejpam-5709	240	31	explicit	explicit	ADJ
ejpam-5709	240	32	shapes	shape	NOUN
ejpam-5709	240	33	of	of	ADP
ejpam-5709	240	34	the	the	DET
ejpam-5709	240	35	base	base	NOUN
ejpam-5709	240	36	-	-	PUNCT
ejpam-5709	240	37	β(c	β(c	NUM
ejpam-5709	240	38	)	)	PUNCT
ejpam-5709	240	39	representations	representation	NOUN
ejpam-5709	240	40	of	of	ADP
ejpam-5709	240	41	η	η	PROPN
ejpam-5709	240	42	for	for	ADP
ejpam-5709	240	43	the	the	DET
ejpam-5709	240	44	case	case	NOUN
ejpam-5709	240	45	γi	γi	X
ejpam-5709	240	46	̸∈	̸∈	PROPN
ejpam-5709	240	47	c	c	PROPN
ejpam-5709	240	48	for	for	ADP
ejpam-5709	240	49	all	all	PRON
ejpam-5709	240	50	i	i	PRON
ejpam-5709	240	51	∈	∈	VERB
ejpam-5709	240	52	n	n	PART
ejpam-5709	240	53	∪	∪	X
ejpam-5709	240	54	{	{	PUNCT
ejpam-5709	240	55	0	0	NUM
ejpam-5709	240	56	}	}	PUNCT
ejpam-5709	240	57	.	.	PUNCT
ejpam-5709	241	1	example	example	NOUN
ejpam-5709	242	1	2	2	NUM
ejpam-5709	242	2	.	.	PUNCT
ejpam-5709	242	3	let	let	VERB
ejpam-5709	242	4	k	k	NOUN
ejpam-5709	243	1	=	=	PUNCT
ejpam-5709	243	2	q	q	ADJ
ejpam-5709	243	3	(	(	PUNCT
ejpam-5709	243	4	√	√	NUM
ejpam-5709	243	5	−5	−5	NOUN
ejpam-5709	243	6	)	)	PUNCT
ejpam-5709	243	7	,	,	PUNCT
ejpam-5709	243	8	β	β	X
ejpam-5709	243	9	=	=	SYM
ejpam-5709	243	10	4	4	NUM
ejpam-5709	243	11	+	+	SYM
ejpam-5709	243	12	2	2	NUM
ejpam-5709	243	13	√	√	NUM
ejpam-5709	243	14	−5	−5	ADV
ejpam-5709	243	15	,	,	PUNCT
ejpam-5709	243	16	and	and	CCONJ
ejpam-5709	243	17	η	η	X
ejpam-5709	243	18	=	=	PRON
ejpam-5709	243	19	−5	−5	PROPN
ejpam-5709	244	1	+	+	NUM
ejpam-5709	244	2	10	10	NUM
ejpam-5709	244	3	√	√	NUM
ejpam-5709	244	4	−5	−5	NOUN
ejpam-5709	244	5	.	.	PUNCT
ejpam-5709	245	1	then	then	ADV
ejpam-5709	245	2	d	d	X
ejpam-5709	245	3	=	=	SYM
ejpam-5709	245	4	2	2	NUM
ejpam-5709	245	5	,	,	PUNCT
ejpam-5709	245	6	n(β	n(β	NUM
ejpam-5709	245	7	)	)	PUNCT
ejpam-5709	245	8	=	=	SYM
ejpam-5709	245	9	36	36	NUM
ejpam-5709	245	10	,	,	PUNCT
ejpam-5709	245	11	and	and	CCONJ
ejpam-5709	245	12	thus	thus	ADV
ejpam-5709	245	13	c	c	X
ejpam-5709	245	14	=	=	PRON
ejpam-5709	245	15	{	{	PUNCT
ejpam-5709	245	16	x+	x+	PROPN
ejpam-5709	245	17	y	y	PROPN
ejpam-5709	245	18	√	√	VERB
ejpam-5709	245	19	−5	−5	NOUN
ejpam-5709	246	1	|	|	ADV
ejpam-5709	246	2	x	x	NOUN
ejpam-5709	246	3	=	=	SYM
ejpam-5709	246	4	0	0	NUM
ejpam-5709	246	5	,	,	PUNCT
ejpam-5709	246	6	1	1	NUM
ejpam-5709	246	7	,	,	PUNCT
ejpam-5709	246	8	.	.	PUNCT
ejpam-5709	246	9	.	.	PUNCT
ejpam-5709	246	10	.	.	PUNCT
ejpam-5709	247	1	,	,	PUNCT
ejpam-5709	247	2	17	17	NUM
ejpam-5709	247	3	and	and	CCONJ
ejpam-5709	247	4	y	y	PROPN
ejpam-5709	247	5	=	=	SYM
ejpam-5709	247	6	0	0	NUM
ejpam-5709	247	7	,	,	PUNCT
ejpam-5709	247	8	1	1	NUM
ejpam-5709	247	9	}	}	PUNCT
ejpam-5709	247	10	is	be	AUX
ejpam-5709	247	11	a	a	DET
ejpam-5709	247	12	crs(β	crs(β	NOUN
ejpam-5709	247	13	)	)	PUNCT
ejpam-5709	247	14	.	.	PUNCT
ejpam-5709	248	1	we	we	PRON
ejpam-5709	248	2	have	have	VERB
ejpam-5709	248	3	|β|	|β|	NOUN
ejpam-5709	248	4	=	=	SYM
ejpam-5709	248	5	√	√	PROPN
ejpam-5709	248	6	36	36	NUM
ejpam-5709	248	7	>	>	SYM
ejpam-5709	248	8	2	2	NUM
ejpam-5709	248	9	and	and	CCONJ
ejpam-5709	248	10	|η|	|η|	NOUN
ejpam-5709	248	11	=	=	PUNCT
ejpam-5709	248	12	√	√	PROPN
ejpam-5709	248	13	525	525	NUM
ejpam-5709	248	14	>	>	PUNCT
ejpam-5709	248	15	√	√	NUM
ejpam-5709	248	16	294	294	NUM
ejpam-5709	248	17	=	=	SYM
ejpam-5709	248	18	s.	s.	PROPN
ejpam-5709	248	19	one	one	NUM
ejpam-5709	248	20	can	can	AUX
ejpam-5709	248	21	compute	compute	VERB
ejpam-5709	248	22	that	that	PRON
ejpam-5709	248	23	η	η	NOUN
ejpam-5709	248	24	=	=	SYM
ejpam-5709	248	25	(	(	PUNCT
ejpam-5709	248	26	1	1	NUM
ejpam-5709	248	27	+	+	SYM
ejpam-5709	248	28	2	2	NUM
ejpam-5709	248	29	√	√	NUM
ejpam-5709	248	30	−5)β	−5)β	NOUN
ejpam-5709	248	31	+	+	NOUN
ejpam-5709	248	32	11	11	NUM
ejpam-5709	248	33	,	,	PUNCT
ejpam-5709	248	34	p.	p.	NOUN
ejpam-5709	248	35	phetnun	phetnun	PROPN
ejpam-5709	248	36	,	,	PUNCT
ejpam-5709	248	37	n.r	n.r	PROPN
ejpam-5709	248	38	.	.	PROPN
ejpam-5709	248	39	kanasri	kanasri	PROPN
ejpam-5709	248	40	/	/	SYM
ejpam-5709	248	41	eur	eur	PROPN
ejpam-5709	248	42	.	.	PUNCT
ejpam-5709	249	1	j.	j.	PROPN
ejpam-5709	249	2	pure	pure	PROPN
ejpam-5709	249	3	appl	appl	PROPN
ejpam-5709	249	4	.	.	PROPN
ejpam-5709	249	5	math	math	PROPN
ejpam-5709	249	6	,	,	PUNCT
ejpam-5709	249	7	18	18	NUM
ejpam-5709	249	8	(	(	PUNCT
ejpam-5709	249	9	1	1	NUM
ejpam-5709	249	10	)	)	PUNCT
ejpam-5709	249	11	(	(	PUNCT
ejpam-5709	249	12	2025	2025	NUM
ejpam-5709	249	13	)	)	PUNCT
ejpam-5709	249	14	,	,	PUNCT
ejpam-5709	249	15	5709	5709	NUM
ejpam-5709	249	16	9	9	NUM
ejpam-5709	249	17	of	of	ADP
ejpam-5709	249	18	15	15	NUM
ejpam-5709	249	19	1	1	NUM
ejpam-5709	249	20	+	+	CCONJ
ejpam-5709	249	21	2	2	NUM
ejpam-5709	249	22	√	√	NUM
ejpam-5709	249	23	−5	−5	NOUN
ejpam-5709	249	24	=	=	PUNCT
ejpam-5709	249	25	(	(	PUNCT
ejpam-5709	249	26	−1	−1	NOUN
ejpam-5709	249	27	+	+	CCONJ
ejpam-5709	249	28	√	√	INTJ
ejpam-5709	249	29	−5)β	−5)β	NOUN
ejpam-5709	249	30	+	+	NOUN
ejpam-5709	249	31	15	15	NUM
ejpam-5709	249	32	,	,	PUNCT
ejpam-5709	249	33	−1	−1	NOUN
ejpam-5709	249	34	+	+	NOUN
ejpam-5709	249	35	√	√	ADJ
ejpam-5709	249	36	−5	−5	NOUN
ejpam-5709	249	37	=	=	PUNCT
ejpam-5709	249	38	(	(	PUNCT
ejpam-5709	249	39	−2	−2	NOUN
ejpam-5709	249	40	+	+	CCONJ
ejpam-5709	249	41	√	√	INTJ
ejpam-5709	249	42	−5)β	−5)β	NOUN
ejpam-5709	249	43	+	+	CCONJ
ejpam-5709	249	44	(	(	PUNCT
ejpam-5709	249	45	17	17	NUM
ejpam-5709	249	46	+	+	CCONJ
ejpam-5709	249	47	√	√	NUM
ejpam-5709	249	48	−5	−5	NOUN
ejpam-5709	249	49	)	)	PUNCT
ejpam-5709	249	50	,	,	PUNCT
ejpam-5709	249	51	−2	−2	NOUN
ejpam-5709	250	1	+	+	CCONJ
ejpam-5709	250	2	√	√	ADJ
ejpam-5709	250	3	−5	−5	NOUN
ejpam-5709	250	4	=	=	PUNCT
ejpam-5709	250	5	(	(	PUNCT
ejpam-5709	250	6	−2	−2	NOUN
ejpam-5709	251	1	+	+	CCONJ
ejpam-5709	251	2	√	√	INTJ
ejpam-5709	251	3	−5)β	−5)β	NOUN
ejpam-5709	251	4	+	+	CCONJ
ejpam-5709	251	5	(	(	PUNCT
ejpam-5709	251	6	16	16	NUM
ejpam-5709	251	7	+	+	CCONJ
ejpam-5709	251	8	√	√	NUM
ejpam-5709	251	9	−5	−5	NOUN
ejpam-5709	251	10	)	)	PUNCT
ejpam-5709	251	11	.	.	PUNCT
ejpam-5709	252	1	thus	thus	ADV
ejpam-5709	252	2	γ0	γ0	NOUN
ejpam-5709	252	3	=	=	SYM
ejpam-5709	252	4	1	1	NUM
ejpam-5709	252	5	+	+	NUM
ejpam-5709	252	6	2	2	NUM
ejpam-5709	252	7	√	√	NUM
ejpam-5709	252	8	−5	−5	ADV
ejpam-5709	252	9	,	,	PUNCT
ejpam-5709	252	10	γ1	γ1	NOUN
ejpam-5709	252	11	=	=	SYM
ejpam-5709	252	12	−1	−1	PROPN
ejpam-5709	252	13	+	+	CCONJ
ejpam-5709	252	14	√	√	NOUN
ejpam-5709	252	15	−5	−5	NOUN
ejpam-5709	252	16	,	,	PUNCT
ejpam-5709	252	17	γ2	γ2	NOUN
ejpam-5709	252	18	=	=	SYM
ejpam-5709	252	19	−2	−2	NOUN
ejpam-5709	252	20	+	+	CCONJ
ejpam-5709	252	21	√	√	ADJ
ejpam-5709	252	22	−5	−5	NOUN
ejpam-5709	252	23	=	=	SYM
ejpam-5709	252	24	γ3	γ3	NOUN
ejpam-5709	252	25	,	,	PUNCT
ejpam-5709	252	26	and	and	CCONJ
ejpam-5709	252	27	so	so	ADV
ejpam-5709	252	28	γi	γi	X
ejpam-5709	252	29	=	=	SYM
ejpam-5709	252	30	γ2	γ2	PROPN
ejpam-5709	252	31	for	for	ADP
ejpam-5709	252	32	all	all	PRON
ejpam-5709	252	33	i	i	PRON
ejpam-5709	252	34	≥	≥	VERB
ejpam-5709	252	35	2	2	NUM
ejpam-5709	252	36	.	.	X
ejpam-5709	252	37	one	one	PRON
ejpam-5709	252	38	can	can	AUX
ejpam-5709	252	39	see	see	VERB
ejpam-5709	252	40	that	that	SCONJ
ejpam-5709	252	41	γi	γi	PROPN
ejpam-5709	252	42	̸∈	̸∈	PROPN
ejpam-5709	252	43	c	c	PROPN
ejpam-5709	252	44	for	for	ADP
ejpam-5709	252	45	all	all	DET
ejpam-5709	252	46	i	i	PRON
ejpam-5709	252	47	∈	∈	VERB
ejpam-5709	252	48	n	n	PART
ejpam-5709	252	49	∪	∪	X
ejpam-5709	252	50	{	{	PUNCT
ejpam-5709	252	51	0	0	NUM
ejpam-5709	252	52	}	}	PUNCT
ejpam-5709	252	53	.	.	PUNCT
ejpam-5709	253	1	note	note	VERB
ejpam-5709	253	2	that	that	SCONJ
ejpam-5709	253	3	α0	α0	ADJ
ejpam-5709	253	4	=	=	SYM
ejpam-5709	253	5	11	11	NUM
ejpam-5709	253	6	,	,	PUNCT
ejpam-5709	253	7	α1	α1	PROPN
ejpam-5709	253	8	=	=	SYM
ejpam-5709	253	9	15	15	NUM
ejpam-5709	253	10	,	,	PUNCT
ejpam-5709	253	11	α2	α2	NOUN
ejpam-5709	253	12	=	=	SYM
ejpam-5709	253	13	17	17	NUM
ejpam-5709	253	14	+	+	CCONJ
ejpam-5709	253	15	√	√	NOUN
ejpam-5709	253	16	−5	−5	NOUN
ejpam-5709	253	17	,	,	PUNCT
ejpam-5709	253	18	and	and	CCONJ
ejpam-5709	253	19	α3	α3	NOUN
ejpam-5709	253	20	=	=	SYM
ejpam-5709	253	21	16	16	NUM
ejpam-5709	254	1	+	+	CCONJ
ejpam-5709	254	2	√	√	ADJ
ejpam-5709	254	3	−5	−5	NOUN
ejpam-5709	254	4	=	=	PUNCT
ejpam-5709	254	5	αi	αi	NOUN
ejpam-5709	254	6	for	for	ADP
ejpam-5709	254	7	all	all	PRON
ejpam-5709	254	8	i	i	PRON
ejpam-5709	254	9	≥	≥	VERB
ejpam-5709	254	10	3	3	NUM
ejpam-5709	254	11	.	.	PUNCT
ejpam-5709	254	12	using	use	VERB
ejpam-5709	254	13	theorem	theorem	NOUN
ejpam-5709	254	14	2	2	NUM
ejpam-5709	254	15	with	with	ADP
ejpam-5709	254	16	u	u	NOUN
ejpam-5709	254	17	=	=	SYM
ejpam-5709	254	18	2	2	NUM
ejpam-5709	254	19	and	and	CCONJ
ejpam-5709	254	20	v	v	NOUN
ejpam-5709	254	21	=	=	SYM
ejpam-5709	254	22	3	3	NUM
ejpam-5709	254	23	,	,	PUNCT
ejpam-5709	254	24	we	we	PRON
ejpam-5709	254	25	conclude	conclude	VERB
ejpam-5709	254	26	that	that	SCONJ
ejpam-5709	254	27	the	the	DET
ejpam-5709	254	28	base	base	NOUN
ejpam-5709	254	29	-	-	PUNCT
ejpam-5709	254	30	β(c	β(c	NUM
ejpam-5709	254	31	)	)	PUNCT
ejpam-5709	254	32	representations	representation	NOUN
ejpam-5709	254	33	of	of	ADP
ejpam-5709	254	34	η	η	PROPN
ejpam-5709	254	35	are	be	AUX
ejpam-5709	254	36	η	η	NOUN
ejpam-5709	254	37	=	=	PUNCT
ejpam-5709	254	38	γkβ	γkβ	NOUN
ejpam-5709	254	39	k+1	k+1	X
ejpam-5709	255	1	+	+	CCONJ
ejpam-5709	255	2	αkβ	αkβ	NOUN
ejpam-5709	255	3	k	k	X
ejpam-5709	256	1	+	+	CCONJ
ejpam-5709	256	2	·	·	PUNCT
ejpam-5709	256	3	·	·	PUNCT
ejpam-5709	256	4	·	·	PUNCT
ejpam-5709	256	5	+	+	NUM
ejpam-5709	256	6	α1β	α1β	X
ejpam-5709	256	7	+	+	CCONJ
ejpam-5709	256	8	α0	α0	ADJ
ejpam-5709	256	9	(	(	PUNCT
ejpam-5709	256	10	0	0	NUM
ejpam-5709	256	11	≤	≤	NUM
ejpam-5709	256	12	k	k	X
ejpam-5709	256	13	≤	≤	NUM
ejpam-5709	256	14	2	2	NUM
ejpam-5709	256	15	)	)	PUNCT
ejpam-5709	256	16	and	and	CCONJ
ejpam-5709	256	17	η	η	PROPN
ejpam-5709	256	18	=	=	PROPN
ejpam-5709	256	19	γ2β	γ2β	PROPN
ejpam-5709	256	20	n+3	n+3	PROPN
ejpam-5709	256	21	+	+	CCONJ
ejpam-5709	256	22	α3β	α3β	NOUN
ejpam-5709	256	23	n+2	n+2	PROPN
ejpam-5709	256	24	+	+	CCONJ
ejpam-5709	256	25	·	·	PUNCT
ejpam-5709	256	26	·	·	PUNCT
ejpam-5709	256	27	·	·	PUNCT
ejpam-5709	257	1	+	+	CCONJ
ejpam-5709	257	2	α3β	α3β	ADJ
ejpam-5709	257	3	3	3	NUM
ejpam-5709	257	4	+	+	NUM
ejpam-5709	257	5	α2β	α2β	PROPN
ejpam-5709	257	6	2	2	NUM
ejpam-5709	257	7	+	+	NUM
ejpam-5709	257	8	α1β	α1β	X
ejpam-5709	258	1	+	+	CCONJ
ejpam-5709	258	2	α0	α0	ADJ
ejpam-5709	258	3	for	for	ADP
ejpam-5709	258	4	all	all	PRON
ejpam-5709	258	5	n	n	PRON
ejpam-5709	258	6	∈	∈	PROPN
ejpam-5709	258	7	n.	n.	NOUN
ejpam-5709	258	8	example	example	NOUN
ejpam-5709	258	9	3	3	X
ejpam-5709	258	10	.	.	PUNCT
ejpam-5709	259	1	let	let	VERB
ejpam-5709	259	2	k	k	NOUN
ejpam-5709	259	3	=	=	PUNCT
ejpam-5709	259	4	q	q	ADJ
ejpam-5709	259	5	(	(	PUNCT
ejpam-5709	259	6	√	√	NUM
ejpam-5709	259	7	−7	−7	NOUN
ejpam-5709	259	8	)	)	PUNCT
ejpam-5709	259	9	,	,	PUNCT
ejpam-5709	259	10	β	β	X
ejpam-5709	259	11	=	=	SYM
ejpam-5709	259	12	2	2	NUM
ejpam-5709	259	13	+	+	SYM
ejpam-5709	259	14	3σ−7	3σ−7	NUM
ejpam-5709	259	15	,	,	PUNCT
ejpam-5709	259	16	and	and	CCONJ
ejpam-5709	259	17	η	η	PROPN
ejpam-5709	259	18	=	=	PROPN
ejpam-5709	259	19	3112	3112	NUM
ejpam-5709	259	20	−	−	PROPN
ejpam-5709	259	21	13810σ−7	13810σ−7	NUM
ejpam-5709	259	22	.	.	PUNCT
ejpam-5709	260	1	then	then	ADV
ejpam-5709	260	2	d	d	X
ejpam-5709	260	3	=	=	SYM
ejpam-5709	260	4	1	1	NUM
ejpam-5709	260	5	,	,	PUNCT
ejpam-5709	260	6	n(β	n(β	NUM
ejpam-5709	260	7	)	)	PUNCT
ejpam-5709	260	8	=	=	SYM
ejpam-5709	260	9	28	28	NUM
ejpam-5709	260	10	,	,	PUNCT
ejpam-5709	260	11	and	and	CCONJ
ejpam-5709	260	12	thus	thus	ADV
ejpam-5709	260	13	c	c	X
ejpam-5709	260	14	=	=	SYM
ejpam-5709	260	15	{	{	PUNCT
ejpam-5709	260	16	0	0	NUM
ejpam-5709	260	17	,	,	PUNCT
ejpam-5709	260	18	1	1	NUM
ejpam-5709	260	19	,	,	PUNCT
ejpam-5709	260	20	.	.	PUNCT
ejpam-5709	260	21	.	.	PUNCT
ejpam-5709	260	22	.	.	PUNCT
ejpam-5709	261	1	,	,	PUNCT
ejpam-5709	261	2	27	27	NUM
ejpam-5709	261	3	}	}	PUNCT
ejpam-5709	261	4	is	be	AUX
ejpam-5709	261	5	a	a	DET
ejpam-5709	261	6	crs(β	crs(β	NOUN
ejpam-5709	261	7	)	)	PUNCT
ejpam-5709	261	8	.	.	PUNCT
ejpam-5709	262	1	we	we	PRON
ejpam-5709	262	2	have	have	VERB
ejpam-5709	262	3	|β|	|β|	NOUN
ejpam-5709	262	4	=	=	SYM
ejpam-5709	262	5	√	√	NUM
ejpam-5709	263	1	28	28	NUM
ejpam-5709	263	2	>	>	SYM
ejpam-5709	263	3	2	2	NUM
ejpam-5709	263	4	and	and	CCONJ
ejpam-5709	263	5	|η|	|η|	NOUN
ejpam-5709	263	6	=	=	PUNCT
ejpam-5709	264	1	√	√	PROPN
ejpam-5709	264	2	348140024	348140024	NUM
ejpam-5709	264	3	>	>	SYM
ejpam-5709	264	4	27	27	NUM
ejpam-5709	264	5	=	=	SYM
ejpam-5709	264	6	s.	s.	PROPN
ejpam-5709	264	7	one	one	NUM
ejpam-5709	264	8	can	can	AUX
ejpam-5709	264	9	compute	compute	VERB
ejpam-5709	264	10	that	that	PRON
ejpam-5709	264	11	η	η	NOUN
ejpam-5709	264	12	=	=	PROPN
ejpam-5709	264	13	(	(	PUNCT
ejpam-5709	264	14	−2405−	−2405−	PROPN
ejpam-5709	264	15	1319σ−7)β	1319σ−7)β	NUM
ejpam-5709	264	16	+	+	CCONJ
ejpam-5709	264	17	8	8	NUM
ejpam-5709	264	18	,	,	PUNCT
ejpam-5709	264	19	−2405−	−2405−	NOUN
ejpam-5709	264	20	1319σ−7	1319σ−7	NUM
ejpam-5709	265	1	=	=	SYM
ejpam-5709	266	1	(	(	PUNCT
ejpam-5709	266	2	−713	−713	NOUN
ejpam-5709	266	3	+	+	CCONJ
ejpam-5709	267	1	164σ−7)β	164σ−7)β	NUM
ejpam-5709	268	1	+	+	CCONJ
ejpam-5709	268	2	5	5	NUM
ejpam-5709	268	3	,	,	PUNCT
ejpam-5709	268	4	−713	−713	NOUN
ejpam-5709	268	5	+	+	CCONJ
ejpam-5709	268	6	164σ−7	164σ−7	NUM
ejpam-5709	268	7	=	=	SYM
ejpam-5709	268	8	(	(	PUNCT
ejpam-5709	268	9	−97	−97	PROPN
ejpam-5709	268	10	+	+	X
ejpam-5709	268	11	91σ−7)β	91σ−7)β	NUM
ejpam-5709	268	12	+	+	NUM
ejpam-5709	268	13	27	27	NUM
ejpam-5709	268	14	,	,	PUNCT
ejpam-5709	268	15	−97	−97	PROPN
ejpam-5709	268	16	+	+	X
ejpam-5709	268	17	91σ−7	91σ−7	PROPN
ejpam-5709	268	18	=	=	SYM
ejpam-5709	268	19	(	(	PUNCT
ejpam-5709	268	20	2	2	NUM
ejpam-5709	268	21	+	+	SYM
ejpam-5709	268	22	17σ−7)β	17σ−7)β	NUM
ejpam-5709	269	1	+	+	CCONJ
ejpam-5709	269	2	1	1	NUM
ejpam-5709	269	3	,	,	PUNCT
ejpam-5709	269	4	2	2	NUM
ejpam-5709	270	1	+	+	SYM
ejpam-5709	270	2	17σ−7	17σ−7	NOUN
ejpam-5709	270	3	=	=	SYM
ejpam-5709	270	4	(	(	PUNCT
ejpam-5709	270	5	4	4	NUM
ejpam-5709	270	6	+	+	NUM
ejpam-5709	270	7	σ−7)β	σ−7)β	NOUN
ejpam-5709	270	8	,	,	PUNCT
ejpam-5709	270	9	4	4	NUM
ejpam-5709	270	10	+	+	NOUN
ejpam-5709	270	11	σ−7	σ−7	PROPN
ejpam-5709	270	12	=	=	SYM
ejpam-5709	270	13	(	(	PUNCT
ejpam-5709	270	14	−3	−3	PROPN
ejpam-5709	271	1	+	+	NUM
ejpam-5709	271	2	2σ−7)β	2σ−7)β	NUM
ejpam-5709	271	3	+	+	CCONJ
ejpam-5709	271	4	22	22	NUM
ejpam-5709	271	5	,	,	PUNCT
ejpam-5709	271	6	−3	−3	PROPN
ejpam-5709	271	7	+	+	CCONJ
ejpam-5709	271	8	2σ−7	2σ−7	X
ejpam-5709	271	9	=	=	SYM
ejpam-5709	271	10	(	(	PUNCT
ejpam-5709	271	11	−1	−1	NOUN
ejpam-5709	271	12	+	+	CCONJ
ejpam-5709	271	13	σ−7)β	σ−7)β	PROPN
ejpam-5709	271	14	+	+	CCONJ
ejpam-5709	271	15	5	5	NUM
ejpam-5709	271	16	,	,	PUNCT
ejpam-5709	271	17	−1	−1	NOUN
ejpam-5709	271	18	+	+	NOUN
ejpam-5709	271	19	σ−7	σ−7	PROPN
ejpam-5709	271	20	=	=	SYM
ejpam-5709	271	21	(	(	PUNCT
ejpam-5709	271	22	−3	−3	PROPN
ejpam-5709	271	23	+	+	NUM
ejpam-5709	271	24	2σ−7)β	2σ−7)β	NUM
ejpam-5709	271	25	+	+	NUM
ejpam-5709	271	26	17	17	NUM
ejpam-5709	271	27	,	,	PUNCT
ejpam-5709	271	28	−3	−3	PROPN
ejpam-5709	271	29	+	+	CCONJ
ejpam-5709	271	30	2σ−7	2σ−7	X
ejpam-5709	271	31	=	=	SYM
ejpam-5709	271	32	(	(	PUNCT
ejpam-5709	271	33	−1	−1	NOUN
ejpam-5709	271	34	+	+	CCONJ
ejpam-5709	271	35	σ−7)β	σ−7)β	PROPN
ejpam-5709	271	36	+	+	CCONJ
ejpam-5709	271	37	5	5	NUM
ejpam-5709	271	38	.	.	PUNCT
ejpam-5709	271	39	thus	thus	ADV
ejpam-5709	271	40	γ0	γ0	NOUN
ejpam-5709	271	41	=	=	SYM
ejpam-5709	271	42	−2405	−2405	PROPN
ejpam-5709	271	43	−	−	PROPN
ejpam-5709	271	44	1319σ−7	1319σ−7	NUM
ejpam-5709	271	45	,	,	PUNCT
ejpam-5709	271	46	γ1	γ1	NOUN
ejpam-5709	271	47	=	=	SYM
ejpam-5709	271	48	−713	−713	NOUN
ejpam-5709	271	49	+	+	CCONJ
ejpam-5709	271	50	164σ−7	164σ−7	NUM
ejpam-5709	271	51	,	,	PUNCT
ejpam-5709	271	52	γ2	γ2	PROPN
ejpam-5709	271	53	=	=	SYM
ejpam-5709	271	54	−97	−97	PROPN
ejpam-5709	271	55	+	+	CCONJ
ejpam-5709	271	56	91σ−7	91σ−7	NUM
ejpam-5709	271	57	,	,	PUNCT
ejpam-5709	271	58	γ3	γ3	NOUN
ejpam-5709	271	59	=	=	SYM
ejpam-5709	271	60	2	2	NUM
ejpam-5709	271	61	+	+	CCONJ
ejpam-5709	271	62	17σ−7	17σ−7	ADJ
ejpam-5709	271	63	,	,	PUNCT
ejpam-5709	271	64	γ4	γ4	NOUN
ejpam-5709	271	65	=	=	SYM
ejpam-5709	271	66	4	4	NUM
ejpam-5709	271	67	+	+	CCONJ
ejpam-5709	271	68	σ−7	σ−7	ADJ
ejpam-5709	271	69	,	,	PUNCT
ejpam-5709	271	70	γ5	γ5	NOUN
ejpam-5709	271	71	=	=	PUNCT
ejpam-5709	271	72	−3	−3	PROPN
ejpam-5709	272	1	+	+	NUM
ejpam-5709	272	2	2σ−7	2σ−7	X
ejpam-5709	272	3	=	=	SYM
ejpam-5709	272	4	γ7	γ7	ADJ
ejpam-5709	272	5	=	=	NOUN
ejpam-5709	272	6	γ9	γ9	PROPN
ejpam-5709	272	7	=	=	SYM
ejpam-5709	272	8	·	·	PUNCT
ejpam-5709	272	9	·	·	PUNCT
ejpam-5709	272	10	·	·	PUNCT
ejpam-5709	272	11	,	,	PUNCT
ejpam-5709	272	12	and	and	CCONJ
ejpam-5709	272	13	γ6	γ6	PROPN
ejpam-5709	272	14	=	=	SYM
ejpam-5709	272	15	−1	−1	NOUN
ejpam-5709	272	16	+	+	CCONJ
ejpam-5709	272	17	σ−7	σ−7	PROPN
ejpam-5709	272	18	=	=	SYM
ejpam-5709	272	19	γ8	γ8	NOUN
ejpam-5709	272	20	=	=	SYM
ejpam-5709	272	21	γ10	γ10	NOUN
ejpam-5709	272	22	=	=	SYM
ejpam-5709	272	23	·	·	PUNCT
ejpam-5709	272	24	·	·	PUNCT
ejpam-5709	272	25	·	·	PUNCT
ejpam-5709	272	26	.	.	PUNCT
ejpam-5709	273	1	one	one	PRON
ejpam-5709	273	2	can	can	AUX
ejpam-5709	273	3	see	see	VERB
ejpam-5709	273	4	that	that	SCONJ
ejpam-5709	273	5	γi	γi	PROPN
ejpam-5709	273	6	̸∈	̸∈	PROPN
ejpam-5709	273	7	c	c	PROPN
ejpam-5709	273	8	for	for	ADP
ejpam-5709	273	9	all	all	DET
ejpam-5709	273	10	i	i	PRON
ejpam-5709	273	11	∈	∈	VERB
ejpam-5709	273	12	n	n	PART
ejpam-5709	273	13	∪	∪	X
ejpam-5709	273	14	{	{	PUNCT
ejpam-5709	273	15	0	0	NUM
ejpam-5709	273	16	}	}	PUNCT
ejpam-5709	273	17	.	.	PUNCT
ejpam-5709	274	1	we	we	PRON
ejpam-5709	274	2	also	also	ADV
ejpam-5709	274	3	have	have	VERB
ejpam-5709	274	4	α0	α0	ADJ
ejpam-5709	274	5	=	=	SYM
ejpam-5709	274	6	8	8	NUM
ejpam-5709	274	7	,	,	PUNCT
ejpam-5709	274	8	α1	α1	NOUN
ejpam-5709	274	9	=	=	SYM
ejpam-5709	274	10	5	5	NUM
ejpam-5709	274	11	,	,	PUNCT
ejpam-5709	274	12	α2	α2	NOUN
ejpam-5709	274	13	=	=	SYM
ejpam-5709	274	14	27	27	NUM
ejpam-5709	274	15	,	,	PUNCT
ejpam-5709	274	16	α3	α3	NOUN
ejpam-5709	274	17	=	=	SYM
ejpam-5709	274	18	1	1	NUM
ejpam-5709	274	19	,	,	PUNCT
ejpam-5709	274	20	α4	α4	NOUN
ejpam-5709	274	21	=	=	SYM
ejpam-5709	274	22	0	0	NUM
ejpam-5709	274	23	,	,	PUNCT
ejpam-5709	274	24	α5	α5	NOUN
ejpam-5709	274	25	=	=	SYM
ejpam-5709	274	26	22	22	NUM
ejpam-5709	274	27	,	,	PUNCT
ejpam-5709	274	28	α6	α6	NOUN
ejpam-5709	274	29	=	=	SYM
ejpam-5709	274	30	5	5	NUM
ejpam-5709	274	31	=	=	SYM
ejpam-5709	274	32	α8	α8	NOUN
ejpam-5709	274	33	=	=	SYM
ejpam-5709	274	34	α10	α10	PROPN
ejpam-5709	274	35	=	=	SYM
ejpam-5709	274	36	·	·	PUNCT
ejpam-5709	274	37	·	·	PUNCT
ejpam-5709	274	38	·	·	PUNCT
ejpam-5709	274	39	,	,	PUNCT
ejpam-5709	274	40	and	and	CCONJ
ejpam-5709	274	41	α7	α7	NOUN
ejpam-5709	274	42	=	=	SYM
ejpam-5709	274	43	17	17	NUM
ejpam-5709	274	44	=	=	SYM
ejpam-5709	274	45	α9	α9	NOUN
ejpam-5709	274	46	=	=	SYM
ejpam-5709	274	47	α11	α11	NOUN
ejpam-5709	274	48	=	=	SYM
ejpam-5709	274	49	·	·	PUNCT
ejpam-5709	274	50	·	·	PUNCT
ejpam-5709	274	51	·	·	PUNCT
ejpam-5709	274	52	.	.	PUNCT
ejpam-5709	275	1	using	use	VERB
ejpam-5709	275	2	theorem	theorem	NOUN
ejpam-5709	275	3	2	2	NUM
ejpam-5709	275	4	with	with	ADP
ejpam-5709	275	5	u	u	NOUN
ejpam-5709	275	6	=	=	SYM
ejpam-5709	275	7	5	5	NUM
ejpam-5709	275	8	and	and	CCONJ
ejpam-5709	275	9	v	v	NOUN
ejpam-5709	275	10	=	=	SYM
ejpam-5709	275	11	7	7	NUM
ejpam-5709	275	12	,	,	PUNCT
ejpam-5709	275	13	we	we	PRON
ejpam-5709	275	14	conclude	conclude	VERB
ejpam-5709	275	15	that	that	SCONJ
ejpam-5709	275	16	the	the	DET
ejpam-5709	275	17	base	base	NOUN
ejpam-5709	275	18	-	-	PUNCT
ejpam-5709	275	19	β(c	β(c	NUM
ejpam-5709	275	20	)	)	PUNCT
ejpam-5709	275	21	representations	representation	NOUN
ejpam-5709	275	22	of	of	ADP
ejpam-5709	275	23	η	η	PROPN
ejpam-5709	275	24	are	be	AUX
ejpam-5709	275	25	η	η	NOUN
ejpam-5709	275	26	=	=	PUNCT
ejpam-5709	275	27	γkβ	γkβ	NOUN
ejpam-5709	275	28	k+1	k+1	X
ejpam-5709	276	1	+	+	CCONJ
ejpam-5709	276	2	αkβ	αkβ	NOUN
ejpam-5709	276	3	k	k	X
ejpam-5709	277	1	+	+	CCONJ
ejpam-5709	277	2	·	·	PUNCT
ejpam-5709	277	3	·	·	PUNCT
ejpam-5709	277	4	·	·	PUNCT
ejpam-5709	277	5	+	+	NUM
ejpam-5709	277	6	α1β	α1β	X
ejpam-5709	277	7	+	+	CCONJ
ejpam-5709	277	8	α0	α0	ADJ
ejpam-5709	277	9	(	(	PUNCT
ejpam-5709	277	10	0	0	NUM
ejpam-5709	277	11	≤	≤	NUM
ejpam-5709	277	12	k	k	X
ejpam-5709	277	13	≤	≤	NUM
ejpam-5709	277	14	5	5	NUM
ejpam-5709	277	15	)	)	PUNCT
ejpam-5709	277	16	,	,	PUNCT
ejpam-5709	277	17	η	η	X
ejpam-5709	277	18	=	=	PRON
ejpam-5709	277	19	γ6β	γ6β	PROPN
ejpam-5709	277	20	7	7	NUM
ejpam-5709	277	21	+	+	CCONJ
ejpam-5709	277	22	α6β	α6β	PROPN
ejpam-5709	277	23	6	6	NUM
ejpam-5709	277	24	+	+	CCONJ
ejpam-5709	277	25	α5β	α5β	NOUN
ejpam-5709	277	26	5	5	NUM
ejpam-5709	277	27	+	+	CCONJ
ejpam-5709	277	28	α4β	α4β	PROPN
ejpam-5709	277	29	4	4	NUM
ejpam-5709	277	30	+	+	CCONJ
ejpam-5709	277	31	α3β	α3β	ADJ
ejpam-5709	277	32	3	3	NUM
ejpam-5709	277	33	+	+	NUM
ejpam-5709	277	34	α2β	α2β	PROPN
ejpam-5709	277	35	2	2	NUM
ejpam-5709	277	36	+	+	NUM
ejpam-5709	277	37	α1β	α1β	ADJ
ejpam-5709	277	38	+	+	CCONJ
ejpam-5709	277	39	α0	α0	ADJ
ejpam-5709	277	40	,	,	PUNCT
ejpam-5709	277	41	η	η	PROPN
ejpam-5709	277	42	=	=	PROPN
ejpam-5709	277	43	γ5β	γ5β	PROPN
ejpam-5709	277	44	2n+6	2n+6	PROPN
ejpam-5709	277	45	+	+	CCONJ
ejpam-5709	277	46	α7β	α7β	PROPN
ejpam-5709	277	47	2n+5	2n+5	PROPN
ejpam-5709	277	48	+	+	NUM
ejpam-5709	277	49	α6β	α6β	PROPN
ejpam-5709	277	50	2n+4	2n+4	PROPN
ejpam-5709	277	51	+	+	CCONJ
ejpam-5709	277	52	·	·	PUNCT
ejpam-5709	277	53	·	·	PUNCT
ejpam-5709	277	54	·	·	PUNCT
ejpam-5709	278	1	+	+	NUM
ejpam-5709	278	2	α7β	α7β	NOUN
ejpam-5709	278	3	7	7	NUM
ejpam-5709	278	4	+	+	CCONJ
ejpam-5709	278	5	α6β	α6β	PROPN
ejpam-5709	278	6	6	6	NUM
ejpam-5709	278	7	+	+	CCONJ
ejpam-5709	278	8	α5β	α5β	NOUN
ejpam-5709	278	9	5	5	NUM
ejpam-5709	278	10	+	+	CCONJ
ejpam-5709	278	11	α4β	α4β	PROPN
ejpam-5709	278	12	4	4	NUM
ejpam-5709	278	13	+	+	CCONJ
ejpam-5709	278	14	α3β	α3β	ADJ
ejpam-5709	278	15	3	3	NUM
ejpam-5709	278	16	+	+	NUM
ejpam-5709	278	17	α2β	α2β	PROPN
ejpam-5709	278	18	2	2	NUM
ejpam-5709	278	19	+	+	NUM
ejpam-5709	278	20	α1β	α1β	X
ejpam-5709	278	21	+	+	CCONJ
ejpam-5709	278	22	α0	α0	ADJ
ejpam-5709	278	23	(	(	PUNCT
ejpam-5709	278	24	n	n	NOUN
ejpam-5709	278	25	∈	∈	PROPN
ejpam-5709	278	26	n	n	CCONJ
ejpam-5709	278	27	)	)	PUNCT
ejpam-5709	278	28	,	,	PUNCT
ejpam-5709	278	29	η	η	PROPN
ejpam-5709	278	30	=	=	X
ejpam-5709	278	31	γ6β	γ6β	PROPN
ejpam-5709	278	32	2n+7	2n+7	PROPN
ejpam-5709	278	33	+	+	CCONJ
ejpam-5709	278	34	α6β	α6β	PROPN
ejpam-5709	278	35	2n+6	2n+6	PROPN
ejpam-5709	278	36	+	+	CCONJ
ejpam-5709	278	37	α7β	α7β	PROPN
ejpam-5709	278	38	2n+5	2n+5	PROPN
ejpam-5709	278	39	+	+	NUM
ejpam-5709	278	40	α6β	α6β	PROPN
ejpam-5709	278	41	2n+4	2n+4	PROPN
ejpam-5709	278	42	+	+	CCONJ
ejpam-5709	278	43	·	·	PUNCT
ejpam-5709	278	44	·	·	PUNCT
ejpam-5709	278	45	·	·	PUNCT
ejpam-5709	278	46	+	+	NUM
ejpam-5709	278	47	α7β	α7β	NOUN
ejpam-5709	278	48	7	7	NUM
ejpam-5709	278	49	+	+	CCONJ
ejpam-5709	278	50	α6β	α6β	PROPN
ejpam-5709	278	51	6	6	NUM
ejpam-5709	278	52	+	+	CCONJ
ejpam-5709	278	53	α5β	α5β	NOUN
ejpam-5709	278	54	5	5	NUM
ejpam-5709	278	55	+	+	CCONJ
ejpam-5709	278	56	α4β	α4β	PROPN
ejpam-5709	278	57	4	4	NUM
ejpam-5709	278	58	+	+	CCONJ
ejpam-5709	278	59	α3β	α3β	ADJ
ejpam-5709	278	60	3	3	NUM
ejpam-5709	278	61	+	+	NUM
ejpam-5709	278	62	α2β	α2β	PROPN
ejpam-5709	278	63	2	2	NUM
ejpam-5709	278	64	+	+	NUM
ejpam-5709	278	65	α1β	α1β	X
ejpam-5709	278	66	+	+	CCONJ
ejpam-5709	278	67	α0	α0	ADJ
ejpam-5709	278	68	(	(	PUNCT
ejpam-5709	278	69	n	n	NOUN
ejpam-5709	278	70	∈	∈	PROPN
ejpam-5709	278	71	n	n	CCONJ
ejpam-5709	278	72	)	)	PUNCT
ejpam-5709	278	73	.	.	PUNCT
ejpam-5709	279	1	p.	p.	NOUN
ejpam-5709	279	2	phetnun	phetnun	PROPN
ejpam-5709	279	3	,	,	PUNCT
ejpam-5709	279	4	n.r	n.r	PROPN
ejpam-5709	279	5	.	.	PROPN
ejpam-5709	279	6	kanasri	kanasri	PROPN
ejpam-5709	279	7	/	/	SYM
ejpam-5709	279	8	eur	eur	PROPN
ejpam-5709	279	9	.	.	PUNCT
ejpam-5709	280	1	j.	j.	PROPN
ejpam-5709	280	2	pure	pure	PROPN
ejpam-5709	280	3	appl	appl	PROPN
ejpam-5709	280	4	.	.	PROPN
ejpam-5709	280	5	math	math	PROPN
ejpam-5709	280	6	,	,	PUNCT
ejpam-5709	280	7	18	18	NUM
ejpam-5709	280	8	(	(	PUNCT
ejpam-5709	280	9	1	1	NUM
ejpam-5709	280	10	)	)	PUNCT
ejpam-5709	280	11	(	(	PUNCT
ejpam-5709	280	12	2025	2025	NUM
ejpam-5709	280	13	)	)	PUNCT
ejpam-5709	280	14	,	,	PUNCT
ejpam-5709	280	15	5709	5709	NUM
ejpam-5709	280	16	10	10	NUM
ejpam-5709	280	17	of	of	ADP
ejpam-5709	280	18	15	15	NUM
ejpam-5709	280	19	3	3	NUM
ejpam-5709	280	20	.	.	PUNCT
ejpam-5709	281	1	generalizations	generalization	NOUN
ejpam-5709	281	2	of	of	ADP
ejpam-5709	281	3	irreducibility	irreducibility	NOUN
ejpam-5709	281	4	criteria	criterion	NOUN
ejpam-5709	281	5	for	for	ADP
ejpam-5709	281	6	polynomials	polynomial	NOUN
ejpam-5709	281	7	in	in	ADP
ejpam-5709	281	8	ok	ok	ADJ
ejpam-5709	281	9	[	[	X
ejpam-5709	281	10	x	x	X
ejpam-5709	281	11	]	]	X
ejpam-5709	281	12	let	let	VERB
ejpam-5709	281	13	k	k	X
ejpam-5709	281	14	=	=	PUNCT
ejpam-5709	281	15	q	q	ADJ
ejpam-5709	281	16	(	(	PUNCT
ejpam-5709	281	17	√	√	NUM
ejpam-5709	281	18	m	m	VERB
ejpam-5709	281	19	)	)	PUNCT
ejpam-5709	281	20	be	be	AUX
ejpam-5709	281	21	an	an	DET
ejpam-5709	281	22	imaginary	imaginary	ADJ
ejpam-5709	281	23	quadratic	quadratic	ADJ
ejpam-5709	281	24	field	field	NOUN
ejpam-5709	281	25	.	.	PUNCT
ejpam-5709	282	1	in	in	ADP
ejpam-5709	282	2	this	this	DET
ejpam-5709	282	3	section	section	NOUN
ejpam-5709	282	4	,	,	PUNCT
ejpam-5709	282	5	we	we	PRON
ejpam-5709	282	6	extend	extend	VERB
ejpam-5709	282	7	theorem	theorem	ADJ
ejpam-5709	282	8	b	b	NOUN
ejpam-5709	282	9	to	to	ADP
ejpam-5709	282	10	any	any	DET
ejpam-5709	282	11	imaginary	imaginary	ADJ
ejpam-5709	282	12	quadratic	quadratic	ADJ
ejpam-5709	282	13	field	field	NOUN
ejpam-5709	282	14	,	,	PUNCT
ejpam-5709	282	15	which	which	PRON
ejpam-5709	282	16	in	in	ADP
ejpam-5709	282	17	turn	turn	NOUN
ejpam-5709	282	18	generalize	generalize	VERB
ejpam-5709	282	19	theorem	theorem	ADJ
ejpam-5709	282	20	c	c	PROPN
ejpam-5709	282	21	and	and	CCONJ
ejpam-5709	282	22	theorem	theorem	VERB
ejpam-5709	282	23	d	d	X
ejpam-5709	282	24	by	by	ADP
ejpam-5709	282	25	considering	consider	VERB
ejpam-5709	282	26	ωπ	ωπ	INTJ
ejpam-5709	282	27	instead	instead	ADV
ejpam-5709	282	28	of	of	ADP
ejpam-5709	282	29	π	π	PROPN
ejpam-5709	282	30	,	,	PUNCT
ejpam-5709	282	31	where	where	SCONJ
ejpam-5709	282	32	ω	ω	PROPN
ejpam-5709	282	33	∈	∈	PROPN
ejpam-5709	282	34	ok\{0	ok\{0	PROPN
ejpam-5709	282	35	}	}	PUNCT
ejpam-5709	282	36	and	and	CCONJ
ejpam-5709	282	37	π	π	PROPN
ejpam-5709	282	38	is	be	AUX
ejpam-5709	282	39	a	a	DET
ejpam-5709	282	40	prime	prime	ADJ
ejpam-5709	282	41	element	element	NOUN
ejpam-5709	282	42	.	.	PUNCT
ejpam-5709	283	1	to	to	PART
ejpam-5709	283	2	prove	prove	VERB
ejpam-5709	283	3	these	these	PRON
ejpam-5709	283	4	,	,	PUNCT
ejpam-5709	283	5	we	we	PRON
ejpam-5709	283	6	first	first	ADV
ejpam-5709	283	7	recall	recall	VERB
ejpam-5709	283	8	the	the	DET
ejpam-5709	283	9	essential	essential	ADJ
ejpam-5709	283	10	three	three	NUM
ejpam-5709	283	11	lemmas	lemmas	ADJ
ejpam-5709	283	12	.	.	PUNCT
ejpam-5709	284	1	lemma	lemma	PROPN
ejpam-5709	284	2	2	2	NUM
ejpam-5709	284	3	.	.	PUNCT
ejpam-5709	285	1	[	[	X
ejpam-5709	285	2	12	12	NUM
ejpam-5709	285	3	]	]	PUNCT
ejpam-5709	285	4	let	let	VERB
ejpam-5709	285	5	f(x	f(x	PROPN
ejpam-5709	285	6	)	)	PUNCT
ejpam-5709	286	1	=	=	PUNCT
ejpam-5709	286	2	αnx	αnx	PROPN
ejpam-5709	286	3	n	n	PROPN
ejpam-5709	286	4	+	+	CCONJ
ejpam-5709	286	5	αn−1x	αn−1x	NUM
ejpam-5709	286	6	n−1	n−1	PROPN
ejpam-5709	286	7	+	+	CCONJ
ejpam-5709	286	8	·	·	PUNCT
ejpam-5709	286	9	·	·	PUNCT
ejpam-5709	287	1	·	·	PUNCT
ejpam-5709	287	2	+	+	NUM
ejpam-5709	287	3	α1x+	α1x+	NOUN
ejpam-5709	287	4	α0	α0	ADJ
ejpam-5709	287	5	∈	∈	PROPN
ejpam-5709	287	6	c[x	c[x	NOUN
ejpam-5709	287	7	]	]	PUNCT
ejpam-5709	287	8	be	be	VERB
ejpam-5709	287	9	such	such	ADJ
ejpam-5709	287	10	that	that	SCONJ
ejpam-5709	287	11	n	n	CCONJ
ejpam-5709	287	12	≥	≥	NOUN
ejpam-5709	287	13	2	2	NUM
ejpam-5709	287	14	and	and	CCONJ
ejpam-5709	287	15	|αi|	|αi|	PRON
ejpam-5709	287	16	≤	≤	NUM
ejpam-5709	287	17	m	m	VERB
ejpam-5709	287	18	(	(	PUNCT
ejpam-5709	287	19	0	0	NUM
ejpam-5709	287	20	≤	≤	NUM
ejpam-5709	288	1	i	i	PRON
ejpam-5709	288	2	≤	≤	ADJ
ejpam-5709	288	3	n−	n−	NOUN
ejpam-5709	288	4	2	2	NUM
ejpam-5709	288	5	)	)	PUNCT
ejpam-5709	288	6	for	for	ADP
ejpam-5709	288	7	some	some	DET
ejpam-5709	288	8	positive	positive	ADJ
ejpam-5709	288	9	real	real	ADJ
ejpam-5709	288	10	number	number	NOUN
ejpam-5709	288	11	m	m	NOUN
ejpam-5709	288	12	.	.	PUNCT
ejpam-5709	289	1	if	if	SCONJ
ejpam-5709	289	2	f(x	f(x	PROPN
ejpam-5709	289	3	)	)	PUNCT
ejpam-5709	289	4	satisfies	satisfie	NOUN
ejpam-5709	289	5	(	(	PUNCT
ejpam-5709	289	6	i	i	NOUN
ejpam-5709	289	7	)	)	PUNCT
ejpam-5709	289	8	re(αn	re(αn	PROPN
ejpam-5709	289	9	)	)	PUNCT
ejpam-5709	289	10	≥	≥	NOUN
ejpam-5709	289	11	1	1	NUM
ejpam-5709	289	12	,	,	PUNCT
ejpam-5709	289	13	re(αn−1	re(αn−1	PROPN
ejpam-5709	289	14	)	)	PUNCT
ejpam-5709	289	15	≥	≥	NOUN
ejpam-5709	289	16	0	0	NUM
ejpam-5709	289	17	,	,	PUNCT
ejpam-5709	289	18	and	and	CCONJ
ejpam-5709	289	19	im(αn−1	im(αn−1	X
ejpam-5709	289	20	)	)	PUNCT
ejpam-5709	289	21	≥	≥	NOUN
ejpam-5709	289	22	0	0	NUM
ejpam-5709	289	23	;	;	PUNCT
ejpam-5709	289	24	and	and	CCONJ
ejpam-5709	289	25	(	(	PUNCT
ejpam-5709	289	26	ii	ii	NOUN
ejpam-5709	289	27	)	)	PUNCT
ejpam-5709	289	28	re(αn−1	re(αn−1	PROPN
ejpam-5709	289	29	)	)	PUNCT
ejpam-5709	289	30	im(αn	im(αn	NOUN
ejpam-5709	289	31	)	)	PUNCT
ejpam-5709	289	32	≥	≥	NOUN
ejpam-5709	289	33	re(αn	re(αn	NOUN
ejpam-5709	289	34	)	)	PUNCT
ejpam-5709	289	35	im(αn−1	im(αn−1	PROPN
ejpam-5709	289	36	)	)	PUNCT
ejpam-5709	289	37	,	,	PUNCT
ejpam-5709	289	38	then	then	ADV
ejpam-5709	289	39	any	any	DET
ejpam-5709	289	40	complex	complex	ADJ
ejpam-5709	289	41	zero	zero	NUM
ejpam-5709	289	42	α	α	NOUN
ejpam-5709	289	43	of	of	ADP
ejpam-5709	289	44	f(x	f(x	PROPN
ejpam-5709	289	45	)	)	PUNCT
ejpam-5709	289	46	satisfies	satisfie	NOUN
ejpam-5709	289	47	either	either	CCONJ
ejpam-5709	289	48	re(α	re(α	NOUN
ejpam-5709	289	49	)	)	PUNCT
ejpam-5709	289	50	<	<	X
ejpam-5709	289	51	0	0	PUNCT
ejpam-5709	290	1	or	or	CCONJ
ejpam-5709	290	2	|α|	|α|	PRON
ejpam-5709	290	3	<	<	X
ejpam-5709	290	4	(	(	PUNCT
ejpam-5709	290	5	1	1	NUM
ejpam-5709	290	6	+	+	CCONJ
ejpam-5709	290	7	√	√	NUM
ejpam-5709	290	8	1	1	NUM
ejpam-5709	290	9	+	+	NUM
ejpam-5709	290	10	4	4	NUM
ejpam-5709	290	11	m	m	NOUN
ejpam-5709	290	12	)	)	PUNCT
ejpam-5709	290	13	/2	/2	PUNCT
ejpam-5709	290	14	.	.	PUNCT
ejpam-5709	291	1	lemma	lemma	PROPN
ejpam-5709	291	2	3	3	X
ejpam-5709	291	3	.	.	PUNCT
ejpam-5709	292	1	[	[	X
ejpam-5709	292	2	8	8	NUM
ejpam-5709	292	3	]	]	PUNCT
ejpam-5709	292	4	let	let	VERB
ejpam-5709	292	5	k	k	PROPN
ejpam-5709	292	6	=	=	PUNCT
ejpam-5709	292	7	q	q	ADJ
ejpam-5709	292	8	(	(	PUNCT
ejpam-5709	292	9	√	√	NUM
ejpam-5709	292	10	m	m	VERB
ejpam-5709	292	11	)	)	PUNCT
ejpam-5709	292	12	be	be	AUX
ejpam-5709	292	13	an	an	DET
ejpam-5709	292	14	imaginary	imaginary	ADJ
ejpam-5709	292	15	quadratic	quadratic	ADJ
ejpam-5709	292	16	field	field	NOUN
ejpam-5709	292	17	with	with	ADP
ejpam-5709	292	18	m	m	PROPN
ejpam-5709	292	19	̸≡	̸≡	ADJ
ejpam-5709	292	20	1	1	NUM
ejpam-5709	292	21	(	(	PUNCT
ejpam-5709	292	22	mod	mod	NOUN
ejpam-5709	292	23	4	4	NUM
ejpam-5709	292	24	)	)	PUNCT
ejpam-5709	292	25	.	.	PUNCT
ejpam-5709	293	1	let	let	VERB
ejpam-5709	293	2	β	β	X
ejpam-5709	293	3	=	=	PUNCT
ejpam-5709	293	4	a+	a+	PUNCT
ejpam-5709	293	5	b	b	X
ejpam-5709	293	6	√	√	NOUN
ejpam-5709	293	7	m	m	VERB
ejpam-5709	293	8	∈	∈	NOUN
ejpam-5709	293	9	ok	ok	INTJ
ejpam-5709	293	10	be	be	AUX
ejpam-5709	293	11	such	such	ADJ
ejpam-5709	293	12	that	that	SCONJ
ejpam-5709	293	13	a	a	DET
ejpam-5709	293	14	≥	≥	NUM
ejpam-5709	293	15	1	1	NUM
ejpam-5709	293	16	+	+	CCONJ
ejpam-5709	293	17	√	√	PROPN
ejpam-5709	293	18	1−m	1−m	NUM
ejpam-5709	293	19	and	and	CCONJ
ejpam-5709	293	20	m	m	NOUN
ejpam-5709	293	21	=	=	ADJ
ejpam-5709	293	22	√	√	PROPN
ejpam-5709	293	23	(	(	PUNCT
ejpam-5709	293	24	max{a	max{a	PROPN
ejpam-5709	293	25	,	,	PUNCT
ejpam-5709	293	26	|b|	|b|	VERB
ejpam-5709	293	27	}	}	PUNCT
ejpam-5709	293	28	−	−	PROPN
ejpam-5709	293	29	1)2	1)2	NUM
ejpam-5709	293	30	−m(d−	−m(d−	NOUN
ejpam-5709	293	31	1)2	1)2	NUM
ejpam-5709	293	32	,	,	PUNCT
ejpam-5709	293	33	(	(	PUNCT
ejpam-5709	293	34	19	19	NUM
ejpam-5709	293	35	)	)	PUNCT
ejpam-5709	294	1	where	where	SCONJ
ejpam-5709	294	2	d	d	NOUN
ejpam-5709	294	3	=	=	SYM
ejpam-5709	294	4	gcd(a	gcd(a	PROPN
ejpam-5709	294	5	,	,	PUNCT
ejpam-5709	294	6	b	b	NOUN
ejpam-5709	294	7	)	)	PUNCT
ejpam-5709	294	8	.	.	PUNCT
ejpam-5709	295	1	then	then	ADV
ejpam-5709	295	2	√	√	INTJ
ejpam-5709	295	3	1−m(|β|	1−m(|β|	NUM
ejpam-5709	296	1	−	−	PROPN
ejpam-5709	296	2	1	1	NUM
ejpam-5709	296	3	)	)	PUNCT
ejpam-5709	296	4	≥	≥	NOUN
ejpam-5709	296	5	m	m	PROPN
ejpam-5709	296	6	.	.	PUNCT
ejpam-5709	297	1	lemma	lemma	PROPN
ejpam-5709	297	2	4	4	NUM
ejpam-5709	297	3	.	.	PUNCT
ejpam-5709	298	1	[	[	X
ejpam-5709	298	2	8	8	NUM
ejpam-5709	298	3	]	]	PUNCT
ejpam-5709	298	4	let	let	VERB
ejpam-5709	298	5	k	k	PROPN
ejpam-5709	298	6	=	=	PUNCT
ejpam-5709	298	7	q	q	ADJ
ejpam-5709	298	8	(	(	PUNCT
ejpam-5709	298	9	√	√	NUM
ejpam-5709	298	10	m	m	VERB
ejpam-5709	298	11	)	)	PUNCT
ejpam-5709	298	12	be	be	AUX
ejpam-5709	298	13	an	an	DET
ejpam-5709	298	14	imaginary	imaginary	ADJ
ejpam-5709	298	15	quadratic	quadratic	ADJ
ejpam-5709	298	16	field	field	NOUN
ejpam-5709	298	17	with	with	ADP
ejpam-5709	298	18	m	m	PROPN
ejpam-5709	298	19	≡	≡	PROPN
ejpam-5709	298	20	1	1	NUM
ejpam-5709	298	21	(	(	PUNCT
ejpam-5709	298	22	mod	mod	NOUN
ejpam-5709	298	23	4	4	NUM
ejpam-5709	298	24	)	)	PUNCT
ejpam-5709	298	25	.	.	PUNCT
ejpam-5709	299	1	let	let	VERB
ejpam-5709	299	2	β	β	X
ejpam-5709	299	3	=	=	PRON
ejpam-5709	299	4	a+	a+	PUNCT
ejpam-5709	299	5	bσm	bσm	NOUN
ejpam-5709	299	6	∈	∈	NOUN
ejpam-5709	299	7	ok	ok	INTJ
ejpam-5709	299	8	be	be	AUX
ejpam-5709	299	9	such	such	ADJ
ejpam-5709	299	10	that	that	SCONJ
ejpam-5709	299	11	a	a	DET
ejpam-5709	299	12	≥	≥	NOUN
ejpam-5709	299	13	1	1	NUM
ejpam-5709	299	14	and	and	CCONJ
ejpam-5709	299	15	m	m	PROPN
ejpam-5709	299	16	=	=	ADJ
ejpam-5709	299	17	√	√	PROPN
ejpam-5709	299	18	(	(	PUNCT
ejpam-5709	299	19	max{a	max{a	PROPN
ejpam-5709	299	20	,	,	PUNCT
ejpam-5709	299	21	|b|	|b|	VERB
ejpam-5709	299	22	}	}	PUNCT
ejpam-5709	299	23	−	−	PROPN
ejpam-5709	299	24	1)2	1)2	NUM
ejpam-5709	299	25	+	+	CCONJ
ejpam-5709	299	26	(	(	PUNCT
ejpam-5709	299	27	max{a	max{a	PROPN
ejpam-5709	299	28	,	,	PUNCT
ejpam-5709	299	29	|b|	|b|	VERB
ejpam-5709	299	30	}	}	PUNCT
ejpam-5709	299	31	−	−	PROPN
ejpam-5709	299	32	1)(d−	1)(d−	NUM
ejpam-5709	299	33	1	1	NUM
ejpam-5709	299	34	)	)	PUNCT
ejpam-5709	300	1	+	+	CCONJ
ejpam-5709	300	2	(	(	PUNCT
ejpam-5709	300	3	d−	d−	PROPN
ejpam-5709	300	4	1)2	1)2	NUM
ejpam-5709	300	5	(	(	PUNCT
ejpam-5709	300	6	1−m	1−m	NUM
ejpam-5709	300	7	4	4	NUM
ejpam-5709	300	8	)	)	PUNCT
ejpam-5709	300	9	,	,	PUNCT
ejpam-5709	300	10	(	(	PUNCT
ejpam-5709	300	11	20	20	NUM
ejpam-5709	300	12	)	)	PUNCT
ejpam-5709	300	13	where	where	SCONJ
ejpam-5709	300	14	d	d	NOUN
ejpam-5709	300	15	=	=	SYM
ejpam-5709	300	16	gcd(a	gcd(a	PROPN
ejpam-5709	300	17	,	,	PUNCT
ejpam-5709	300	18	b	b	NOUN
ejpam-5709	300	19	)	)	PUNCT
ejpam-5709	300	20	.	.	PUNCT
ejpam-5709	301	1	then	then	ADV
ejpam-5709	301	2	√	√	INTJ
ejpam-5709	301	3	(	(	PUNCT
ejpam-5709	301	4	9−m)/4(|β|	9−m)/4(|β|	NUM
ejpam-5709	301	5	−	−	NOUN
ejpam-5709	301	6	1	1	NUM
ejpam-5709	301	7	)	)	PUNCT
ejpam-5709	301	8	≥	≥	NOUN
ejpam-5709	301	9	m	m	NOUN
ejpam-5709	301	10	.	.	PUNCT
ejpam-5709	302	1	if	if	SCONJ
ejpam-5709	302	2	f(x	f(x	PROPN
ejpam-5709	302	3	)	)	PUNCT
ejpam-5709	302	4	is	be	AUX
ejpam-5709	302	5	a	a	DET
ejpam-5709	302	6	nonconstant	nonconstant	ADJ
ejpam-5709	302	7	polynomial	polynomial	NOUN
ejpam-5709	302	8	in	in	ADP
ejpam-5709	302	9	ok	ok	ADJ
ejpam-5709	302	10	[	[	X
ejpam-5709	302	11	x	x	X
ejpam-5709	302	12	]	]	X
ejpam-5709	302	13	,	,	PUNCT
ejpam-5709	302	14	we	we	PRON
ejpam-5709	302	15	say	say	VERB
ejpam-5709	302	16	that	that	SCONJ
ejpam-5709	302	17	f(x	f(x	PROPN
ejpam-5709	302	18	)	)	PUNCT
ejpam-5709	302	19	=	=	SYM
ejpam-5709	302	20	g(x)h(x	g(x)h(x	X
ejpam-5709	302	21	)	)	PUNCT
ejpam-5709	302	22	in	in	ADP
ejpam-5709	302	23	ok	ok	ADJ
ejpam-5709	302	24	[	[	X
ejpam-5709	302	25	x	x	X
ejpam-5709	302	26	]	]	X
ejpam-5709	302	27	is	be	AUX
ejpam-5709	302	28	a	a	DET
ejpam-5709	302	29	proper	proper	ADJ
ejpam-5709	302	30	factorization	factorization	NOUN
ejpam-5709	302	31	if	if	SCONJ
ejpam-5709	302	32	both	both	PRON
ejpam-5709	302	33	g(x	g(x	NOUN
ejpam-5709	302	34	)	)	PUNCT
ejpam-5709	302	35	and	and	CCONJ
ejpam-5709	302	36	h(x	h(x	PROPN
ejpam-5709	302	37	)	)	PUNCT
ejpam-5709	302	38	have	have	VERB
ejpam-5709	302	39	a	a	DET
ejpam-5709	302	40	smaller	small	ADJ
ejpam-5709	302	41	degree	degree	NOUN
ejpam-5709	302	42	than	than	ADP
ejpam-5709	302	43	f(x	f(x	PROPN
ejpam-5709	302	44	)	)	PUNCT
ejpam-5709	302	45	.	.	PUNCT
ejpam-5709	303	1	we	we	PRON
ejpam-5709	303	2	now	now	ADV
ejpam-5709	303	3	proceed	proceed	VERB
ejpam-5709	303	4	to	to	ADP
ejpam-5709	303	5	our	our	PRON
ejpam-5709	303	6	second	second	ADJ
ejpam-5709	303	7	main	main	ADJ
ejpam-5709	303	8	result	result	NOUN
ejpam-5709	303	9	and	and	CCONJ
ejpam-5709	303	10	start	start	VERB
ejpam-5709	303	11	with	with	ADP
ejpam-5709	303	12	the	the	DET
ejpam-5709	303	13	case	case	NOUN
ejpam-5709	303	14	m	m	VERB
ejpam-5709	303	15	̸≡	̸≡	NOUN
ejpam-5709	303	16	1	1	NUM
ejpam-5709	303	17	(	(	PUNCT
ejpam-5709	303	18	mod	mod	PROPN
ejpam-5709	303	19	4	4	NUM
ejpam-5709	303	20	)	)	PUNCT
ejpam-5709	303	21	.	.	PUNCT
ejpam-5709	304	1	theorem	theorem	NOUN
ejpam-5709	304	2	3	3	X
ejpam-5709	304	3	.	.	PUNCT
ejpam-5709	305	1	let	let	VERB
ejpam-5709	305	2	k	k	NOUN
ejpam-5709	305	3	=	=	PUNCT
ejpam-5709	305	4	q	q	ADJ
ejpam-5709	305	5	(	(	PUNCT
ejpam-5709	305	6	√	√	NUM
ejpam-5709	305	7	m	m	VERB
ejpam-5709	305	8	)	)	PUNCT
ejpam-5709	305	9	be	be	AUX
ejpam-5709	305	10	an	an	DET
ejpam-5709	305	11	imaginary	imaginary	ADJ
ejpam-5709	305	12	quadratic	quadratic	ADJ
ejpam-5709	305	13	field	field	NOUN
ejpam-5709	305	14	with	with	ADP
ejpam-5709	305	15	m	m	PROPN
ejpam-5709	305	16	̸≡	̸≡	ADJ
ejpam-5709	305	17	1	1	NUM
ejpam-5709	305	18	(	(	PUNCT
ejpam-5709	305	19	mod	mod	NOUN
ejpam-5709	305	20	4	4	NUM
ejpam-5709	305	21	)	)	PUNCT
ejpam-5709	305	22	.	.	PUNCT
ejpam-5709	306	1	let	let	VERB
ejpam-5709	306	2	β	β	NOUN
ejpam-5709	306	3	=	=	PUNCT
ejpam-5709	306	4	a	a	DET
ejpam-5709	306	5	+	+	X
ejpam-5709	306	6	b	b	NOUN
ejpam-5709	306	7	√	√	NUM
ejpam-5709	306	8	m	m	VERB
ejpam-5709	306	9	∈	∈	NOUN
ejpam-5709	306	10	ok	ok	ADJ
ejpam-5709	306	11	and	and	CCONJ
ejpam-5709	306	12	ω	ω	NUM
ejpam-5709	306	13	∈	∈	PROPN
ejpam-5709	306	14	ok\{0	ok\{0	PROPN
ejpam-5709	306	15	}	}	PUNCT
ejpam-5709	306	16	be	be	AUX
ejpam-5709	306	17	such	such	ADJ
ejpam-5709	306	18	that	that	SCONJ
ejpam-5709	306	19	|β|	|β|	PRON
ejpam-5709	306	20	≥	≥	VERB
ejpam-5709	306	21	2|ω|	2|ω|	NUM
ejpam-5709	306	22	+	+	CCONJ
ejpam-5709	306	23	√	√	NUM
ejpam-5709	306	24	1−m	1−m	NUM
ejpam-5709	306	25	and	and	CCONJ
ejpam-5709	306	26	a	a	DET
ejpam-5709	306	27	≥	≥	NOUN
ejpam-5709	306	28	|ω|+	|ω|+	NOUN
ejpam-5709	306	29	√	√	NUM
ejpam-5709	306	30	1−m	1−m	NUM
ejpam-5709	306	31	.	.	PUNCT
ejpam-5709	307	1	for	for	ADP
ejpam-5709	307	2	a	a	DET
ejpam-5709	307	3	prime	prime	ADJ
ejpam-5709	307	4	element	element	NOUN
ejpam-5709	307	5	π	π	PROPN
ejpam-5709	307	6	of	of	ADP
ejpam-5709	307	7	ok	ok	INTJ
ejpam-5709	307	8	with	with	ADP
ejpam-5709	307	9	|π|	|π|	NOUN
ejpam-5709	307	10	≥	≥	NOUN
ejpam-5709	307	11	|β|	|β|	NOUN
ejpam-5709	307	12	,	,	PUNCT
ejpam-5709	307	13	if	if	SCONJ
ejpam-5709	307	14	ωπ	ωπ	PRON
ejpam-5709	307	15	=	=	PUNCT
ejpam-5709	307	16	αnβ	αnβ	PROPN
ejpam-5709	307	17	n	n	PROPN
ejpam-5709	307	18	+	+	CCONJ
ejpam-5709	307	19	αn−1β	αn−1β	PROPN
ejpam-5709	307	20	n−1	n−1	PROPN
ejpam-5709	307	21	+	+	NUM
ejpam-5709	307	22	·	·	PUNCT
ejpam-5709	307	23	·	·	PUNCT
ejpam-5709	307	24	·	·	PUNCT
ejpam-5709	307	25	+	+	NUM
ejpam-5709	307	26	α1β	α1β	X
ejpam-5709	308	1	+	+	CCONJ
ejpam-5709	308	2	α0	α0	ADJ
ejpam-5709	308	3	=	=	NOUN
ejpam-5709	308	4	:	:	PUNCT
ejpam-5709	308	5	f(β	f(β	NOUN
ejpam-5709	308	6	)	)	PUNCT
ejpam-5709	308	7	is	be	AUX
ejpam-5709	308	8	a	a	DET
ejpam-5709	308	9	base	base	NOUN
ejpam-5709	308	10	-	-	PUNCT
ejpam-5709	308	11	β(c′	β(c′	NUM
ejpam-5709	308	12	)	)	PUNCT
ejpam-5709	308	13	representation	representation	NOUN
ejpam-5709	308	14	with	with	ADP
ejpam-5709	308	15	n	n	PRON
ejpam-5709	308	16	≥	≥	NUM
ejpam-5709	308	17	2	2	NUM
ejpam-5709	308	18	and	and	CCONJ
ejpam-5709	308	19	re(αn	re(αn	NOUN
ejpam-5709	308	20	)	)	PUNCT
ejpam-5709	308	21	≥	≥	NOUN
ejpam-5709	308	22	1	1	NUM
ejpam-5709	308	23	satisfying	satisfy	VERB
ejpam-5709	308	24	the	the	DET
ejpam-5709	308	25	condition	condition	NOUN
ejpam-5709	308	26	(	(	PUNCT
ejpam-5709	308	27	ii	ii	NOUN
ejpam-5709	308	28	)	)	PUNCT
ejpam-5709	308	29	of	of	ADP
ejpam-5709	308	30	lemma	lemma	PROPN
ejpam-5709	308	31	2	2	NUM
ejpam-5709	308	32	,	,	PUNCT
ejpam-5709	308	33	then	then	ADV
ejpam-5709	308	34	f(x	f(x	PROPN
ejpam-5709	308	35	)	)	PUNCT
ejpam-5709	308	36	has	have	VERB
ejpam-5709	308	37	no	no	DET
ejpam-5709	308	38	proper	proper	ADJ
ejpam-5709	308	39	factorization	factorization	NOUN
ejpam-5709	308	40	in	in	ADP
ejpam-5709	308	41	ok	ok	ADJ
ejpam-5709	308	42	[	[	X
ejpam-5709	308	43	x	x	X
ejpam-5709	308	44	]	]	X
ejpam-5709	308	45	.	.	PUNCT
ejpam-5709	309	1	moreover	moreover	ADV
ejpam-5709	309	2	,	,	PUNCT
ejpam-5709	309	3	(	(	PUNCT
ejpam-5709	309	4	i	i	NOUN
ejpam-5709	309	5	)	)	PUNCT
ejpam-5709	309	6	if	if	SCONJ
ejpam-5709	309	7	δ	δ	PROPN
ejpam-5709	309	8	∤	∤	PROPN
ejpam-5709	309	9	f(x	f(x	PROPN
ejpam-5709	309	10	)	)	PUNCT
ejpam-5709	309	11	for	for	ADP
ejpam-5709	309	12	all	all	DET
ejpam-5709	309	13	δ	δ	PROPN
ejpam-5709	309	14	∈	∈	PROPN
ejpam-5709	309	15	ok\u(ok	ok\u(ok	NOUN
ejpam-5709	309	16	)	)	PUNCT
ejpam-5709	309	17	,	,	PUNCT
ejpam-5709	309	18	then	then	ADV
ejpam-5709	309	19	f(x	f(x	PROPN
ejpam-5709	309	20	)	)	PUNCT
ejpam-5709	309	21	is	be	AUX
ejpam-5709	309	22	irreducible	irreducible	ADJ
ejpam-5709	309	23	in	in	ADP
ejpam-5709	309	24	ok	ok	ADJ
ejpam-5709	309	25	[	[	X
ejpam-5709	309	26	x	x	X
ejpam-5709	309	27	]	]	X
ejpam-5709	309	28	;	;	PUNCT
ejpam-5709	309	29	(	(	PUNCT
ejpam-5709	309	30	ii	ii	NOUN
ejpam-5709	309	31	)	)	PUNCT
ejpam-5709	309	32	if	if	SCONJ
ejpam-5709	309	33	ok	ok	ADJ
ejpam-5709	309	34	is	be	AUX
ejpam-5709	309	35	a	a	DET
ejpam-5709	309	36	unique	unique	ADJ
ejpam-5709	309	37	factorization	factorization	NOUN
ejpam-5709	309	38	domain	domain	NOUN
ejpam-5709	309	39	,	,	PUNCT
ejpam-5709	309	40	then	then	ADV
ejpam-5709	309	41	f(x	f(x	PROPN
ejpam-5709	309	42	)	)	PUNCT
ejpam-5709	309	43	is	be	AUX
ejpam-5709	309	44	irreducible	irreducible	ADJ
ejpam-5709	309	45	over	over	ADP
ejpam-5709	309	46	k.	k.	PROPN
ejpam-5709	309	47	p.	p.	PROPN
ejpam-5709	309	48	phetnun	phetnun	PROPN
ejpam-5709	309	49	,	,	PUNCT
ejpam-5709	309	50	n.r	n.r	PROPN
ejpam-5709	309	51	.	.	PROPN
ejpam-5709	309	52	kanasri	kanasri	PROPN
ejpam-5709	309	53	/	/	SYM
ejpam-5709	309	54	eur	eur	PROPN
ejpam-5709	309	55	.	.	PUNCT
ejpam-5709	310	1	j.	j.	PROPN
ejpam-5709	310	2	pure	pure	PROPN
ejpam-5709	310	3	appl	appl	PROPN
ejpam-5709	310	4	.	.	PROPN
ejpam-5709	310	5	math	math	PROPN
ejpam-5709	310	6	,	,	PUNCT
ejpam-5709	310	7	18	18	NUM
ejpam-5709	310	8	(	(	PUNCT
ejpam-5709	310	9	1	1	NUM
ejpam-5709	310	10	)	)	PUNCT
ejpam-5709	310	11	(	(	PUNCT
ejpam-5709	310	12	2025	2025	NUM
ejpam-5709	310	13	)	)	PUNCT
ejpam-5709	310	14	,	,	PUNCT
ejpam-5709	310	15	5709	5709	NUM
ejpam-5709	310	16	11	11	NUM
ejpam-5709	310	17	of	of	ADP
ejpam-5709	310	18	15	15	NUM
ejpam-5709	310	19	proof	proof	NOUN
ejpam-5709	310	20	.	.	PUNCT
ejpam-5709	311	1	suppose	suppose	VERB
ejpam-5709	311	2	to	to	ADP
ejpam-5709	311	3	the	the	DET
ejpam-5709	311	4	contrary	contrary	NOUN
ejpam-5709	311	5	that	that	SCONJ
ejpam-5709	311	6	f(x	f(x	PROPN
ejpam-5709	311	7	)	)	PUNCT
ejpam-5709	311	8	has	have	VERB
ejpam-5709	311	9	proper	proper	ADJ
ejpam-5709	311	10	factorization	factorization	NOUN
ejpam-5709	311	11	in	in	ADP
ejpam-5709	311	12	ok	ok	ADJ
ejpam-5709	311	13	[	[	X
ejpam-5709	311	14	x	x	X
ejpam-5709	311	15	]	]	X
ejpam-5709	311	16	.	.	PUNCT
ejpam-5709	312	1	then	then	ADV
ejpam-5709	312	2	f(x	f(x	PROPN
ejpam-5709	312	3	)	)	PUNCT
ejpam-5709	312	4	=	=	SYM
ejpam-5709	312	5	g(x)h(x	g(x)h(x	X
ejpam-5709	312	6	)	)	PUNCT
ejpam-5709	312	7	for	for	ADP
ejpam-5709	312	8	some	some	DET
ejpam-5709	312	9	nonconstant	nonconstant	ADJ
ejpam-5709	312	10	polynomials	polynomial	NOUN
ejpam-5709	312	11	g(x	g(x	NOUN
ejpam-5709	312	12	)	)	PUNCT
ejpam-5709	312	13	and	and	CCONJ
ejpam-5709	312	14	h(x	h(x	PROPN
ejpam-5709	312	15	)	)	PUNCT
ejpam-5709	312	16	in	in	ADP
ejpam-5709	312	17	ok	ok	ADJ
ejpam-5709	312	18	[	[	X
ejpam-5709	312	19	x	x	X
ejpam-5709	312	20	]	]	X
ejpam-5709	312	21	,	,	PUNCT
ejpam-5709	312	22	so	so	CCONJ
ejpam-5709	312	23	ωπ	ωπ	PROPN
ejpam-5709	312	24	=	=	PROPN
ejpam-5709	312	25	g(β)h(β	g(β)h(β	NUM
ejpam-5709	312	26	)	)	PUNCT
ejpam-5709	312	27	.	.	PUNCT
ejpam-5709	313	1	since	since	SCONJ
ejpam-5709	313	2	π	π	PROPN
ejpam-5709	313	3	is	be	AUX
ejpam-5709	313	4	a	a	DET
ejpam-5709	313	5	prime	prime	ADJ
ejpam-5709	313	6	element	element	NOUN
ejpam-5709	313	7	,	,	PUNCT
ejpam-5709	313	8	either	either	CCONJ
ejpam-5709	313	9	π	π	PROPN
ejpam-5709	313	10	|	|	ADV
ejpam-5709	313	11	g(β	g(β	PROPN
ejpam-5709	313	12	)	)	PUNCT
ejpam-5709	313	13	and	and	CCONJ
ejpam-5709	313	14	h(β	h(β	PROPN
ejpam-5709	313	15	)	)	PUNCT
ejpam-5709	313	16	|	|	ADV
ejpam-5709	313	17	ω	ω	NUM
ejpam-5709	313	18	or	or	CCONJ
ejpam-5709	313	19	π	π	PROPN
ejpam-5709	313	20	|	|	ADV
ejpam-5709	313	21	h(β	h(β	NOUN
ejpam-5709	313	22	)	)	PUNCT
ejpam-5709	313	23	and	and	CCONJ
ejpam-5709	313	24	g(β	g(β	PROPN
ejpam-5709	313	25	)	)	PUNCT
ejpam-5709	314	1	|	|	ADV
ejpam-5709	314	2	ω	ω	X
ejpam-5709	314	3	.	.	PUNCT
ejpam-5709	315	1	this	this	PRON
ejpam-5709	315	2	implies	imply	VERB
ejpam-5709	315	3	that	that	SCONJ
ejpam-5709	315	4	either	either	CCONJ
ejpam-5709	315	5	|ω|	|ω|	VERB
ejpam-5709	315	6	≥	≥	PRON
ejpam-5709	315	7	|h(β)|	|h(β)|	NOUN
ejpam-5709	315	8	or	or	CCONJ
ejpam-5709	315	9	|ω|	|ω|	VERB
ejpam-5709	315	10	≥	≥	NUM
ejpam-5709	315	11	|g(β)|	|g(β)|	NOUN
ejpam-5709	315	12	.	.	NOUN
ejpam-5709	315	13	without	without	ADP
ejpam-5709	315	14	loss	loss	NOUN
ejpam-5709	315	15	of	of	ADP
ejpam-5709	315	16	generality	generality	NOUN
ejpam-5709	315	17	,	,	PUNCT
ejpam-5709	315	18	we	we	PRON
ejpam-5709	315	19	may	may	AUX
ejpam-5709	315	20	assume	assume	VERB
ejpam-5709	315	21	that	that	SCONJ
ejpam-5709	315	22	|ω|	|ω|	VERB
ejpam-5709	315	23	≥	≥	PRON
ejpam-5709	315	24	|g(β)|	|g(β)|	NOUN
ejpam-5709	315	25	.	.	PROPN
ejpam-5709	315	26	since	since	SCONJ
ejpam-5709	315	27	αi	αi	PROPN
ejpam-5709	315	28	∈	∈	PROPN
ejpam-5709	315	29	c′	c′	PROPN
ejpam-5709	315	30	for	for	ADP
ejpam-5709	315	31	all	all	PRON
ejpam-5709	315	32	i	i	PRON
ejpam-5709	315	33	∈	∈	PROPN
ejpam-5709	315	34	{	{	PUNCT
ejpam-5709	315	35	0	0	NUM
ejpam-5709	315	36	,	,	PUNCT
ejpam-5709	315	37	1	1	NUM
ejpam-5709	315	38	,	,	PUNCT
ejpam-5709	315	39	.	.	PUNCT
ejpam-5709	315	40	.	.	PUNCT
ejpam-5709	315	41	.	.	PUNCT
ejpam-5709	316	1	,	,	PUNCT
ejpam-5709	316	2	n−	n−	NOUN
ejpam-5709	316	3	1	1	NUM
ejpam-5709	316	4	}	}	PUNCT
ejpam-5709	316	5	,	,	PUNCT
ejpam-5709	316	6	we	we	PRON
ejpam-5709	316	7	have	have	VERB
ejpam-5709	316	8	|αi|	|αi|	NUM
ejpam-5709	316	9	≤	≤	NUM
ejpam-5709	316	10	m	m	VERB
ejpam-5709	316	11	for	for	ADP
ejpam-5709	316	12	all	all	PRON
ejpam-5709	316	13	i	i	PRON
ejpam-5709	316	14	∈	∈	PROPN
ejpam-5709	316	15	{	{	PUNCT
ejpam-5709	316	16	0	0	NUM
ejpam-5709	316	17	,	,	PUNCT
ejpam-5709	316	18	1	1	NUM
ejpam-5709	316	19	,	,	PUNCT
ejpam-5709	316	20	.	.	PUNCT
ejpam-5709	316	21	.	.	PUNCT
ejpam-5709	317	1	.	.	PUNCT
ejpam-5709	318	1	,	,	PUNCT
ejpam-5709	318	2	n−	n−	NOUN
ejpam-5709	318	3	1	1	NUM
ejpam-5709	318	4	}	}	PUNCT
ejpam-5709	318	5	,	,	PUNCT
ejpam-5709	318	6	where	where	SCONJ
ejpam-5709	318	7	m	m	NOUN
ejpam-5709	318	8	is	be	AUX
ejpam-5709	318	9	defined	define	VERB
ejpam-5709	318	10	as	as	ADP
ejpam-5709	318	11	in	in	ADP
ejpam-5709	318	12	(	(	PUNCT
ejpam-5709	318	13	19	19	NUM
ejpam-5709	318	14	)	)	PUNCT
ejpam-5709	318	15	.	.	PUNCT
ejpam-5709	319	1	we	we	PRON
ejpam-5709	319	2	now	now	ADV
ejpam-5709	319	3	show	show	VERB
ejpam-5709	319	4	that	that	SCONJ
ejpam-5709	319	5	|β|	|β|	PRON
ejpam-5709	319	6	−	−	PROPN
ejpam-5709	319	7	1	1	NUM
ejpam-5709	319	8	+	+	CCONJ
ejpam-5709	319	9	√	√	NUM
ejpam-5709	319	10	1	1	NUM
ejpam-5709	320	1	+	+	NUM
ejpam-5709	320	2	4	4	NUM
ejpam-5709	320	3	m	m	NUM
ejpam-5709	320	4	2	2	NUM
ejpam-5709	320	5	≥	≥	NOUN
ejpam-5709	320	6	|ω|	|ω|	PROPN
ejpam-5709	320	7	.	.	PUNCT
ejpam-5709	321	1	(	(	PUNCT
ejpam-5709	321	2	21	21	NUM
ejpam-5709	321	3	)	)	PUNCT
ejpam-5709	321	4	since	since	SCONJ
ejpam-5709	321	5	|β|	|β|	PRON
ejpam-5709	321	6	≥	≥	VERB
ejpam-5709	321	7	2|ω|	2|ω|	NUM
ejpam-5709	321	8	+	+	CCONJ
ejpam-5709	321	9	√	√	NUM
ejpam-5709	321	10	1−m	1−m	NUM
ejpam-5709	321	11	,	,	PUNCT
ejpam-5709	321	12	we	we	PRON
ejpam-5709	321	13	have	have	VERB
ejpam-5709	321	14	|β|2	|β|2	VERB
ejpam-5709	321	15	−	−	PROPN
ejpam-5709	322	1	(	(	PUNCT
ejpam-5709	322	2	2|ω|+	2|ω|+	NUM
ejpam-5709	322	3	1	1	NUM
ejpam-5709	322	4	+	+	CCONJ
ejpam-5709	322	5	√	√	NUM
ejpam-5709	322	6	1−m	1−m	NUM
ejpam-5709	322	7	)	)	PUNCT
ejpam-5709	322	8	|β|	|β|	PROPN
ejpam-5709	322	9	+	+	CCONJ
ejpam-5709	322	10	2|ω|	2|ω|	NUM
ejpam-5709	322	11	+	+	CCONJ
ejpam-5709	322	12	√	√	NUM
ejpam-5709	322	13	1−m	1−m	NUM
ejpam-5709	322	14	=	=	SYM
ejpam-5709	322	15	(	(	PUNCT
ejpam-5709	322	16	|β|	|β|	NOUN
ejpam-5709	322	17	−	−	PROPN
ejpam-5709	322	18	1	1	NUM
ejpam-5709	322	19	)	)	PUNCT
ejpam-5709	322	20	[	[	PUNCT
ejpam-5709	322	21	|β|	|β|	NOUN
ejpam-5709	322	22	−	−	PROPN
ejpam-5709	322	23	(	(	PUNCT
ejpam-5709	322	24	2|ω|+	2|ω|+	NUM
ejpam-5709	322	25	√	√	NUM
ejpam-5709	322	26	1−m	1−m	NUM
ejpam-5709	322	27	)	)	PUNCT
ejpam-5709	322	28	]	]	PUNCT
ejpam-5709	322	29	≥	≥	NOUN
ejpam-5709	322	30	0	0	NUM
ejpam-5709	322	31	.	.	PUNCT
ejpam-5709	323	1	as	as	ADP
ejpam-5709	323	2	|ω|2	|ω|2	PROPN
ejpam-5709	323	3	+	+	CCONJ
ejpam-5709	323	4	|ω|	|ω|	NUM
ejpam-5709	323	5	≥	≥	NOUN
ejpam-5709	323	6	2|ω|	2|ω|	NUM
ejpam-5709	323	7	,	,	PUNCT
ejpam-5709	323	8	we	we	PRON
ejpam-5709	323	9	obtain	obtain	VERB
ejpam-5709	323	10	4	4	NUM
ejpam-5709	323	11	[	[	PUNCT
ejpam-5709	323	12	|β|2	|β|2	ADJ
ejpam-5709	323	13	−	−	PROPN
ejpam-5709	323	14	(	(	PUNCT
ejpam-5709	323	15	2|ω|+	2|ω|+	NUM
ejpam-5709	323	16	1	1	NUM
ejpam-5709	323	17	+	+	CCONJ
ejpam-5709	323	18	√	√	NUM
ejpam-5709	323	19	1−m	1−m	NUM
ejpam-5709	323	20	)	)	PUNCT
ejpam-5709	323	21	|β|+	|β|+	PROPN
ejpam-5709	323	22	|ω|2	|ω|2	PROPN
ejpam-5709	323	23	+	+	CCONJ
ejpam-5709	323	24	|ω|+	|ω|+	PROPN
ejpam-5709	323	25	√	√	PROPN
ejpam-5709	323	26	1−m	1−m	NUM
ejpam-5709	323	27	]	]	PUNCT
ejpam-5709	323	28	≥	≥	X
ejpam-5709	323	29	0	0	NUM
ejpam-5709	324	1	and	and	CCONJ
ejpam-5709	324	2	so	so	ADV
ejpam-5709	324	3	[	[	X
ejpam-5709	324	4	2|β|	2|β|	NUM
ejpam-5709	324	5	−	−	NOUN
ejpam-5709	324	6	(	(	PUNCT
ejpam-5709	324	7	2|ω|+	2|ω|+	NUM
ejpam-5709	324	8	1)]2	1)]2	NUM
ejpam-5709	324	9	=	=	SYM
ejpam-5709	324	10	4|β|2−4(2|ω|+1)|β|+4|ω|2	4|β|2−4(2|ω|+1)|β|+4|ω|2	NUM
ejpam-5709	325	1	+	+	NOUN
ejpam-5709	325	2	4|ω|+1	4|ω|+1	PROPN
ejpam-5709	325	3	≥	≥	NUM
ejpam-5709	325	4	1	1	NUM
ejpam-5709	325	5	+	+	NOUN
ejpam-5709	325	6	4	4	NUM
ejpam-5709	325	7	√	√	NUM
ejpam-5709	325	8	1−m	1−m	NUM
ejpam-5709	325	9	(	(	PUNCT
ejpam-5709	325	10	|β|	|β|	NOUN
ejpam-5709	325	11	−	−	PROPN
ejpam-5709	325	12	1	1	NUM
ejpam-5709	325	13	)	)	PUNCT
ejpam-5709	325	14	.	.	PUNCT
ejpam-5709	326	1	again	again	ADV
ejpam-5709	326	2	,	,	PUNCT
ejpam-5709	326	3	|β|	|β|	PRON
ejpam-5709	326	4	≥	≥	NOUN
ejpam-5709	327	1	2|ω|+	2|ω|+	NUM
ejpam-5709	327	2	√	√	NUM
ejpam-5709	327	3	1−m	1−m	NUM
ejpam-5709	327	4	implies	imply	VERB
ejpam-5709	327	5	2|β|	2|β|	NUM
ejpam-5709	327	6	−	−	PROPN
ejpam-5709	327	7	(	(	PUNCT
ejpam-5709	327	8	2|ω|+1	2|ω|+1	NUM
ejpam-5709	327	9	)	)	PUNCT
ejpam-5709	327	10	>	>	X
ejpam-5709	327	11	0	0	PUNCT
ejpam-5709	327	12	and	and	CCONJ
ejpam-5709	327	13	thus	thus	ADV
ejpam-5709	327	14	,	,	PUNCT
ejpam-5709	327	15	2|β|	2|β|	NUM
ejpam-5709	327	16	−	−	PROPN
ejpam-5709	327	17	(	(	PUNCT
ejpam-5709	327	18	2|ω|+1	2|ω|+1	NUM
ejpam-5709	327	19	)	)	PUNCT
ejpam-5709	327	20	≥	≥	NOUN
ejpam-5709	328	1	√	√	NUM
ejpam-5709	328	2	1	1	NUM
ejpam-5709	329	1	+	+	CCONJ
ejpam-5709	329	2	4	4	NUM
ejpam-5709	329	3	√	√	NUM
ejpam-5709	329	4	1−m	1−m	NUM
ejpam-5709	329	5	(	(	PUNCT
ejpam-5709	329	6	|β|	|β|	NOUN
ejpam-5709	329	7	−	−	PROPN
ejpam-5709	329	8	1	1	NUM
ejpam-5709	329	9	)	)	PUNCT
ejpam-5709	329	10	.	.	PUNCT
ejpam-5709	330	1	hence	hence	ADV
ejpam-5709	330	2	|β|	|β|	PRON
ejpam-5709	330	3	≥	≥	NUM
ejpam-5709	330	4	2|ω|+	2|ω|+	NUM
ejpam-5709	330	5	1	1	NUM
ejpam-5709	330	6	+	+	CCONJ
ejpam-5709	330	7	√	√	NUM
ejpam-5709	330	8	1	1	NUM
ejpam-5709	330	9	+	+	CCONJ
ejpam-5709	330	10	4	4	NUM
ejpam-5709	330	11	√	√	NUM
ejpam-5709	330	12	1−m	1−m	NUM
ejpam-5709	330	13	(	(	PUNCT
ejpam-5709	330	14	|β|	|β|	NOUN
ejpam-5709	330	15	−	−	PROPN
ejpam-5709	330	16	1	1	NUM
ejpam-5709	330	17	)	)	PUNCT
ejpam-5709	330	18	2	2	NUM
ejpam-5709	330	19	.	.	PUNCT
ejpam-5709	331	1	as	as	ADP
ejpam-5709	331	2	a	a	DET
ejpam-5709	331	3	≥	≥	NOUN
ejpam-5709	331	4	|ω|+	|ω|+	NOUN
ejpam-5709	331	5	√	√	NOUN
ejpam-5709	331	6	1−m	1−m	NUM
ejpam-5709	331	7	,	,	PUNCT
ejpam-5709	331	8	it	it	PRON
ejpam-5709	331	9	follows	follow	VERB
ejpam-5709	331	10	from	from	ADP
ejpam-5709	331	11	lemma	lemma	PROPN
ejpam-5709	331	12	3	3	NUM
ejpam-5709	331	13	that	that	PRON
ejpam-5709	331	14	√	√	VERB
ejpam-5709	331	15	1−m	1−m	NUM
ejpam-5709	331	16	(	(	PUNCT
ejpam-5709	331	17	|β|	|β|	NOUN
ejpam-5709	331	18	−	−	PROPN
ejpam-5709	331	19	1	1	NUM
ejpam-5709	331	20	)	)	PUNCT
ejpam-5709	331	21	≥	≥	NOUN
ejpam-5709	331	22	m	m	PROPN
ejpam-5709	331	23	.	.	PUNCT
ejpam-5709	332	1	therefore	therefore	ADV
ejpam-5709	332	2	,	,	PUNCT
ejpam-5709	332	3	|β|	|β|	X
ejpam-5709	332	4	≥	≥	NOUN
ejpam-5709	332	5	2|ω|+	2|ω|+	NUM
ejpam-5709	332	6	1	1	NUM
ejpam-5709	332	7	+	+	CCONJ
ejpam-5709	332	8	√	√	NUM
ejpam-5709	332	9	1	1	NUM
ejpam-5709	333	1	+	+	NUM
ejpam-5709	333	2	4	4	NUM
ejpam-5709	333	3	m	m	NOUN
ejpam-5709	333	4	2	2	NUM
ejpam-5709	333	5	=	=	SYM
ejpam-5709	333	6	|ω|+	|ω|+	NOUN
ejpam-5709	333	7	1	1	NUM
ejpam-5709	333	8	+	+	CCONJ
ejpam-5709	333	9	√	√	NUM
ejpam-5709	333	10	1	1	NUM
ejpam-5709	333	11	+	+	NUM
ejpam-5709	333	12	4	4	NUM
ejpam-5709	333	13	m	m	NUM
ejpam-5709	333	14	2	2	NUM
ejpam-5709	333	15	,	,	PUNCT
ejpam-5709	333	16	which	which	PRON
ejpam-5709	333	17	proves	prove	VERB
ejpam-5709	333	18	(	(	PUNCT
ejpam-5709	333	19	21	21	NUM
ejpam-5709	333	20	)	)	PUNCT
ejpam-5709	333	21	.	.	PUNCT
ejpam-5709	334	1	now	now	ADV
ejpam-5709	334	2	,	,	PUNCT
ejpam-5709	334	3	we	we	PRON
ejpam-5709	334	4	have	have	VERB
ejpam-5709	334	5	that	that	DET
ejpam-5709	334	6	deg	deg	NOUN
ejpam-5709	334	7	g(x	g(x	NOUN
ejpam-5709	334	8	)	)	PUNCT
ejpam-5709	334	9	≥	≥	NOUN
ejpam-5709	334	10	1	1	NUM
ejpam-5709	334	11	,	,	PUNCT
ejpam-5709	334	12	so	so	ADV
ejpam-5709	334	13	g(x	g(x	NOUN
ejpam-5709	334	14	)	)	PUNCT
ejpam-5709	334	15	can	can	AUX
ejpam-5709	334	16	be	be	AUX
ejpam-5709	334	17	expressed	express	VERB
ejpam-5709	334	18	in	in	ADP
ejpam-5709	334	19	the	the	DET
ejpam-5709	334	20	form	form	NOUN
ejpam-5709	334	21	g(x	g(x	NOUN
ejpam-5709	334	22	)	)	PUNCT
ejpam-5709	335	1	=	=	SYM
ejpam-5709	335	2	ε	ε	PROPN
ejpam-5709	335	3	∏	∏	PROPN
ejpam-5709	335	4	i(x−	i(x−	PROPN
ejpam-5709	335	5	γi	γi	NOUN
ejpam-5709	335	6	)	)	PUNCT
ejpam-5709	335	7	,	,	PUNCT
ejpam-5709	335	8	where	where	SCONJ
ejpam-5709	335	9	ε	ε	PROPN
ejpam-5709	335	10	∈	∈	PROPN
ejpam-5709	335	11	ok	ok	INTJ
ejpam-5709	335	12	is	be	AUX
ejpam-5709	335	13	the	the	DET
ejpam-5709	335	14	leading	leading	ADJ
ejpam-5709	335	15	coefficient	coefficient	NOUN
ejpam-5709	335	16	of	of	ADP
ejpam-5709	335	17	g(x	g(x	NOUN
ejpam-5709	335	18	)	)	PUNCT
ejpam-5709	335	19	and	and	CCONJ
ejpam-5709	335	20	the	the	DET
ejpam-5709	335	21	product	product	NOUN
ejpam-5709	335	22	is	be	AUX
ejpam-5709	335	23	over	over	ADP
ejpam-5709	335	24	the	the	DET
ejpam-5709	335	25	set	set	NOUN
ejpam-5709	335	26	of	of	ADP
ejpam-5709	335	27	complex	complex	ADJ
ejpam-5709	335	28	zeros	zero	NOUN
ejpam-5709	335	29	of	of	ADP
ejpam-5709	335	30	g(x	g(x	NOUN
ejpam-5709	335	31	)	)	PUNCT
ejpam-5709	335	32	.	.	PUNCT
ejpam-5709	336	1	it	it	PRON
ejpam-5709	336	2	follows	follow	VERB
ejpam-5709	336	3	from	from	ADP
ejpam-5709	336	4	lemma	lemma	PROPN
ejpam-5709	336	5	2	2	NUM
ejpam-5709	336	6	that	that	PRON
ejpam-5709	336	7	any	any	DET
ejpam-5709	336	8	zero	zero	NUM
ejpam-5709	336	9	γ	γ	NOUN
ejpam-5709	336	10	of	of	ADP
ejpam-5709	336	11	g(x	g(x	NOUN
ejpam-5709	336	12	)	)	PUNCT
ejpam-5709	336	13	satisfies	satisfie	NOUN
ejpam-5709	336	14	either	either	DET
ejpam-5709	336	15	re(γ	re(γ	NOUN
ejpam-5709	336	16	)	)	PUNCT
ejpam-5709	336	17	<	<	X
ejpam-5709	336	18	0	0	PUNCT
ejpam-5709	337	1	or	or	CCONJ
ejpam-5709	337	2	|γ|	|γ|	PRON
ejpam-5709	337	3	<	<	X
ejpam-5709	337	4	1	1	NUM
ejpam-5709	337	5	+	+	CCONJ
ejpam-5709	337	6	√	√	NUM
ejpam-5709	337	7	1	1	NUM
ejpam-5709	337	8	+	+	NUM
ejpam-5709	337	9	4	4	NUM
ejpam-5709	337	10	m	m	NUM
ejpam-5709	337	11	2	2	NUM
ejpam-5709	337	12	.	.	PUNCT
ejpam-5709	338	1	(	(	PUNCT
ejpam-5709	338	2	22	22	NUM
ejpam-5709	338	3	)	)	PUNCT
ejpam-5709	338	4	in	in	ADP
ejpam-5709	338	5	the	the	DET
ejpam-5709	338	6	former	former	ADJ
ejpam-5709	338	7	case	case	NOUN
ejpam-5709	338	8	,	,	PUNCT
ejpam-5709	338	9	we	we	PRON
ejpam-5709	338	10	have	have	VERB
ejpam-5709	338	11	|β	|β	NOUN
ejpam-5709	338	12	−	−	PROPN
ejpam-5709	338	13	γ|	γ|	PROPN
ejpam-5709	338	14	≥	≥	X
ejpam-5709	338	15	re(β	re(β	VERB
ejpam-5709	338	16	−	−	PROPN
ejpam-5709	338	17	γ	γ	X
ejpam-5709	338	18	)	)	PUNCT
ejpam-5709	338	19	=	=	PUNCT
ejpam-5709	338	20	re(β	re(β	X
ejpam-5709	338	21	)	)	PUNCT
ejpam-5709	338	22	−	−	NOUN
ejpam-5709	338	23	re(γ	re(γ	NOUN
ejpam-5709	338	24	)	)	PUNCT
ejpam-5709	338	25	>	>	X
ejpam-5709	338	26	re(β	re(β	X
ejpam-5709	338	27	)	)	PUNCT
ejpam-5709	338	28	=	=	PUNCT
ejpam-5709	338	29	a	a	DET
ejpam-5709	338	30	≥	≥	NOUN
ejpam-5709	338	31	|ω|+	|ω|+	NOUN
ejpam-5709	338	32	√	√	PROPN
ejpam-5709	338	33	1−m	1−m	NUM
ejpam-5709	338	34	>	>	SYM
ejpam-5709	338	35	|ω|	|ω|	PROPN
ejpam-5709	338	36	.	.	PROPN
ejpam-5709	339	1	in	in	ADP
ejpam-5709	339	2	the	the	DET
ejpam-5709	339	3	latter	latter	ADJ
ejpam-5709	339	4	case	case	NOUN
ejpam-5709	339	5	,	,	PUNCT
ejpam-5709	339	6	we	we	PRON
ejpam-5709	339	7	obtain	obtain	VERB
ejpam-5709	339	8	by	by	ADP
ejpam-5709	339	9	using	use	VERB
ejpam-5709	339	10	(	(	PUNCT
ejpam-5709	339	11	21	21	NUM
ejpam-5709	339	12	)	)	PUNCT
ejpam-5709	339	13	and	and	CCONJ
ejpam-5709	339	14	(	(	PUNCT
ejpam-5709	339	15	22	22	NUM
ejpam-5709	339	16	)	)	PUNCT
ejpam-5709	339	17	that	that	PRON
ejpam-5709	339	18	|β	|β	VERB
ejpam-5709	339	19	−	−	PROPN
ejpam-5709	339	20	γ|	γ|	PROPN
ejpam-5709	339	21	≥	≥	NOUN
ejpam-5709	339	22	|β|	|β|	NOUN
ejpam-5709	340	1	−	−	PROPN
ejpam-5709	341	1	|γ|	|γ|	INTJ
ejpam-5709	341	2	>	>	X
ejpam-5709	341	3	|β|	|β|	NOUN
ejpam-5709	341	4	−	−	NUM
ejpam-5709	341	5	1	1	NUM
ejpam-5709	342	1	+	+	CCONJ
ejpam-5709	342	2	√	√	NUM
ejpam-5709	342	3	1	1	NUM
ejpam-5709	342	4	+	+	NUM
ejpam-5709	342	5	4	4	NUM
ejpam-5709	342	6	m	m	NUM
ejpam-5709	342	7	2	2	NUM
ejpam-5709	342	8	≥	≥	NOUN
ejpam-5709	342	9	|ω|	|ω|	PROPN
ejpam-5709	342	10	.	.	PROPN
ejpam-5709	342	11	from	from	ADP
ejpam-5709	342	12	both	both	DET
ejpam-5709	342	13	cases	case	NOUN
ejpam-5709	342	14	and	and	CCONJ
ejpam-5709	342	15	|ε|	|ε|	DET
ejpam-5709	342	16	≥	≥	NOUN
ejpam-5709	342	17	1	1	NUM
ejpam-5709	342	18	,	,	PUNCT
ejpam-5709	342	19	we	we	PRON
ejpam-5709	342	20	deduce	deduce	VERB
ejpam-5709	342	21	that	that	SCONJ
ejpam-5709	342	22	|ω|	|ω|	VERB
ejpam-5709	342	23	≥	≥	NUM
ejpam-5709	342	24	|g(β)|	|g(β)|	PROPN
ejpam-5709	342	25	=	=	SYM
ejpam-5709	342	26	|ε|	|ε|	SYM
ejpam-5709	342	27	∏	∏	PROPN
ejpam-5709	342	28	i	i	PROPN
ejpam-5709	342	29	|β	|β	VERB
ejpam-5709	342	30	−	−	PROPN
ejpam-5709	342	31	γi|	γi|	NOUN
ejpam-5709	342	32	≥	≥	VERB
ejpam-5709	342	33	∏	∏	NUM
ejpam-5709	342	34	i	i	PRON
ejpam-5709	342	35	|β	|β	VERB
ejpam-5709	342	36	−	−	PROPN
ejpam-5709	342	37	γi|	γi|	NOUN
ejpam-5709	342	38	>	>	X
ejpam-5709	342	39	|ω|	|ω|	PROPN
ejpam-5709	342	40	,	,	PUNCT
ejpam-5709	342	41	which	which	PRON
ejpam-5709	342	42	is	be	AUX
ejpam-5709	342	43	a	a	DET
ejpam-5709	342	44	contradiction	contradiction	NOUN
ejpam-5709	342	45	.	.	PUNCT
ejpam-5709	343	1	we	we	PRON
ejpam-5709	343	2	have	have	VERB
ejpam-5709	343	3	that	that	DET
ejpam-5709	343	4	f(x	f(x	PROPN
ejpam-5709	343	5	)	)	PUNCT
ejpam-5709	343	6	has	have	VERB
ejpam-5709	343	7	no	no	DET
ejpam-5709	343	8	proper	proper	ADJ
ejpam-5709	343	9	factorization	factorization	NOUN
ejpam-5709	343	10	in	in	ADP
ejpam-5709	343	11	ok	ok	ADJ
ejpam-5709	343	12	[	[	X
ejpam-5709	343	13	x	x	X
ejpam-5709	343	14	]	]	X
ejpam-5709	343	15	.	.	PUNCT
ejpam-5709	344	1	then	then	ADV
ejpam-5709	344	2	,	,	PUNCT
ejpam-5709	344	3	(	(	PUNCT
ejpam-5709	344	4	i	i	NOUN
ejpam-5709	344	5	)	)	PUNCT
ejpam-5709	344	6	is	be	AUX
ejpam-5709	344	7	true	true	ADJ
ejpam-5709	344	8	.	.	PUNCT
ejpam-5709	345	1	if	if	SCONJ
ejpam-5709	345	2	ok	ok	ADJ
ejpam-5709	345	3	is	be	AUX
ejpam-5709	345	4	a	a	DET
ejpam-5709	345	5	unique	unique	ADJ
ejpam-5709	345	6	factorization	factorization	NOUN
ejpam-5709	345	7	domain	domain	NOUN
ejpam-5709	345	8	,	,	PUNCT
ejpam-5709	345	9	f(x	f(x	PROPN
ejpam-5709	345	10	)	)	PUNCT
ejpam-5709	345	11	is	be	AUX
ejpam-5709	345	12	irreducible	irreducible	ADJ
ejpam-5709	345	13	over	over	ADP
ejpam-5709	345	14	k.	k.	PROPN
ejpam-5709	345	15	to	to	PART
ejpam-5709	345	16	see	see	VERB
ejpam-5709	345	17	this	this	PRON
ejpam-5709	345	18	,	,	PUNCT
ejpam-5709	345	19	suppose	suppose	VERB
ejpam-5709	345	20	to	to	ADP
ejpam-5709	345	21	the	the	DET
ejpam-5709	345	22	p.	p.	PROPN
ejpam-5709	345	23	phetnun	phetnun	PROPN
ejpam-5709	345	24	,	,	PUNCT
ejpam-5709	345	25	n.r	n.r	PROPN
ejpam-5709	345	26	.	.	PROPN
ejpam-5709	345	27	kanasri	kanasri	PROPN
ejpam-5709	345	28	/	/	SYM
ejpam-5709	345	29	eur	eur	PROPN
ejpam-5709	345	30	.	.	PUNCT
ejpam-5709	346	1	j.	j.	PROPN
ejpam-5709	346	2	pure	pure	PROPN
ejpam-5709	346	3	appl	appl	PROPN
ejpam-5709	346	4	.	.	PROPN
ejpam-5709	346	5	math	math	PROPN
ejpam-5709	346	6	,	,	PUNCT
ejpam-5709	346	7	18	18	NUM
ejpam-5709	346	8	(	(	PUNCT
ejpam-5709	346	9	1	1	NUM
ejpam-5709	346	10	)	)	PUNCT
ejpam-5709	346	11	(	(	PUNCT
ejpam-5709	346	12	2025	2025	NUM
ejpam-5709	346	13	)	)	PUNCT
ejpam-5709	346	14	,	,	PUNCT
ejpam-5709	346	15	5709	5709	NUM
ejpam-5709	346	16	12	12	NUM
ejpam-5709	346	17	of	of	ADP
ejpam-5709	346	18	15	15	NUM
ejpam-5709	346	19	contrary	contrary	ADJ
ejpam-5709	346	20	that	that	SCONJ
ejpam-5709	346	21	f(x	f(x	PROPN
ejpam-5709	346	22	)	)	PUNCT
ejpam-5709	346	23	=	=	PUNCT
ejpam-5709	347	1	g1(x)h1(x	g1(x)h1(x	NOUN
ejpam-5709	347	2	)	)	PUNCT
ejpam-5709	347	3	for	for	ADP
ejpam-5709	347	4	some	some	DET
ejpam-5709	347	5	nonconstant	nonconstant	ADJ
ejpam-5709	347	6	polynomials	polynomial	NOUN
ejpam-5709	347	7	g1(x	g1(x	NOUN
ejpam-5709	347	8	)	)	PUNCT
ejpam-5709	347	9	and	and	CCONJ
ejpam-5709	347	10	h1(x	h1(x	NOUN
ejpam-5709	347	11	)	)	PUNCT
ejpam-5709	347	12	in	in	ADP
ejpam-5709	347	13	k[x	k[x	NOUN
ejpam-5709	347	14	]	]	PUNCT
ejpam-5709	347	15	.	.	PUNCT
ejpam-5709	348	1	since	since	SCONJ
ejpam-5709	348	2	every	every	DET
ejpam-5709	348	3	element	element	NOUN
ejpam-5709	348	4	of	of	ADP
ejpam-5709	348	5	k	k	PROPN
ejpam-5709	348	6	is	be	AUX
ejpam-5709	348	7	of	of	ADP
ejpam-5709	348	8	the	the	DET
ejpam-5709	348	9	form	form	NOUN
ejpam-5709	348	10	α	α	NOUN
ejpam-5709	348	11	/	/	SYM
ejpam-5709	348	12	r	r	NOUN
ejpam-5709	348	13	,	,	PUNCT
ejpam-5709	348	14	where	where	SCONJ
ejpam-5709	348	15	α	α	PROPN
ejpam-5709	348	16	∈	∈	PROPN
ejpam-5709	348	17	ok	ok	INTJ
ejpam-5709	348	18	and	and	CCONJ
ejpam-5709	348	19	r	r	NOUN
ejpam-5709	348	20	∈	∈	PROPN
ejpam-5709	348	21	z\{0	z\{0	NOUN
ejpam-5709	348	22	}	}	PUNCT
ejpam-5709	348	23	,	,	PUNCT
ejpam-5709	348	24	we	we	PRON
ejpam-5709	348	25	may	may	AUX
ejpam-5709	348	26	take	take	VERB
ejpam-5709	348	27	g2(x	g2(x	NOUN
ejpam-5709	348	28	)	)	PUNCT
ejpam-5709	348	29	=	=	SYM
ejpam-5709	348	30	rg1(x	rg1(x	PROPN
ejpam-5709	348	31	)	)	PUNCT
ejpam-5709	348	32	and	and	CCONJ
ejpam-5709	348	33	h2(x	h2(x	NUM
ejpam-5709	348	34	)	)	PUNCT
ejpam-5709	348	35	=	=	SYM
ejpam-5709	348	36	sh1(x	sh1(x	NOUN
ejpam-5709	348	37	)	)	PUNCT
ejpam-5709	348	38	,	,	PUNCT
ejpam-5709	348	39	where	where	SCONJ
ejpam-5709	348	40	g2(x	g2(x	NOUN
ejpam-5709	348	41	)	)	PUNCT
ejpam-5709	348	42	,	,	PUNCT
ejpam-5709	348	43	h2(x	h2(x	X
ejpam-5709	348	44	)	)	PUNCT
ejpam-5709	348	45	∈	∈	NOUN
ejpam-5709	349	1	ok	ok	INTJ
ejpam-5709	350	1	[	[	X
ejpam-5709	350	2	x	x	X
ejpam-5709	350	3	]	]	X
ejpam-5709	350	4	with	with	ADP
ejpam-5709	350	5	deg	deg	PROPN
ejpam-5709	350	6	g2(x	g2(x	PROPN
ejpam-5709	350	7	)	)	PUNCT
ejpam-5709	350	8	≥	≥	NOUN
ejpam-5709	350	9	1	1	NUM
ejpam-5709	350	10	,	,	PUNCT
ejpam-5709	350	11	deg	deg	X
ejpam-5709	350	12	h2(x	h2(x	NUM
ejpam-5709	350	13	)	)	PUNCT
ejpam-5709	350	14	≥	≥	NOUN
ejpam-5709	350	15	1	1	NUM
ejpam-5709	350	16	and	and	CCONJ
ejpam-5709	350	17	r	r	NOUN
ejpam-5709	350	18	,	,	PUNCT
ejpam-5709	350	19	s	s	PART
ejpam-5709	350	20	∈	∈	PROPN
ejpam-5709	350	21	z\{0	z\{0	NOUN
ejpam-5709	350	22	}	}	PUNCT
ejpam-5709	350	23	so	so	SCONJ
ejpam-5709	350	24	that	that	SCONJ
ejpam-5709	350	25	rsf(x	rsf(x	ADP
ejpam-5709	350	26	)	)	PUNCT
ejpam-5709	350	27	=	=	SYM
ejpam-5709	350	28	g2(x)h2(x	g2(x)h2(x	PROPN
ejpam-5709	350	29	)	)	PUNCT
ejpam-5709	350	30	.	.	PUNCT
ejpam-5709	351	1	let	let	VERB
ejpam-5709	351	2	π	π	PRON
ejpam-5709	351	3	be	be	AUX
ejpam-5709	351	4	a	a	DET
ejpam-5709	351	5	prime	prime	ADJ
ejpam-5709	351	6	divisor	divisor	NOUN
ejpam-5709	351	7	of	of	ADP
ejpam-5709	351	8	rs	rs	NOUN
ejpam-5709	351	9	in	in	ADP
ejpam-5709	351	10	ok	ok	INTJ
ejpam-5709	351	11	.	.	PUNCT
ejpam-5709	352	1	considering	consider	VERB
ejpam-5709	352	2	the	the	DET
ejpam-5709	352	3	contents	content	NOUN
ejpam-5709	352	4	of	of	ADP
ejpam-5709	352	5	the	the	DET
ejpam-5709	352	6	above	above	ADJ
ejpam-5709	352	7	polynomials	polynomial	NOUN
ejpam-5709	352	8	and	and	CCONJ
ejpam-5709	352	9	using	use	VERB
ejpam-5709	352	10	gauss	gauss	PROPN
ejpam-5709	352	11	’s	’s	PART
ejpam-5709	352	12	lemma	lemma	PROPN
ejpam-5709	352	13	for	for	ADP
ejpam-5709	352	14	ok	ok	INTJ
ejpam-5709	352	15	,	,	PUNCT
ejpam-5709	352	16	we	we	PRON
ejpam-5709	352	17	get	get	VERB
ejpam-5709	352	18	that	that	SCONJ
ejpam-5709	352	19	π	π	PROPN
ejpam-5709	352	20	divides	divide	VERB
ejpam-5709	352	21	g2(x	g2(x	X
ejpam-5709	352	22	)	)	PUNCT
ejpam-5709	352	23	or	or	CCONJ
ejpam-5709	352	24	h2(x	h2(x	NUM
ejpam-5709	352	25	)	)	PUNCT
ejpam-5709	352	26	,	,	PUNCT
ejpam-5709	352	27	yielding	yield	VERB
ejpam-5709	352	28	rs	rs	PROPN
ejpam-5709	352	29	π	π	PROPN
ejpam-5709	352	30	f(x	f(x	PROPN
ejpam-5709	352	31	)	)	PUNCT
ejpam-5709	352	32	=	=	PUNCT
ejpam-5709	353	1	g3(x)h3(x	g3(x)h3(x	PROPN
ejpam-5709	353	2	)	)	PUNCT
ejpam-5709	353	3	for	for	ADP
ejpam-5709	353	4	some	some	DET
ejpam-5709	353	5	nonconstant	nonconstant	ADJ
ejpam-5709	353	6	polynomials	polynomial	NOUN
ejpam-5709	353	7	g3(x	g3(x	X
ejpam-5709	353	8	)	)	PUNCT
ejpam-5709	353	9	,	,	PUNCT
ejpam-5709	353	10	h3(x	h3(x	PROPN
ejpam-5709	353	11	)	)	PUNCT
ejpam-5709	353	12	∈	∈	NOUN
ejpam-5709	353	13	ok	ok	INTJ
ejpam-5709	354	1	[	[	X
ejpam-5709	355	1	x	x	X
ejpam-5709	355	2	]	]	X
ejpam-5709	355	3	.	.	PUNCT
ejpam-5709	356	1	continuing	continue	VERB
ejpam-5709	356	2	in	in	ADP
ejpam-5709	356	3	the	the	DET
ejpam-5709	356	4	same	same	ADJ
ejpam-5709	356	5	manner	manner	NOUN
ejpam-5709	356	6	,	,	PUNCT
ejpam-5709	356	7	we	we	PRON
ejpam-5709	356	8	finally	finally	ADV
ejpam-5709	356	9	get	get	VERB
ejpam-5709	356	10	f(x	f(x	PROPN
ejpam-5709	356	11	)	)	PUNCT
ejpam-5709	356	12	=	=	SYM
ejpam-5709	356	13	g(x)h(x	g(x)h(x	X
ejpam-5709	356	14	)	)	PUNCT
ejpam-5709	356	15	,	,	PUNCT
ejpam-5709	356	16	where	where	SCONJ
ejpam-5709	356	17	g(x	g(x	NOUN
ejpam-5709	356	18	)	)	PUNCT
ejpam-5709	356	19	and	and	CCONJ
ejpam-5709	356	20	h(x	h(x	PROPN
ejpam-5709	356	21	)	)	PUNCT
ejpam-5709	356	22	are	be	AUX
ejpam-5709	356	23	positive	positive	ADJ
ejpam-5709	356	24	degree	degree	NOUN
ejpam-5709	356	25	polynomials	polynomial	NOUN
ejpam-5709	356	26	in	in	ADP
ejpam-5709	356	27	ok	ok	ADJ
ejpam-5709	356	28	[	[	X
ejpam-5709	356	29	x	x	X
ejpam-5709	356	30	]	]	X
ejpam-5709	356	31	,	,	PUNCT
ejpam-5709	356	32	which	which	PRON
ejpam-5709	356	33	is	be	AUX
ejpam-5709	356	34	a	a	DET
ejpam-5709	356	35	contradiction	contradiction	NOUN
ejpam-5709	356	36	.	.	PUNCT
ejpam-5709	357	1	this	this	PRON
ejpam-5709	357	2	proves	prove	VERB
ejpam-5709	357	3	(	(	PUNCT
ejpam-5709	357	4	ii	ii	NOUN
ejpam-5709	357	5	)	)	PUNCT
ejpam-5709	357	6	and	and	CCONJ
ejpam-5709	357	7	we	we	PRON
ejpam-5709	357	8	complete	complete	VERB
ejpam-5709	357	9	the	the	DET
ejpam-5709	357	10	proof	proof	NOUN
ejpam-5709	357	11	.	.	PUNCT
ejpam-5709	358	1	we	we	PRON
ejpam-5709	358	2	note	note	VERB
ejpam-5709	358	3	that	that	SCONJ
ejpam-5709	358	4	the	the	DET
ejpam-5709	358	5	condition	condition	NOUN
ejpam-5709	358	6	a	a	DET
ejpam-5709	358	7	≥	≥	NOUN
ejpam-5709	358	8	|ω|+	|ω|+	NOUN
ejpam-5709	358	9	√	√	NUM
ejpam-5709	358	10	1−m	1−m	NUM
ejpam-5709	358	11	in	in	ADP
ejpam-5709	358	12	theorem	theorem	NOUN
ejpam-5709	358	13	3	3	NUM
ejpam-5709	358	14	can	can	AUX
ejpam-5709	358	15	be	be	AUX
ejpam-5709	358	16	reduced	reduce	VERB
ejpam-5709	358	17	to	to	ADP
ejpam-5709	358	18	a	a	DET
ejpam-5709	358	19	≥	≥	NOUN
ejpam-5709	358	20	|ω|	|ω|	VERB
ejpam-5709	358	21	for	for	ADP
ejpam-5709	358	22	the	the	DET
ejpam-5709	358	23	case	case	NOUN
ejpam-5709	358	24	of	of	ADP
ejpam-5709	358	25	gaussian	gaussian	ADJ
ejpam-5709	358	26	integers	integer	NOUN
ejpam-5709	358	27	and	and	CCONJ
ejpam-5709	358	28	one	one	PRON
ejpam-5709	358	29	can	can	AUX
ejpam-5709	358	30	see	see	VERB
ejpam-5709	358	31	that	that	SCONJ
ejpam-5709	358	32	z[i	z[i	NOUN
ejpam-5709	358	33	]	]	X
ejpam-5709	358	34	is	be	AUX
ejpam-5709	358	35	a	a	DET
ejpam-5709	358	36	unique	unique	ADJ
ejpam-5709	358	37	factorization	factorization	NOUN
ejpam-5709	358	38	domain	domain	NOUN
ejpam-5709	358	39	.	.	PUNCT
ejpam-5709	359	1	this	this	PRON
ejpam-5709	359	2	implies	imply	VERB
ejpam-5709	359	3	that	that	SCONJ
ejpam-5709	359	4	theorem	theorem	VERB
ejpam-5709	359	5	3	3	NUM
ejpam-5709	359	6	extends	extend	NOUN
ejpam-5709	359	7	theorem	theorem	VERB
ejpam-5709	359	8	b	b	NOUN
ejpam-5709	359	9	to	to	ADP
ejpam-5709	359	10	any	any	DET
ejpam-5709	359	11	imaginary	imaginary	ADJ
ejpam-5709	359	12	quadratic	quadratic	ADJ
ejpam-5709	359	13	field	field	NOUN
ejpam-5709	359	14	with	with	ADP
ejpam-5709	359	15	m	m	PROPN
ejpam-5709	359	16	̸≡	̸≡	ADJ
ejpam-5709	359	17	1	1	NUM
ejpam-5709	359	18	(	(	PUNCT
ejpam-5709	359	19	mod	mod	NOUN
ejpam-5709	359	20	4	4	NUM
ejpam-5709	359	21	)	)	PUNCT
ejpam-5709	359	22	.	.	PUNCT
ejpam-5709	360	1	applying	apply	VERB
ejpam-5709	360	2	theorem	theorem	NOUN
ejpam-5709	360	3	3	3	NUM
ejpam-5709	360	4	,	,	PUNCT
ejpam-5709	360	5	we	we	PRON
ejpam-5709	360	6	can	can	AUX
ejpam-5709	360	7	find	find	VERB
ejpam-5709	360	8	irreducible	irreducible	ADJ
ejpam-5709	360	9	polynomials	polynomial	NOUN
ejpam-5709	360	10	as	as	ADP
ejpam-5709	360	11	the	the	DET
ejpam-5709	360	12	following	follow	VERB
ejpam-5709	360	13	examples	example	NOUN
ejpam-5709	360	14	.	.	PUNCT
ejpam-5709	361	1	example	example	NOUN
ejpam-5709	362	1	4	4	NUM
ejpam-5709	362	2	.	.	PUNCT
ejpam-5709	362	3	let	let	VERB
ejpam-5709	362	4	k	k	NOUN
ejpam-5709	362	5	=	=	PUNCT
ejpam-5709	362	6	q	q	PROPN
ejpam-5709	362	7	(	(	PUNCT
ejpam-5709	362	8	√	√	NUM
ejpam-5709	362	9	−2	−2	NOUN
ejpam-5709	362	10	)	)	PUNCT
ejpam-5709	362	11	,	,	PUNCT
ejpam-5709	362	12	β	β	X
ejpam-5709	362	13	=	=	SYM
ejpam-5709	363	1	9−3	9−3	NUM
ejpam-5709	363	2	√	√	NUM
ejpam-5709	363	3	−2	−2	NOUN
ejpam-5709	363	4	,	,	PUNCT
ejpam-5709	363	5	ω	ω	NOUN
ejpam-5709	363	6	=	=	SYM
ejpam-5709	363	7	2	2	NUM
ejpam-5709	363	8	+	+	NUM
ejpam-5709	363	9	√	√	NOUN
ejpam-5709	363	10	−2	−2	NOUN
ejpam-5709	363	11	,	,	PUNCT
ejpam-5709	363	12	and	and	CCONJ
ejpam-5709	363	13	π	π	X
ejpam-5709	363	14	=	=	PUNCT
ejpam-5709	363	15	−8177−28932	−8177−28932	PROPN
ejpam-5709	363	16	√	√	NUM
ejpam-5709	364	1	−2	−2	NOUN
ejpam-5709	364	2	.	.	PUNCT
ejpam-5709	365	1	then	then	ADV
ejpam-5709	365	2	d	d	X
ejpam-5709	365	3	=	=	SYM
ejpam-5709	365	4	3	3	NUM
ejpam-5709	365	5	and	and	CCONJ
ejpam-5709	365	6	thus	thus	ADV
ejpam-5709	365	7	c′	c′	VERB
ejpam-5709	365	8	=	=	SYM
ejpam-5709	365	9	{	{	PUNCT
ejpam-5709	365	10	x+	x+	PROPN
ejpam-5709	365	11	y	y	NOUN
ejpam-5709	365	12	√	√	PROPN
ejpam-5709	365	13	−2	−2	NOUN
ejpam-5709	366	1	|	|	ADV
ejpam-5709	366	2	x	x	NOUN
ejpam-5709	366	3	=	=	SYM
ejpam-5709	366	4	0	0	NUM
ejpam-5709	366	5	,	,	PUNCT
ejpam-5709	366	6	1	1	NUM
ejpam-5709	366	7	,	,	PUNCT
ejpam-5709	366	8	.	.	PUNCT
ejpam-5709	366	9	.	.	PUNCT
ejpam-5709	366	10	.	.	PUNCT
ejpam-5709	367	1	,	,	PUNCT
ejpam-5709	367	2	8	8	NUM
ejpam-5709	367	3	and	and	CCONJ
ejpam-5709	367	4	y	y	PROPN
ejpam-5709	367	5	=	=	SYM
ejpam-5709	367	6	0	0	NUM
ejpam-5709	367	7	,	,	PUNCT
ejpam-5709	367	8	1	1	NUM
ejpam-5709	367	9	,	,	PUNCT
ejpam-5709	367	10	2	2	NUM
ejpam-5709	367	11	}	}	PUNCT
ejpam-5709	367	12	.	.	PUNCT
ejpam-5709	368	1	one	one	PRON
ejpam-5709	368	2	can	can	AUX
ejpam-5709	368	3	see	see	VERB
ejpam-5709	368	4	that	that	PRON
ejpam-5709	368	5	|β|	|β|	NOUN
ejpam-5709	368	6	=	=	SYM
ejpam-5709	368	7	√	√	PROPN
ejpam-5709	368	8	99	99	NUM
ejpam-5709	368	9	>	>	SYM
ejpam-5709	368	10	2	2	NUM
ejpam-5709	368	11	√	√	NUM
ejpam-5709	368	12	6	6	NUM
ejpam-5709	368	13	+	+	CCONJ
ejpam-5709	368	14	√	√	NUM
ejpam-5709	368	15	3	3	NUM
ejpam-5709	368	16	=	=	SYM
ejpam-5709	368	17	2|ω|	2|ω|	NUM
ejpam-5709	368	18	+	+	CCONJ
ejpam-5709	368	19	√	√	NUM
ejpam-5709	368	20	1−m	1−m	NUM
ejpam-5709	368	21	and	and	CCONJ
ejpam-5709	368	22	a	a	DET
ejpam-5709	368	23	=	=	NOUN
ejpam-5709	368	24	9	9	NUM
ejpam-5709	368	25	>	>	SYM
ejpam-5709	368	26	√	√	NUM
ejpam-5709	368	27	6	6	NUM
ejpam-5709	368	28	+	+	CCONJ
ejpam-5709	368	29	√	√	NUM
ejpam-5709	368	30	3	3	NUM
ejpam-5709	368	31	=	=	SYM
ejpam-5709	368	32	|ω|	|ω|	VERB
ejpam-5709	368	33	+	+	ADV
ejpam-5709	368	34	√	√	PROPN
ejpam-5709	368	35	1−m	1−m	NUM
ejpam-5709	368	36	.	.	PUNCT
ejpam-5709	369	1	since	since	SCONJ
ejpam-5709	369	2	n(π	n(π	NOUN
ejpam-5709	369	3	)	)	PUNCT
ejpam-5709	369	4	=	=	PRON
ejpam-5709	369	5	1740984577	1740984577	NUM
ejpam-5709	369	6	is	be	AUX
ejpam-5709	369	7	a	a	DET
ejpam-5709	369	8	rational	rational	ADJ
ejpam-5709	369	9	prime	prime	NOUN
ejpam-5709	369	10	,	,	PUNCT
ejpam-5709	369	11	we	we	PRON
ejpam-5709	369	12	have	have	VERB
ejpam-5709	369	13	that	that	SCONJ
ejpam-5709	369	14	π	π	PROPN
ejpam-5709	369	15	is	be	AUX
ejpam-5709	369	16	an	an	DET
ejpam-5709	369	17	irreducible	irreducible	ADJ
ejpam-5709	369	18	element	element	NOUN
ejpam-5709	369	19	and	and	CCONJ
ejpam-5709	369	20	so	so	ADV
ejpam-5709	369	21	is	be	AUX
ejpam-5709	369	22	a	a	DET
ejpam-5709	369	23	prime	prime	ADJ
ejpam-5709	369	24	element	element	NOUN
ejpam-5709	369	25	because	because	SCONJ
ejpam-5709	369	26	ok	ok	INTJ
ejpam-5709	369	27	is	be	VERB
ejpam-5709	369	28	a	a	DET
ejpam-5709	369	29	unique	unique	ADJ
ejpam-5709	369	30	factorization	factorization	NOUN
ejpam-5709	369	31	domain	domain	NOUN
ejpam-5709	369	32	.	.	PUNCT
ejpam-5709	370	1	now	now	ADV
ejpam-5709	370	2	,	,	PUNCT
ejpam-5709	370	3	we	we	PRON
ejpam-5709	370	4	have	have	VERB
ejpam-5709	370	5	|π|	|π|	NOUN
ejpam-5709	370	6	>	>	PUNCT
ejpam-5709	370	7	|β|	|β|	PROPN
ejpam-5709	370	8	and	and	CCONJ
ejpam-5709	370	9	ωπ	ωπ	NOUN
ejpam-5709	370	10	=	=	NOUN
ejpam-5709	370	11	41510−	41510−	NUM
ejpam-5709	370	12	66041	66041	NUM
ejpam-5709	370	13	√	√	NOUN
ejpam-5709	371	1	−2	−2	NOUN
ejpam-5709	371	2	=	=	SYM
ejpam-5709	371	3	(	(	PUNCT
ejpam-5709	371	4	8	8	NUM
ejpam-5709	371	5	+	+	CCONJ
ejpam-5709	371	6	4	4	NUM
ejpam-5709	371	7	√	√	NOUN
ejpam-5709	371	8	−2)β4	−2)β4	NUM
ejpam-5709	371	9	+	+	NOUN
ejpam-5709	371	10	6β3	6β3	NUM
ejpam-5709	371	11	+	+	CCONJ
ejpam-5709	371	12	(	(	PUNCT
ejpam-5709	371	13	4	4	NUM
ejpam-5709	371	14	+	+	SYM
ejpam-5709	371	15	2	2	NUM
ejpam-5709	371	16	√	√	NOUN
ejpam-5709	371	17	−2)β2	−2)β2	NOUN
ejpam-5709	371	18	+	+	CCONJ
ejpam-5709	371	19	6β	6β	NOUN
ejpam-5709	371	20	+	+	CCONJ
ejpam-5709	371	21	(	(	PUNCT
ejpam-5709	371	22	2	2	NUM
ejpam-5709	371	23	+	+	NUM
ejpam-5709	371	24	√	√	PROPN
ejpam-5709	371	25	−2	−2	NOUN
ejpam-5709	371	26	)	)	PUNCT
ejpam-5709	371	27	,	,	PUNCT
ejpam-5709	371	28	which	which	PRON
ejpam-5709	371	29	is	be	AUX
ejpam-5709	371	30	a	a	DET
ejpam-5709	371	31	base	base	NOUN
ejpam-5709	371	32	-	-	PUNCT
ejpam-5709	371	33	β(c′	β(c′	NUM
ejpam-5709	371	34	)	)	PUNCT
ejpam-5709	371	35	representation	representation	NOUN
ejpam-5709	371	36	of	of	ADP
ejpam-5709	371	37	ωπ	ωπ	PROPN
ejpam-5709	371	38	.	.	PUNCT
ejpam-5709	372	1	moreover	moreover	ADV
ejpam-5709	372	2	,	,	PUNCT
ejpam-5709	372	3	re(α4	re(α4	X
ejpam-5709	372	4	)	)	PUNCT
ejpam-5709	372	5	=	=	SYM
ejpam-5709	372	6	8	8	NUM
ejpam-5709	372	7	>	>	SYM
ejpam-5709	372	8	1	1	NUM
ejpam-5709	372	9	and	and	CCONJ
ejpam-5709	372	10	re(α3	re(α3	NOUN
ejpam-5709	372	11	)	)	PUNCT
ejpam-5709	372	12	im(α4	im(α4	NUM
ejpam-5709	372	13	)	)	PUNCT
ejpam-5709	372	14	=	=	PUNCT
ejpam-5709	373	1	(	(	PUNCT
ejpam-5709	373	2	6)(4	6)(4	NUM
ejpam-5709	373	3	√	√	NUM
ejpam-5709	373	4	2	2	NUM
ejpam-5709	373	5	)	)	PUNCT
ejpam-5709	373	6	>	>	X
ejpam-5709	373	7	(	(	PUNCT
ejpam-5709	373	8	8)(0	8)(0	NUM
ejpam-5709	373	9	)	)	PUNCT
ejpam-5709	373	10	=	=	SYM
ejpam-5709	373	11	re(α4	re(α4	NOUN
ejpam-5709	373	12	)	)	PUNCT
ejpam-5709	373	13	im(α3	im(α3	NOUN
ejpam-5709	373	14	)	)	PUNCT
ejpam-5709	373	15	.	.	PUNCT
ejpam-5709	374	1	by	by	ADP
ejpam-5709	374	2	theorem	theorem	NOUN
ejpam-5709	374	3	3	3	NUM
ejpam-5709	374	4	,	,	PUNCT
ejpam-5709	374	5	we	we	PRON
ejpam-5709	374	6	obtain	obtain	VERB
ejpam-5709	374	7	that	that	SCONJ
ejpam-5709	374	8	f(x	f(x	NOUN
ejpam-5709	374	9	)	)	PUNCT
ejpam-5709	374	10	=	=	PUNCT
ejpam-5709	375	1	(	(	PUNCT
ejpam-5709	375	2	8	8	NUM
ejpam-5709	375	3	+	+	NOUN
ejpam-5709	375	4	4	4	NUM
ejpam-5709	375	5	√	√	ADP
ejpam-5709	375	6	−2)x4	−2)x4	NOUN
ejpam-5709	375	7	+	+	NOUN
ejpam-5709	375	8	6x3	6x3	NUM
ejpam-5709	375	9	+	+	CCONJ
ejpam-5709	375	10	(	(	PUNCT
ejpam-5709	375	11	4	4	NUM
ejpam-5709	375	12	+	+	SYM
ejpam-5709	375	13	2	2	NUM
ejpam-5709	375	14	√	√	NOUN
ejpam-5709	375	15	−2)x2	−2)x2	NOUN
ejpam-5709	376	1	+	+	NOUN
ejpam-5709	376	2	6x	6x	NUM
ejpam-5709	376	3	+	+	CCONJ
ejpam-5709	376	4	(	(	PUNCT
ejpam-5709	376	5	2	2	NUM
ejpam-5709	376	6	+	+	CCONJ
ejpam-5709	376	7	√	√	NUM
ejpam-5709	376	8	−2	−2	NOUN
ejpam-5709	376	9	)	)	PUNCT
ejpam-5709	376	10	has	have	VERB
ejpam-5709	376	11	no	no	DET
ejpam-5709	376	12	proper	proper	ADJ
ejpam-5709	376	13	factorization	factorization	NOUN
ejpam-5709	376	14	in	in	ADP
ejpam-5709	376	15	ok	ok	ADJ
ejpam-5709	376	16	[	[	X
ejpam-5709	376	17	x	x	X
ejpam-5709	376	18	]	]	X
ejpam-5709	376	19	and	and	CCONJ
ejpam-5709	376	20	so	so	ADV
ejpam-5709	376	21	is	be	AUX
ejpam-5709	376	22	irreducible	irreducible	ADJ
ejpam-5709	376	23	over	over	ADP
ejpam-5709	376	24	k.	k.	PROPN
ejpam-5709	376	25	observe	observe	VERB
ejpam-5709	376	26	that	that	SCONJ
ejpam-5709	376	27	f(x	f(x	PROPN
ejpam-5709	376	28	)	)	PUNCT
ejpam-5709	376	29	is	be	AUX
ejpam-5709	376	30	reducible	reducible	ADJ
ejpam-5709	376	31	in	in	ADP
ejpam-5709	376	32	ok	ok	ADJ
ejpam-5709	376	33	[	[	X
ejpam-5709	376	34	x	x	X
ejpam-5709	376	35	]	]	X
ejpam-5709	376	36	because	because	SCONJ
ejpam-5709	376	37	f(x	f(x	PROPN
ejpam-5709	376	38	)	)	PUNCT
ejpam-5709	377	1	=	=	PUNCT
ejpam-5709	378	1	(	(	PUNCT
ejpam-5709	378	2	2	2	NUM
ejpam-5709	378	3	+	+	NOUN
ejpam-5709	378	4	√	√	PROPN
ejpam-5709	378	5	−2	−2	NOUN
ejpam-5709	378	6	)	)	PUNCT
ejpam-5709	378	7	[	[	PUNCT
ejpam-5709	379	1	4x4	4x4	NUM
ejpam-5709	379	2	+	+	CCONJ
ejpam-5709	379	3	(	(	PUNCT
ejpam-5709	379	4	2−	2−	NUM
ejpam-5709	379	5	√	√	NOUN
ejpam-5709	379	6	−2)x3	−2)x3	NOUN
ejpam-5709	379	7	+	+	CCONJ
ejpam-5709	379	8	2x2	2x2	NUM
ejpam-5709	379	9	+	+	CCONJ
ejpam-5709	379	10	(	(	PUNCT
ejpam-5709	379	11	2−	2−	NUM
ejpam-5709	379	12	√	√	NOUN
ejpam-5709	379	13	−2)x+	−2)x+	VERB
ejpam-5709	379	14	1	1	NUM
ejpam-5709	379	15	]	]	PUNCT
ejpam-5709	379	16	.	.	PUNCT
ejpam-5709	380	1	example	example	NOUN
ejpam-5709	381	1	5	5	NUM
ejpam-5709	381	2	.	.	PUNCT
ejpam-5709	381	3	let	let	VERB
ejpam-5709	381	4	k	k	NOUN
ejpam-5709	382	1	=	=	PUNCT
ejpam-5709	382	2	q	q	PROPN
ejpam-5709	382	3	(	(	PUNCT
ejpam-5709	382	4	√	√	NUM
ejpam-5709	382	5	−1	−1	NOUN
ejpam-5709	382	6	)	)	PUNCT
ejpam-5709	382	7	,	,	PUNCT
ejpam-5709	382	8	β	β	X
ejpam-5709	382	9	=	=	SYM
ejpam-5709	382	10	15	15	NUM
ejpam-5709	382	11	−	−	PROPN
ejpam-5709	382	12	16i	16i	NOUN
ejpam-5709	382	13	,	,	PUNCT
ejpam-5709	382	14	ω	ω	PROPN
ejpam-5709	382	15	=	=	SYM
ejpam-5709	382	16	3	3	NUM
ejpam-5709	382	17	+	+	NUM
ejpam-5709	382	18	2i	2i	NUM
ejpam-5709	382	19	,	,	PUNCT
ejpam-5709	382	20	and	and	CCONJ
ejpam-5709	383	1	π	π	X
ejpam-5709	383	2	=	=	PUNCT
ejpam-5709	383	3	−831565	−831565	PROPN
ejpam-5709	383	4	+	+	CCONJ
ejpam-5709	383	5	715166i	715166i	NUM
ejpam-5709	383	6	.	.	PUNCT
ejpam-5709	384	1	then	then	ADV
ejpam-5709	384	2	d	d	X
ejpam-5709	384	3	=	=	SYM
ejpam-5709	384	4	1	1	NUM
ejpam-5709	384	5	and	and	CCONJ
ejpam-5709	384	6	thus	thus	ADV
ejpam-5709	384	7	c′	c′	VERB
ejpam-5709	384	8	=	=	SYM
ejpam-5709	384	9	{	{	PUNCT
ejpam-5709	384	10	0	0	NUM
ejpam-5709	384	11	,	,	PUNCT
ejpam-5709	384	12	1	1	NUM
ejpam-5709	384	13	,	,	PUNCT
ejpam-5709	384	14	.	.	PUNCT
ejpam-5709	384	15	.	.	PUNCT
ejpam-5709	384	16	.	.	PUNCT
ejpam-5709	385	1	,	,	PUNCT
ejpam-5709	385	2	15	15	NUM
ejpam-5709	385	3	}	}	PUNCT
ejpam-5709	385	4	.	.	PUNCT
ejpam-5709	386	1	note	note	VERB
ejpam-5709	386	2	that	that	SCONJ
ejpam-5709	386	3	|β|	|β|	PRON
ejpam-5709	386	4	=	=	PUNCT
ejpam-5709	386	5	√	√	PROPN
ejpam-5709	386	6	481	481	NUM
ejpam-5709	386	7	>	>	SYM
ejpam-5709	386	8	2	2	NUM
ejpam-5709	386	9	√	√	NUM
ejpam-5709	386	10	13	13	NUM
ejpam-5709	386	11	+	+	CCONJ
ejpam-5709	386	12	√	√	NUM
ejpam-5709	386	13	2	2	NUM
ejpam-5709	386	14	=	=	SYM
ejpam-5709	386	15	2|ω|	2|ω|	NUM
ejpam-5709	387	1	+	+	ADJ
ejpam-5709	387	2	√	√	PROPN
ejpam-5709	387	3	1−m	1−m	NUM
ejpam-5709	387	4	and	and	CCONJ
ejpam-5709	387	5	a	a	DET
ejpam-5709	387	6	=	=	SYM
ejpam-5709	387	7	15	15	NUM
ejpam-5709	387	8	>	>	SYM
ejpam-5709	387	9	√	√	NUM
ejpam-5709	387	10	13	13	NUM
ejpam-5709	387	11	+	+	CCONJ
ejpam-5709	387	12	√	√	NUM
ejpam-5709	387	13	2	2	NUM
ejpam-5709	387	14	=	=	SYM
ejpam-5709	387	15	|ω|	|ω|	VERB
ejpam-5709	387	16	+	+	ADV
ejpam-5709	387	17	√	√	PROPN
ejpam-5709	387	18	1−m	1−m	NUM
ejpam-5709	387	19	.	.	PUNCT
ejpam-5709	388	1	since	since	SCONJ
ejpam-5709	388	2	n(π	n(π	NOUN
ejpam-5709	388	3	)	)	PUNCT
ejpam-5709	388	4	=	=	PUNCT
ejpam-5709	388	5	1202962756781	1202962756781	NUM
ejpam-5709	388	6	is	be	AUX
ejpam-5709	388	7	a	a	DET
ejpam-5709	388	8	rational	rational	ADJ
ejpam-5709	388	9	prime	prime	NOUN
ejpam-5709	388	10	,	,	PUNCT
ejpam-5709	388	11	we	we	PRON
ejpam-5709	388	12	get	get	VERB
ejpam-5709	388	13	that	that	SCONJ
ejpam-5709	388	14	π	π	PROPN
ejpam-5709	388	15	is	be	AUX
ejpam-5709	388	16	a	a	DET
ejpam-5709	388	17	gaussian	gaussian	ADJ
ejpam-5709	388	18	prime	prime	NOUN
ejpam-5709	388	19	.	.	PUNCT
ejpam-5709	389	1	now	now	ADV
ejpam-5709	389	2	,	,	PUNCT
ejpam-5709	389	3	we	we	PRON
ejpam-5709	389	4	have	have	VERB
ejpam-5709	389	5	|π|	|π|	NOUN
ejpam-5709	389	6	>	>	PUNCT
ejpam-5709	389	7	|β|	|β|	PROPN
ejpam-5709	389	8	and	and	CCONJ
ejpam-5709	389	9	ωπ	ωπ	NOUN
ejpam-5709	389	10	=	=	PUNCT
ejpam-5709	389	11	−3925027	−3925027	NOUN
ejpam-5709	389	12	+	+	CCONJ
ejpam-5709	389	13	482368i	482368i	NOUN
ejpam-5709	389	14	=	=	SYM
ejpam-5709	390	1	17β4	17β4	NUM
ejpam-5709	390	2	+	+	NUM
ejpam-5709	390	3	3β3	3β3	NUM
ejpam-5709	390	4	+	+	CCONJ
ejpam-5709	390	5	7β2	7β2	NUM
ejpam-5709	391	1	+	+	NUM
ejpam-5709	391	2	5β	5β	NUM
ejpam-5709	391	3	+	+	CCONJ
ejpam-5709	391	4	13	13	NUM
ejpam-5709	391	5	,	,	PUNCT
ejpam-5709	391	6	which	which	PRON
ejpam-5709	391	7	is	be	AUX
ejpam-5709	391	8	a	a	DET
ejpam-5709	391	9	base	base	NOUN
ejpam-5709	391	10	-	-	PUNCT
ejpam-5709	391	11	β(c′	β(c′	NUM
ejpam-5709	391	12	)	)	PUNCT
ejpam-5709	391	13	representation	representation	NOUN
ejpam-5709	391	14	of	of	ADP
ejpam-5709	391	15	ωπ	ωπ	PROPN
ejpam-5709	391	16	.	.	PUNCT
ejpam-5709	392	1	moreover	moreover	ADV
ejpam-5709	392	2	,	,	PUNCT
ejpam-5709	392	3	re(α4	re(α4	X
ejpam-5709	392	4	)	)	PUNCT
ejpam-5709	392	5	=	=	PUNCT
ejpam-5709	392	6	17	17	NUM
ejpam-5709	392	7	>	>	SYM
ejpam-5709	392	8	1	1	NUM
ejpam-5709	392	9	and	and	CCONJ
ejpam-5709	392	10	re(α3	re(α3	NOUN
ejpam-5709	392	11	)	)	PUNCT
ejpam-5709	392	12	im(α4	im(α4	NUM
ejpam-5709	392	13	)	)	PUNCT
ejpam-5709	393	1	=	=	PRON
ejpam-5709	393	2	(	(	PUNCT
ejpam-5709	393	3	3)(0	3)(0	NUM
ejpam-5709	393	4	)	)	PUNCT
ejpam-5709	393	5	≥	≥	NOUN
ejpam-5709	393	6	(	(	PUNCT
ejpam-5709	393	7	17)(0	17)(0	NUM
ejpam-5709	393	8	)	)	PUNCT
ejpam-5709	393	9	=	=	SYM
ejpam-5709	393	10	re(α4	re(α4	X
ejpam-5709	393	11	)	)	PUNCT
ejpam-5709	393	12	im(α3	im(α3	NOUN
ejpam-5709	393	13	)	)	PUNCT
ejpam-5709	393	14	.	.	PUNCT
ejpam-5709	394	1	by	by	ADP
ejpam-5709	394	2	theorem	theorem	NOUN
ejpam-5709	394	3	3	3	NUM
ejpam-5709	394	4	,	,	PUNCT
ejpam-5709	394	5	we	we	PRON
ejpam-5709	394	6	obtain	obtain	VERB
ejpam-5709	394	7	that	that	SCONJ
ejpam-5709	394	8	f(x	f(x	NOUN
ejpam-5709	394	9	)	)	PUNCT
ejpam-5709	395	1	=	=	PUNCT
ejpam-5709	396	1	17x4	17x4	NUM
ejpam-5709	397	1	+	+	NUM
ejpam-5709	397	2	3x3	3x3	NUM
ejpam-5709	398	1	+	+	NUM
ejpam-5709	398	2	7x2	7x2	NUM
ejpam-5709	398	3	+	+	CCONJ
ejpam-5709	398	4	5x+	5x+	NUM
ejpam-5709	398	5	13	13	NUM
ejpam-5709	398	6	has	have	VERB
ejpam-5709	398	7	no	no	DET
ejpam-5709	398	8	proper	proper	ADJ
ejpam-5709	398	9	factorization	factorization	NOUN
ejpam-5709	398	10	in	in	ADP
ejpam-5709	398	11	z[i][x	z[i][x	NOUN
ejpam-5709	398	12	]	]	PUNCT
ejpam-5709	398	13	and	and	CCONJ
ejpam-5709	398	14	so	so	ADV
ejpam-5709	398	15	is	be	AUX
ejpam-5709	398	16	irreducible	irreducible	ADJ
ejpam-5709	398	17	over	over	ADP
ejpam-5709	398	18	k.	k.	PROPN
ejpam-5709	398	19	since	since	SCONJ
ejpam-5709	398	20	3	3	NUM
ejpam-5709	398	21	and	and	CCONJ
ejpam-5709	398	22	7	7	NUM
ejpam-5709	398	23	are	be	AUX
ejpam-5709	398	24	gaussian	gaussian	ADJ
ejpam-5709	398	25	primes	prime	NOUN
ejpam-5709	398	26	,	,	PUNCT
ejpam-5709	398	27	it	it	PRON
ejpam-5709	398	28	follows	follow	VERB
ejpam-5709	398	29	that	that	SCONJ
ejpam-5709	398	30	f(x	f(x	PROPN
ejpam-5709	398	31	)	)	PUNCT
ejpam-5709	398	32	is	be	AUX
ejpam-5709	398	33	irreducible	irreducible	ADJ
ejpam-5709	398	34	in	in	ADP
ejpam-5709	398	35	z[i][x	z[i][x	NOUN
ejpam-5709	398	36	]	]	X
ejpam-5709	398	37	.	.	PUNCT
ejpam-5709	399	1	p.	p.	NOUN
ejpam-5709	399	2	phetnun	phetnun	PROPN
ejpam-5709	399	3	,	,	PUNCT
ejpam-5709	399	4	n.r	n.r	PROPN
ejpam-5709	399	5	.	.	PROPN
ejpam-5709	399	6	kanasri	kanasri	PROPN
ejpam-5709	399	7	/	/	SYM
ejpam-5709	399	8	eur	eur	PROPN
ejpam-5709	399	9	.	.	PUNCT
ejpam-5709	400	1	j.	j.	PROPN
ejpam-5709	400	2	pure	pure	PROPN
ejpam-5709	400	3	appl	appl	PROPN
ejpam-5709	400	4	.	.	PROPN
ejpam-5709	400	5	math	math	PROPN
ejpam-5709	400	6	,	,	PUNCT
ejpam-5709	400	7	18	18	NUM
ejpam-5709	400	8	(	(	PUNCT
ejpam-5709	400	9	1	1	NUM
ejpam-5709	400	10	)	)	PUNCT
ejpam-5709	400	11	(	(	PUNCT
ejpam-5709	400	12	2025	2025	NUM
ejpam-5709	400	13	)	)	PUNCT
ejpam-5709	400	14	,	,	PUNCT
ejpam-5709	400	15	5709	5709	NUM
ejpam-5709	400	16	13	13	NUM
ejpam-5709	400	17	of	of	ADP
ejpam-5709	400	18	15	15	NUM
ejpam-5709	400	19	by	by	ADP
ejpam-5709	400	20	taking	take	VERB
ejpam-5709	400	21	ω	ω	PROPN
ejpam-5709	400	22	∈	∈	PROPN
ejpam-5709	400	23	u(ok	u(ok	NOUN
ejpam-5709	400	24	)	)	PUNCT
ejpam-5709	400	25	in	in	ADP
ejpam-5709	400	26	theorem	theorem	NOUN
ejpam-5709	400	27	3	3	NUM
ejpam-5709	400	28	,	,	PUNCT
ejpam-5709	400	29	we	we	PRON
ejpam-5709	400	30	obtain	obtain	VERB
ejpam-5709	400	31	that	that	SCONJ
ejpam-5709	400	32	the	the	DET
ejpam-5709	400	33	polynomial	polynomial	ADJ
ejpam-5709	400	34	f(x	f(x	PROPN
ejpam-5709	400	35	)	)	PUNCT
ejpam-5709	400	36	has	have	VERB
ejpam-5709	400	37	no	no	DET
ejpam-5709	400	38	proper	proper	ADJ
ejpam-5709	400	39	factorization	factorization	NOUN
ejpam-5709	400	40	in	in	ADP
ejpam-5709	400	41	ok	ok	ADJ
ejpam-5709	400	42	[	[	X
ejpam-5709	400	43	x	x	X
ejpam-5709	400	44	]	]	X
ejpam-5709	400	45	.	.	PUNCT
ejpam-5709	401	1	since	since	SCONJ
ejpam-5709	401	2	π	π	PROPN
ejpam-5709	401	3	is	be	AUX
ejpam-5709	401	4	also	also	ADV
ejpam-5709	401	5	an	an	DET
ejpam-5709	401	6	irreducible	irreducible	ADJ
ejpam-5709	401	7	element	element	NOUN
ejpam-5709	401	8	,	,	PUNCT
ejpam-5709	401	9	then	then	ADV
ejpam-5709	401	10	δ	δ	PROPN
ejpam-5709	401	11	∤	∤	PROPN
ejpam-5709	401	12	f(x	f(x	PROPN
ejpam-5709	401	13	)	)	PUNCT
ejpam-5709	401	14	for	for	ADP
ejpam-5709	401	15	all	all	DET
ejpam-5709	401	16	δ	δ	PROPN
ejpam-5709	401	17	∈	∈	PROPN
ejpam-5709	401	18	ok\u(ok	ok\u(ok	NOUN
ejpam-5709	401	19	)	)	PUNCT
ejpam-5709	401	20	,	,	PUNCT
ejpam-5709	401	21	by	by	ADP
ejpam-5709	401	22	lemma	lemma	PROPN
ejpam-5709	401	23	4	4	NUM
ejpam-5709	401	24	in	in	ADP
ejpam-5709	401	25	[	[	X
ejpam-5709	401	26	8	8	NUM
ejpam-5709	401	27	]	]	PUNCT
ejpam-5709	401	28	.	.	PUNCT
ejpam-5709	402	1	this	this	PRON
ejpam-5709	402	2	shows	show	VERB
ejpam-5709	402	3	that	that	SCONJ
ejpam-5709	402	4	f(x	f(x	PROPN
ejpam-5709	402	5	)	)	PUNCT
ejpam-5709	402	6	is	be	AUX
ejpam-5709	402	7	irreducible	irreducible	ADJ
ejpam-5709	402	8	in	in	ADP
ejpam-5709	402	9	ok	ok	ADJ
ejpam-5709	402	10	[	[	X
ejpam-5709	402	11	x	x	X
ejpam-5709	402	12	]	]	X
ejpam-5709	402	13	.	.	PUNCT
ejpam-5709	403	1	therefore	therefore	ADV
ejpam-5709	403	2	,	,	PUNCT
ejpam-5709	403	3	theorem	theorem	VERB
ejpam-5709	403	4	3	3	NUM
ejpam-5709	403	5	is	be	AUX
ejpam-5709	403	6	a	a	DET
ejpam-5709	403	7	generalization	generalization	NOUN
ejpam-5709	403	8	of	of	ADP
ejpam-5709	403	9	theorem	theorem	ADJ
ejpam-5709	403	10	c	c	NOUN
ejpam-5709	403	11	by	by	ADP
ejpam-5709	403	12	considering	consider	VERB
ejpam-5709	403	13	ωπ	ωπ	INTJ
ejpam-5709	403	14	instead	instead	ADV
ejpam-5709	403	15	of	of	ADP
ejpam-5709	403	16	π	π	PROPN
ejpam-5709	403	17	,	,	PUNCT
ejpam-5709	403	18	where	where	SCONJ
ejpam-5709	403	19	ω	ω	PROPN
ejpam-5709	403	20	∈	∈	PROPN
ejpam-5709	403	21	ok\{0	ok\{0	PROPN
ejpam-5709	403	22	}	}	PUNCT
ejpam-5709	403	23	and	and	CCONJ
ejpam-5709	403	24	π	π	PROPN
ejpam-5709	403	25	is	be	AUX
ejpam-5709	403	26	a	a	DET
ejpam-5709	403	27	prime	prime	ADJ
ejpam-5709	403	28	element	element	NOUN
ejpam-5709	403	29	.	.	PUNCT
ejpam-5709	404	1	for	for	ADP
ejpam-5709	404	2	the	the	DET
ejpam-5709	404	3	case	case	NOUN
ejpam-5709	404	4	m	m	NOUN
ejpam-5709	404	5	≡	≡	PROPN
ejpam-5709	404	6	1	1	NUM
ejpam-5709	404	7	(	(	PUNCT
ejpam-5709	404	8	mod	mod	NOUN
ejpam-5709	404	9	4	4	NUM
ejpam-5709	404	10	)	)	PUNCT
ejpam-5709	404	11	,	,	PUNCT
ejpam-5709	404	12	we	we	PRON
ejpam-5709	404	13	now	now	ADV
ejpam-5709	404	14	prove	prove	VERB
ejpam-5709	404	15	the	the	DET
ejpam-5709	404	16	following	following	NOUN
ejpam-5709	404	17	.	.	PUNCT
ejpam-5709	405	1	theorem	theorem	ADJ
ejpam-5709	405	2	4	4	NUM
ejpam-5709	405	3	.	.	PUNCT
ejpam-5709	406	1	let	let	VERB
ejpam-5709	406	2	k	k	NOUN
ejpam-5709	406	3	=	=	PUNCT
ejpam-5709	406	4	q	q	ADJ
ejpam-5709	406	5	(	(	PUNCT
ejpam-5709	406	6	√	√	NUM
ejpam-5709	406	7	m	m	VERB
ejpam-5709	406	8	)	)	PUNCT
ejpam-5709	406	9	be	be	AUX
ejpam-5709	406	10	an	an	DET
ejpam-5709	406	11	imaginary	imaginary	ADJ
ejpam-5709	406	12	quadratic	quadratic	ADJ
ejpam-5709	406	13	field	field	NOUN
ejpam-5709	406	14	with	with	ADP
ejpam-5709	406	15	m	m	PROPN
ejpam-5709	406	16	≡	≡	PROPN
ejpam-5709	406	17	1	1	NUM
ejpam-5709	406	18	(	(	PUNCT
ejpam-5709	406	19	mod	mod	NOUN
ejpam-5709	406	20	4	4	NUM
ejpam-5709	406	21	)	)	PUNCT
ejpam-5709	406	22	.	.	PUNCT
ejpam-5709	407	1	let	let	VERB
ejpam-5709	407	2	β	β	NOUN
ejpam-5709	407	3	=	=	PUNCT
ejpam-5709	407	4	a	a	DET
ejpam-5709	407	5	+	+	NUM
ejpam-5709	407	6	bσm	bσm	NOUN
ejpam-5709	407	7	∈	∈	NOUN
ejpam-5709	407	8	ok	ok	ADJ
ejpam-5709	407	9	and	and	CCONJ
ejpam-5709	407	10	ω	ω	NUM
ejpam-5709	407	11	∈	∈	PROPN
ejpam-5709	407	12	ok\{0	ok\{0	PROPN
ejpam-5709	407	13	}	}	PUNCT
ejpam-5709	407	14	be	be	AUX
ejpam-5709	407	15	such	such	ADJ
ejpam-5709	407	16	that	that	SCONJ
ejpam-5709	407	17	|β|	|β|	PRON
ejpam-5709	407	18	≥	≥	VERB
ejpam-5709	407	19	2|ω|	2|ω|	NUM
ejpam-5709	407	20	+	+	CCONJ
ejpam-5709	407	21	√	√	PROPN
ejpam-5709	407	22	(	(	PUNCT
ejpam-5709	407	23	9−m)/4	9−m)/4	NUM
ejpam-5709	407	24	,	,	PUNCT
ejpam-5709	407	25	a	a	DET
ejpam-5709	407	26	≥	≥	NOUN
ejpam-5709	407	27	1	1	NUM
ejpam-5709	407	28	,	,	PUNCT
ejpam-5709	407	29	and	and	CCONJ
ejpam-5709	407	30	a+	a+	X
ejpam-5709	407	31	(	(	PUNCT
ejpam-5709	407	32	b/2	b/2	NUM
ejpam-5709	407	33	)	)	PUNCT
ejpam-5709	407	34	≥	≥	PRON
ejpam-5709	407	35	|ω|	|ω|	PROPN
ejpam-5709	407	36	.	.	PROPN
ejpam-5709	407	37	for	for	ADP
ejpam-5709	407	38	a	a	DET
ejpam-5709	407	39	prime	prime	ADJ
ejpam-5709	407	40	element	element	NOUN
ejpam-5709	407	41	π	π	PROPN
ejpam-5709	407	42	of	of	ADP
ejpam-5709	407	43	ok	ok	INTJ
ejpam-5709	407	44	with	with	ADP
ejpam-5709	407	45	|π|	|π|	NOUN
ejpam-5709	407	46	>	>	SYM
ejpam-5709	407	47	√	√	PROPN
ejpam-5709	407	48	(	(	PUNCT
ejpam-5709	407	49	9−m)/4	9−m)/4	NUM
ejpam-5709	407	50	(	(	PUNCT
ejpam-5709	407	51	|β|	|β|	NOUN
ejpam-5709	407	52	−	−	PROPN
ejpam-5709	407	53	1	1	NUM
ejpam-5709	407	54	)	)	PUNCT
ejpam-5709	407	55	,	,	PUNCT
ejpam-5709	407	56	if	if	SCONJ
ejpam-5709	407	57	ωπ	ωπ	PRON
ejpam-5709	407	58	=	=	PUNCT
ejpam-5709	407	59	αnβ	αnβ	PROPN
ejpam-5709	407	60	n	n	PROPN
ejpam-5709	407	61	+	+	CCONJ
ejpam-5709	407	62	αn−1β	αn−1β	PROPN
ejpam-5709	407	63	n−1	n−1	PROPN
ejpam-5709	407	64	+	+	NUM
ejpam-5709	407	65	·	·	PUNCT
ejpam-5709	407	66	·	·	PUNCT
ejpam-5709	407	67	·	·	PUNCT
ejpam-5709	407	68	+	+	NUM
ejpam-5709	407	69	α1β	α1β	X
ejpam-5709	407	70	+	+	CCONJ
ejpam-5709	407	71	α0	α0	ADJ
ejpam-5709	407	72	=	=	NOUN
ejpam-5709	407	73	:	:	PUNCT
ejpam-5709	407	74	f(β	f(β	NOUN
ejpam-5709	407	75	)	)	PUNCT
ejpam-5709	407	76	is	be	AUX
ejpam-5709	407	77	a	a	DET
ejpam-5709	407	78	base	base	NOUN
ejpam-5709	407	79	-	-	PUNCT
ejpam-5709	407	80	β(c′	β(c′	NUM
ejpam-5709	407	81	)	)	PUNCT
ejpam-5709	407	82	representation	representation	NOUN
ejpam-5709	407	83	with	with	ADP
ejpam-5709	407	84	n	n	PRON
ejpam-5709	407	85	≥	≥	NUM
ejpam-5709	407	86	2	2	NUM
ejpam-5709	407	87	and	and	CCONJ
ejpam-5709	407	88	re(αn	re(αn	NOUN
ejpam-5709	407	89	)	)	PUNCT
ejpam-5709	407	90	≥	≥	NOUN
ejpam-5709	407	91	1	1	NUM
ejpam-5709	407	92	satisfying	satisfy	VERB
ejpam-5709	407	93	the	the	DET
ejpam-5709	407	94	condition	condition	NOUN
ejpam-5709	407	95	(	(	PUNCT
ejpam-5709	407	96	ii	ii	NOUN
ejpam-5709	407	97	)	)	PUNCT
ejpam-5709	407	98	of	of	ADP
ejpam-5709	407	99	lemma	lemma	PROPN
ejpam-5709	407	100	2	2	NUM
ejpam-5709	407	101	,	,	PUNCT
ejpam-5709	407	102	then	then	ADV
ejpam-5709	407	103	f(x	f(x	PROPN
ejpam-5709	407	104	)	)	PUNCT
ejpam-5709	407	105	has	have	VERB
ejpam-5709	407	106	no	no	DET
ejpam-5709	407	107	proper	proper	ADJ
ejpam-5709	407	108	factorization	factorization	NOUN
ejpam-5709	407	109	in	in	ADP
ejpam-5709	407	110	ok	ok	ADJ
ejpam-5709	407	111	[	[	X
ejpam-5709	407	112	x	x	X
ejpam-5709	407	113	]	]	X
ejpam-5709	407	114	.	.	PUNCT
ejpam-5709	408	1	moreover	moreover	ADV
ejpam-5709	408	2	,	,	PUNCT
ejpam-5709	408	3	(	(	PUNCT
ejpam-5709	408	4	i	i	NOUN
ejpam-5709	408	5	)	)	PUNCT
ejpam-5709	408	6	if	if	SCONJ
ejpam-5709	408	7	δ	δ	PROPN
ejpam-5709	408	8	∤	∤	PROPN
ejpam-5709	408	9	f(x	f(x	PROPN
ejpam-5709	408	10	)	)	PUNCT
ejpam-5709	408	11	for	for	ADP
ejpam-5709	408	12	all	all	DET
ejpam-5709	408	13	δ	δ	PROPN
ejpam-5709	408	14	∈	∈	PROPN
ejpam-5709	408	15	ok\u(ok	ok\u(ok	NOUN
ejpam-5709	408	16	)	)	PUNCT
ejpam-5709	408	17	,	,	PUNCT
ejpam-5709	408	18	then	then	ADV
ejpam-5709	408	19	f(x	f(x	PROPN
ejpam-5709	408	20	)	)	PUNCT
ejpam-5709	408	21	is	be	AUX
ejpam-5709	408	22	irreducible	irreducible	ADJ
ejpam-5709	408	23	in	in	ADP
ejpam-5709	408	24	ok	ok	ADJ
ejpam-5709	408	25	[	[	X
ejpam-5709	408	26	x	x	X
ejpam-5709	408	27	]	]	X
ejpam-5709	408	28	;	;	PUNCT
ejpam-5709	408	29	(	(	PUNCT
ejpam-5709	408	30	ii	ii	NOUN
ejpam-5709	408	31	)	)	PUNCT
ejpam-5709	408	32	if	if	SCONJ
ejpam-5709	408	33	ok	ok	ADJ
ejpam-5709	408	34	is	be	AUX
ejpam-5709	408	35	a	a	DET
ejpam-5709	408	36	unique	unique	ADJ
ejpam-5709	408	37	factorization	factorization	NOUN
ejpam-5709	408	38	domain	domain	NOUN
ejpam-5709	408	39	,	,	PUNCT
ejpam-5709	408	40	then	then	ADV
ejpam-5709	408	41	f(x	f(x	PROPN
ejpam-5709	408	42	)	)	PUNCT
ejpam-5709	408	43	is	be	AUX
ejpam-5709	408	44	irreducible	irreducible	ADJ
ejpam-5709	408	45	over	over	ADP
ejpam-5709	408	46	k.	k.	PROPN
ejpam-5709	408	47	proof	proof	PROPN
ejpam-5709	408	48	.	.	PUNCT
ejpam-5709	409	1	suppose	suppose	VERB
ejpam-5709	409	2	to	to	ADP
ejpam-5709	409	3	the	the	DET
ejpam-5709	409	4	contrary	contrary	NOUN
ejpam-5709	409	5	that	that	SCONJ
ejpam-5709	409	6	f(x	f(x	PROPN
ejpam-5709	409	7	)	)	PUNCT
ejpam-5709	409	8	has	have	VERB
ejpam-5709	409	9	proper	proper	ADJ
ejpam-5709	409	10	factorization	factorization	NOUN
ejpam-5709	409	11	in	in	ADP
ejpam-5709	409	12	ok	ok	ADJ
ejpam-5709	409	13	[	[	X
ejpam-5709	409	14	x	x	X
ejpam-5709	409	15	]	]	X
ejpam-5709	409	16	,	,	PUNCT
ejpam-5709	409	17	namely	namely	ADV
ejpam-5709	409	18	f(x	f(x	PROPN
ejpam-5709	409	19	)	)	PUNCT
ejpam-5709	409	20	=	=	SYM
ejpam-5709	409	21	g(x)h(x	g(x)h(x	X
ejpam-5709	409	22	)	)	PUNCT
ejpam-5709	409	23	for	for	ADP
ejpam-5709	409	24	some	some	DET
ejpam-5709	409	25	nonconstant	nonconstant	ADJ
ejpam-5709	409	26	polynomials	polynomial	NOUN
ejpam-5709	409	27	g(x	g(x	NOUN
ejpam-5709	409	28	)	)	PUNCT
ejpam-5709	409	29	and	and	CCONJ
ejpam-5709	409	30	h(x	h(x	PROPN
ejpam-5709	409	31	)	)	PUNCT
ejpam-5709	409	32	in	in	ADP
ejpam-5709	409	33	ok	ok	ADJ
ejpam-5709	409	34	[	[	X
ejpam-5709	409	35	x	x	X
ejpam-5709	409	36	]	]	X
ejpam-5709	409	37	.	.	PUNCT
ejpam-5709	410	1	by	by	ADP
ejpam-5709	410	2	the	the	DET
ejpam-5709	410	3	same	same	ADJ
ejpam-5709	410	4	proof	proof	NOUN
ejpam-5709	410	5	as	as	ADP
ejpam-5709	410	6	in	in	ADP
ejpam-5709	410	7	theorem	theorem	NOUN
ejpam-5709	410	8	3	3	NUM
ejpam-5709	410	9	,	,	PUNCT
ejpam-5709	410	10	we	we	PRON
ejpam-5709	410	11	have	have	VERB
ejpam-5709	410	12	that	that	SCONJ
ejpam-5709	410	13	either	either	CCONJ
ejpam-5709	410	14	|ω|	|ω|	VERB
ejpam-5709	410	15	≥	≥	PRON
ejpam-5709	410	16	|h(β)|	|h(β)|	NOUN
ejpam-5709	410	17	or	or	CCONJ
ejpam-5709	410	18	|ω|	|ω|	VERB
ejpam-5709	410	19	≥	≥	NUM
ejpam-5709	410	20	|g(β)|	|g(β)|	NOUN
ejpam-5709	410	21	.	.	NOUN
ejpam-5709	410	22	without	without	ADP
ejpam-5709	410	23	loss	loss	NOUN
ejpam-5709	410	24	of	of	ADP
ejpam-5709	410	25	generality	generality	NOUN
ejpam-5709	410	26	,	,	PUNCT
ejpam-5709	410	27	we	we	PRON
ejpam-5709	410	28	may	may	AUX
ejpam-5709	410	29	assume	assume	VERB
ejpam-5709	410	30	that	that	SCONJ
ejpam-5709	410	31	|ω|	|ω|	VERB
ejpam-5709	410	32	≥	≥	PRON
ejpam-5709	410	33	|g(β)|	|g(β)|	NOUN
ejpam-5709	410	34	.	.	PROPN
ejpam-5709	410	35	since	since	SCONJ
ejpam-5709	410	36	αi	αi	PROPN
ejpam-5709	410	37	∈	∈	PROPN
ejpam-5709	410	38	c′	c′	PROPN
ejpam-5709	410	39	for	for	ADP
ejpam-5709	410	40	all	all	PRON
ejpam-5709	410	41	i	i	PRON
ejpam-5709	410	42	∈	∈	PROPN
ejpam-5709	410	43	{	{	PUNCT
ejpam-5709	410	44	0	0	NUM
ejpam-5709	410	45	,	,	PUNCT
ejpam-5709	410	46	1	1	NUM
ejpam-5709	410	47	,	,	PUNCT
ejpam-5709	410	48	.	.	PUNCT
ejpam-5709	410	49	.	.	PUNCT
ejpam-5709	411	1	.	.	PUNCT
ejpam-5709	412	1	,	,	PUNCT
ejpam-5709	412	2	n−	n−	NOUN
ejpam-5709	412	3	1	1	NUM
ejpam-5709	412	4	}	}	PUNCT
ejpam-5709	412	5	,	,	PUNCT
ejpam-5709	412	6	we	we	PRON
ejpam-5709	412	7	have	have	VERB
ejpam-5709	412	8	|αi|	|αi|	NUM
ejpam-5709	412	9	≤	≤	NUM
ejpam-5709	412	10	m	m	VERB
ejpam-5709	412	11	for	for	ADP
ejpam-5709	412	12	all	all	PRON
ejpam-5709	412	13	i	i	PRON
ejpam-5709	412	14	∈	∈	PROPN
ejpam-5709	412	15	{	{	PUNCT
ejpam-5709	412	16	0	0	NUM
ejpam-5709	412	17	,	,	PUNCT
ejpam-5709	412	18	1	1	NUM
ejpam-5709	412	19	,	,	PUNCT
ejpam-5709	412	20	.	.	PUNCT
ejpam-5709	412	21	.	.	PUNCT
ejpam-5709	413	1	.	.	PUNCT
ejpam-5709	414	1	,	,	PUNCT
ejpam-5709	414	2	n−	n−	NOUN
ejpam-5709	414	3	1	1	NUM
ejpam-5709	414	4	}	}	PUNCT
ejpam-5709	414	5	,	,	PUNCT
ejpam-5709	414	6	where	where	SCONJ
ejpam-5709	414	7	m	m	NOUN
ejpam-5709	414	8	is	be	AUX
ejpam-5709	414	9	defined	define	VERB
ejpam-5709	414	10	as	as	ADP
ejpam-5709	414	11	in	in	ADP
ejpam-5709	414	12	(	(	PUNCT
ejpam-5709	414	13	20	20	NUM
ejpam-5709	414	14	)	)	PUNCT
ejpam-5709	414	15	.	.	PUNCT
ejpam-5709	415	1	we	we	PRON
ejpam-5709	415	2	now	now	ADV
ejpam-5709	415	3	show	show	VERB
ejpam-5709	415	4	that	that	SCONJ
ejpam-5709	415	5	|β|	|β|	PRON
ejpam-5709	415	6	−	−	PROPN
ejpam-5709	415	7	1	1	NUM
ejpam-5709	415	8	+	+	CCONJ
ejpam-5709	415	9	√	√	NUM
ejpam-5709	415	10	1	1	NUM
ejpam-5709	416	1	+	+	NUM
ejpam-5709	416	2	4	4	NUM
ejpam-5709	416	3	m	m	NUM
ejpam-5709	416	4	2	2	NUM
ejpam-5709	416	5	≥	≥	NOUN
ejpam-5709	416	6	|ω|	|ω|	PROPN
ejpam-5709	416	7	.	.	PUNCT
ejpam-5709	417	1	(	(	PUNCT
ejpam-5709	417	2	23	23	NUM
ejpam-5709	417	3	)	)	PUNCT
ejpam-5709	417	4	as	as	ADP
ejpam-5709	417	5	|β|	|β|	PRON
ejpam-5709	417	6	≥	≥	NOUN
ejpam-5709	417	7	2|ω|+	2|ω|+	NUM
ejpam-5709	417	8	√	√	NUM
ejpam-5709	417	9	(	(	PUNCT
ejpam-5709	417	10	9−m)/4	9−m)/4	NUM
ejpam-5709	417	11	,	,	PUNCT
ejpam-5709	417	12	we	we	PRON
ejpam-5709	417	13	have	have	VERB
ejpam-5709	417	14	|β|2−	|β|2−	NOUN
ejpam-5709	417	15	(	(	PUNCT
ejpam-5709	417	16	2|ω|+	2|ω|+	NUM
ejpam-5709	417	17	1	1	NUM
ejpam-5709	417	18	+	+	CCONJ
ejpam-5709	417	19	√	√	PROPN
ejpam-5709	417	20	(	(	PUNCT
ejpam-5709	417	21	9−m)/4	9−m)/4	NUM
ejpam-5709	417	22	)	)	PUNCT
ejpam-5709	417	23	|β|+2|ω|+	|β|+2|ω|+	ADP
ejpam-5709	417	24	√	√	INTJ
ejpam-5709	417	25	(	(	PUNCT
ejpam-5709	417	26	9−m)/4	9−m)/4	NUM
ejpam-5709	417	27	=	=	SYM
ejpam-5709	417	28	(	(	PUNCT
ejpam-5709	417	29	|β|	|β|	NOUN
ejpam-5709	417	30	−	−	PROPN
ejpam-5709	417	31	1	1	NUM
ejpam-5709	417	32	)	)	PUNCT
ejpam-5709	417	33	[	[	PUNCT
ejpam-5709	417	34	|β|	|β|	NOUN
ejpam-5709	417	35	−	−	PROPN
ejpam-5709	417	36	(	(	PUNCT
ejpam-5709	417	37	2|ω|+	2|ω|+	NUM
ejpam-5709	417	38	√	√	NUM
ejpam-5709	417	39	(	(	PUNCT
ejpam-5709	417	40	9−m)/4	9−m)/4	NUM
ejpam-5709	417	41	)	)	PUNCT
ejpam-5709	417	42	]	]	PUNCT
ejpam-5709	417	43	≥	≥	NOUN
ejpam-5709	417	44	0	0	NUM
ejpam-5709	417	45	.	.	PUNCT
ejpam-5709	418	1	since	since	SCONJ
ejpam-5709	418	2	|ω|2	|ω|2	PROPN
ejpam-5709	418	3	+	+	CCONJ
ejpam-5709	418	4	|ω|	|ω|	NUM
ejpam-5709	418	5	≥	≥	NOUN
ejpam-5709	418	6	2|ω|	2|ω|	NUM
ejpam-5709	418	7	,	,	PUNCT
ejpam-5709	418	8	it	it	PRON
ejpam-5709	418	9	follows	follow	VERB
ejpam-5709	418	10	that	that	SCONJ
ejpam-5709	418	11	4	4	NUM
ejpam-5709	418	12	[	[	PUNCT
ejpam-5709	418	13	|β|2	|β|2	ADJ
ejpam-5709	418	14	−	−	PROPN
ejpam-5709	418	15	(	(	PUNCT
ejpam-5709	418	16	2|ω|+	2|ω|+	NUM
ejpam-5709	418	17	1	1	NUM
ejpam-5709	418	18	+	+	CCONJ
ejpam-5709	418	19	√	√	PROPN
ejpam-5709	418	20	(	(	PUNCT
ejpam-5709	418	21	9−m)/4	9−m)/4	NUM
ejpam-5709	418	22	)	)	PUNCT
ejpam-5709	418	23	|β|+	|β|+	PROPN
ejpam-5709	418	24	|ω|2	|ω|2	PROPN
ejpam-5709	418	25	+	+	CCONJ
ejpam-5709	418	26	|ω|+	|ω|+	NOUN
ejpam-5709	418	27	√	√	PROPN
ejpam-5709	418	28	(	(	PUNCT
ejpam-5709	418	29	9−m)/4	9−m)/4	NUM
ejpam-5709	418	30	]	]	PUNCT
ejpam-5709	418	31	≥	≥	X
ejpam-5709	418	32	0	0	NUM
ejpam-5709	418	33	and	and	CCONJ
ejpam-5709	418	34	so	so	ADV
ejpam-5709	418	35	[	[	X
ejpam-5709	418	36	2|β|	2|β|	NUM
ejpam-5709	418	37	−	−	NOUN
ejpam-5709	418	38	(	(	PUNCT
ejpam-5709	418	39	2|ω|+	2|ω|+	NUM
ejpam-5709	418	40	1)]2	1)]2	NUM
ejpam-5709	418	41	=	=	SYM
ejpam-5709	419	1	4|β|2	4|β|2	NUM
ejpam-5709	419	2	−	−	NUM
ejpam-5709	419	3	4(2|ω|	4(2|ω|	NUM
ejpam-5709	419	4	+	+	CCONJ
ejpam-5709	419	5	1)|β|	1)|β|	NUM
ejpam-5709	419	6	+	+	CCONJ
ejpam-5709	419	7	4|ω|2	4|ω|2	NUM
ejpam-5709	419	8	+	+	SYM
ejpam-5709	419	9	4|ω|	4|ω|	NUM
ejpam-5709	420	1	+	+	SYM
ejpam-5709	420	2	1	1	NUM
ejpam-5709	420	3	≥	≥	NOUN
ejpam-5709	420	4	1	1	NUM
ejpam-5709	420	5	+	+	CCONJ
ejpam-5709	420	6	4	4	NUM
ejpam-5709	420	7	√	√	NUM
ejpam-5709	420	8	(	(	PUNCT
ejpam-5709	420	9	9−m)/4	9−m)/4	NUM
ejpam-5709	420	10	(	(	PUNCT
ejpam-5709	420	11	|β|	|β|	NOUN
ejpam-5709	420	12	−	−	PROPN
ejpam-5709	420	13	1	1	NUM
ejpam-5709	420	14	)	)	PUNCT
ejpam-5709	420	15	.	.	PUNCT
ejpam-5709	421	1	again	again	ADV
ejpam-5709	421	2	,	,	PUNCT
ejpam-5709	421	3	|β|	|β|	PRON
ejpam-5709	421	4	≥	≥	VERB
ejpam-5709	421	5	2|ω|	2|ω|	NUM
ejpam-5709	421	6	+	+	CCONJ
ejpam-5709	421	7	√	√	PROPN
ejpam-5709	421	8	(	(	PUNCT
ejpam-5709	421	9	9−m)/4	9−m)/4	NUM
ejpam-5709	421	10	shows	show	VERB
ejpam-5709	421	11	that	that	SCONJ
ejpam-5709	421	12	2|β|	2|β|	NUM
ejpam-5709	421	13	−	−	PROPN
ejpam-5709	421	14	(	(	PUNCT
ejpam-5709	421	15	2|ω|	2|ω|	NUM
ejpam-5709	421	16	+	+	CCONJ
ejpam-5709	421	17	1	1	NUM
ejpam-5709	421	18	)	)	PUNCT
ejpam-5709	421	19	>	>	X
ejpam-5709	421	20	0	0	PUNCT
ejpam-5709	421	21	and	and	CCONJ
ejpam-5709	421	22	thus	thus	ADV
ejpam-5709	421	23	,	,	PUNCT
ejpam-5709	421	24	2|β|	2|β|	NUM
ejpam-5709	421	25	−	−	PROPN
ejpam-5709	421	26	(	(	PUNCT
ejpam-5709	421	27	2|ω|	2|ω|	NUM
ejpam-5709	421	28	+	+	CCONJ
ejpam-5709	421	29	1	1	X
ejpam-5709	421	30	)	)	PUNCT
ejpam-5709	421	31	≥√	≥√	NOUN
ejpam-5709	421	32	1	1	NUM
ejpam-5709	422	1	+	+	CCONJ
ejpam-5709	422	2	4	4	NUM
ejpam-5709	422	3	√	√	NUM
ejpam-5709	422	4	(	(	PUNCT
ejpam-5709	422	5	9−m)/4	9−m)/4	NUM
ejpam-5709	422	6	(	(	PUNCT
ejpam-5709	422	7	|β|	|β|	NOUN
ejpam-5709	422	8	−	−	PROPN
ejpam-5709	422	9	1	1	NUM
ejpam-5709	422	10	)	)	PUNCT
ejpam-5709	422	11	.	.	PUNCT
ejpam-5709	423	1	hence	hence	ADV
ejpam-5709	423	2	,	,	PUNCT
ejpam-5709	423	3	|β|	|β|	X
ejpam-5709	423	4	≥	≥	NOUN
ejpam-5709	423	5	2|ω|+	2|ω|+	NUM
ejpam-5709	423	6	1	1	NUM
ejpam-5709	423	7	+	+	CCONJ
ejpam-5709	423	8	√	√	NUM
ejpam-5709	423	9	1	1	NUM
ejpam-5709	423	10	+	+	CCONJ
ejpam-5709	423	11	4	4	NUM
ejpam-5709	423	12	√	√	NUM
ejpam-5709	423	13	(	(	PUNCT
ejpam-5709	423	14	9−m)/4	9−m)/4	NUM
ejpam-5709	423	15	(	(	PUNCT
ejpam-5709	423	16	|β|	|β|	NOUN
ejpam-5709	423	17	−	−	PROPN
ejpam-5709	423	18	1	1	NUM
ejpam-5709	423	19	)	)	PUNCT
ejpam-5709	423	20	2	2	NUM
ejpam-5709	423	21	.	.	PUNCT
ejpam-5709	424	1	since	since	SCONJ
ejpam-5709	424	2	a	a	DET
ejpam-5709	424	3	≥	≥	NOUN
ejpam-5709	424	4	1	1	NUM
ejpam-5709	424	5	,	,	PUNCT
ejpam-5709	424	6	it	it	PRON
ejpam-5709	424	7	follows	follow	VERB
ejpam-5709	424	8	from	from	ADP
ejpam-5709	424	9	lemma	lemma	PROPN
ejpam-5709	424	10	4	4	NUM
ejpam-5709	425	1	that	that	PRON
ejpam-5709	425	2	√	√	VERB
ejpam-5709	425	3	(	(	PUNCT
ejpam-5709	425	4	9−m)/4	9−m)/4	NUM
ejpam-5709	425	5	(	(	PUNCT
ejpam-5709	425	6	|β|	|β|	NOUN
ejpam-5709	425	7	−	−	PROPN
ejpam-5709	425	8	1	1	NUM
ejpam-5709	425	9	)	)	PUNCT
ejpam-5709	425	10	≥	≥	NOUN
ejpam-5709	425	11	m	m	PROPN
ejpam-5709	425	12	.	.	PUNCT
ejpam-5709	425	13	thus	thus	ADV
ejpam-5709	425	14	,	,	PUNCT
ejpam-5709	425	15	|β|	|β|	PRON
ejpam-5709	425	16	≥	≥	VERB
ejpam-5709	425	17	2|ω|+	2|ω|+	NUM
ejpam-5709	425	18	1	1	NUM
ejpam-5709	425	19	+	+	CCONJ
ejpam-5709	425	20	√	√	NUM
ejpam-5709	425	21	1	1	NUM
ejpam-5709	426	1	+	+	NUM
ejpam-5709	426	2	4	4	NUM
ejpam-5709	426	3	m	m	NOUN
ejpam-5709	426	4	2	2	NUM
ejpam-5709	426	5	=	=	SYM
ejpam-5709	426	6	|ω|+	|ω|+	NOUN
ejpam-5709	426	7	1	1	NUM
ejpam-5709	426	8	+	+	CCONJ
ejpam-5709	426	9	√	√	NUM
ejpam-5709	426	10	1	1	NUM
ejpam-5709	426	11	+	+	NUM
ejpam-5709	426	12	4	4	NUM
ejpam-5709	426	13	m	m	NUM
ejpam-5709	426	14	2	2	NUM
ejpam-5709	426	15	,	,	PUNCT
ejpam-5709	426	16	p.	p.	NOUN
ejpam-5709	426	17	phetnun	phetnun	PROPN
ejpam-5709	426	18	,	,	PUNCT
ejpam-5709	426	19	n.r	n.r	PROPN
ejpam-5709	426	20	.	.	PROPN
ejpam-5709	426	21	kanasri	kanasri	PROPN
ejpam-5709	426	22	/	/	SYM
ejpam-5709	426	23	eur	eur	PROPN
ejpam-5709	426	24	.	.	PUNCT
ejpam-5709	427	1	j.	j.	PROPN
ejpam-5709	427	2	pure	pure	PROPN
ejpam-5709	427	3	appl	appl	PROPN
ejpam-5709	427	4	.	.	PROPN
ejpam-5709	427	5	math	math	PROPN
ejpam-5709	427	6	,	,	PUNCT
ejpam-5709	427	7	18	18	NUM
ejpam-5709	427	8	(	(	PUNCT
ejpam-5709	427	9	1	1	NUM
ejpam-5709	427	10	)	)	PUNCT
ejpam-5709	427	11	(	(	PUNCT
ejpam-5709	427	12	2025	2025	NUM
ejpam-5709	427	13	)	)	PUNCT
ejpam-5709	427	14	,	,	PUNCT
ejpam-5709	427	15	5709	5709	NUM
ejpam-5709	427	16	14	14	NUM
ejpam-5709	427	17	of	of	ADP
ejpam-5709	427	18	15	15	NUM
ejpam-5709	427	19	which	which	PRON
ejpam-5709	427	20	proves	prove	VERB
ejpam-5709	427	21	(	(	PUNCT
ejpam-5709	427	22	23	23	NUM
ejpam-5709	427	23	)	)	PUNCT
ejpam-5709	427	24	.	.	PUNCT
ejpam-5709	428	1	the	the	DET
ejpam-5709	428	2	remaining	remain	VERB
ejpam-5709	428	3	proof	proof	NOUN
ejpam-5709	428	4	is	be	AUX
ejpam-5709	428	5	similar	similar	ADJ
ejpam-5709	428	6	to	to	ADP
ejpam-5709	428	7	that	that	PRON
ejpam-5709	428	8	of	of	ADP
ejpam-5709	428	9	theorem	theorem	NOUN
ejpam-5709	428	10	3	3	NUM
ejpam-5709	428	11	,	,	PUNCT
ejpam-5709	428	12	so	so	SCONJ
ejpam-5709	428	13	we	we	PRON
ejpam-5709	428	14	omit	omit	VERB
ejpam-5709	428	15	it	it	PRON
ejpam-5709	428	16	here	here	ADV
ejpam-5709	428	17	.	.	PUNCT
ejpam-5709	429	1	we	we	PRON
ejpam-5709	429	2	end	end	VERB
ejpam-5709	429	3	this	this	DET
ejpam-5709	429	4	paper	paper	NOUN
ejpam-5709	429	5	with	with	ADP
ejpam-5709	429	6	the	the	DET
ejpam-5709	429	7	following	follow	VERB
ejpam-5709	429	8	examples	example	NOUN
ejpam-5709	429	9	,	,	PUNCT
ejpam-5709	429	10	illustrating	illustrate	VERB
ejpam-5709	429	11	the	the	DET
ejpam-5709	429	12	use	use	NOUN
ejpam-5709	429	13	of	of	ADP
ejpam-5709	429	14	theorem	theorem	ADJ
ejpam-5709	429	15	4	4	NUM
ejpam-5709	429	16	.	.	NOUN
ejpam-5709	429	17	example	example	NOUN
ejpam-5709	429	18	6	6	NUM
ejpam-5709	429	19	.	.	PUNCT
ejpam-5709	430	1	let	let	VERB
ejpam-5709	430	2	k	k	NOUN
ejpam-5709	431	1	=	=	PUNCT
ejpam-5709	431	2	q	q	X
ejpam-5709	431	3	(	(	PUNCT
ejpam-5709	431	4	√	√	NUM
ejpam-5709	431	5	−3	−3	NUM
ejpam-5709	431	6	)	)	PUNCT
ejpam-5709	431	7	,	,	PUNCT
ejpam-5709	431	8	β	β	X
ejpam-5709	431	9	=	=	SYM
ejpam-5709	431	10	7	7	NUM
ejpam-5709	431	11	+	+	CCONJ
ejpam-5709	431	12	σ−3	σ−3	PROPN
ejpam-5709	431	13	,	,	PUNCT
ejpam-5709	431	14	ω	ω	NOUN
ejpam-5709	431	15	=	=	SYM
ejpam-5709	431	16	2	2	NUM
ejpam-5709	431	17	,	,	PUNCT
ejpam-5709	431	18	and	and	CCONJ
ejpam-5709	431	19	π	π	X
ejpam-5709	431	20	=	=	SYM
ejpam-5709	431	21	54343	54343	NUM
ejpam-5709	431	22	+	+	CCONJ
ejpam-5709	431	23	63615σ−3	63615σ−3	X
ejpam-5709	431	24	.	.	PUNCT
ejpam-5709	432	1	then	then	ADV
ejpam-5709	432	2	d	d	X
ejpam-5709	432	3	=	=	SYM
ejpam-5709	432	4	1	1	NUM
ejpam-5709	432	5	and	and	CCONJ
ejpam-5709	432	6	thus	thus	ADV
ejpam-5709	432	7	c′	c′	VERB
ejpam-5709	432	8	=	=	SYM
ejpam-5709	432	9	{	{	PUNCT
ejpam-5709	432	10	0	0	NUM
ejpam-5709	432	11	,	,	PUNCT
ejpam-5709	432	12	1	1	NUM
ejpam-5709	432	13	,	,	PUNCT
ejpam-5709	432	14	2	2	NUM
ejpam-5709	432	15	,	,	PUNCT
ejpam-5709	432	16	3	3	NUM
ejpam-5709	432	17	,	,	PUNCT
ejpam-5709	432	18	4	4	NUM
ejpam-5709	432	19	,	,	PUNCT
ejpam-5709	432	20	5	5	NUM
ejpam-5709	432	21	,	,	PUNCT
ejpam-5709	432	22	6	6	NUM
ejpam-5709	432	23	}	}	PUNCT
ejpam-5709	432	24	.	.	PUNCT
ejpam-5709	433	1	one	one	PRON
ejpam-5709	433	2	can	can	AUX
ejpam-5709	433	3	see	see	VERB
ejpam-5709	433	4	that	that	PRON
ejpam-5709	433	5	|β|	|β|	NOUN
ejpam-5709	433	6	=	=	SYM
ejpam-5709	433	7	√	√	PROPN
ejpam-5709	433	8	57	57	NUM
ejpam-5709	433	9	>	>	SYM
ejpam-5709	433	10	4	4	NUM
ejpam-5709	433	11	+	+	CCONJ
ejpam-5709	433	12	√	√	NUM
ejpam-5709	433	13	3	3	NUM
ejpam-5709	433	14	=	=	SYM
ejpam-5709	433	15	2|ω|	2|ω|	NUM
ejpam-5709	433	16	+	+	CCONJ
ejpam-5709	433	17	√	√	PROPN
ejpam-5709	433	18	(	(	PUNCT
ejpam-5709	433	19	9−m)/4	9−m)/4	NUM
ejpam-5709	433	20	,	,	PUNCT
ejpam-5709	433	21	a	a	DET
ejpam-5709	433	22	=	=	SYM
ejpam-5709	433	23	7	7	NUM
ejpam-5709	433	24	>	>	SYM
ejpam-5709	433	25	1	1	NUM
ejpam-5709	433	26	,	,	PUNCT
ejpam-5709	433	27	and	and	CCONJ
ejpam-5709	433	28	a	a	DET
ejpam-5709	433	29	+	+	ADJ
ejpam-5709	433	30	(	(	PUNCT
ejpam-5709	433	31	b/2	b/2	NOUN
ejpam-5709	433	32	)	)	PUNCT
ejpam-5709	433	33	=	=	PUNCT
ejpam-5709	433	34	7	7	NUM
ejpam-5709	434	1	+	+	CCONJ
ejpam-5709	434	2	(	(	PUNCT
ejpam-5709	434	3	1/2	1/2	NUM
ejpam-5709	434	4	)	)	PUNCT
ejpam-5709	434	5	>	>	SYM
ejpam-5709	434	6	2	2	X
ejpam-5709	434	7	=	=	SYM
ejpam-5709	434	8	|ω|	|ω|	PROPN
ejpam-5709	434	9	.	.	PROPN
ejpam-5709	434	10	note	note	NOUN
ejpam-5709	434	11	that	that	SCONJ
ejpam-5709	434	12	ok	ok	INTJ
ejpam-5709	434	13	is	be	AUX
ejpam-5709	434	14	a	a	DET
ejpam-5709	434	15	unique	unique	ADJ
ejpam-5709	434	16	factorization	factorization	NOUN
ejpam-5709	434	17	domain	domain	NOUN
ejpam-5709	434	18	and	and	CCONJ
ejpam-5709	434	19	π	π	PROPN
ejpam-5709	434	20	is	be	AUX
ejpam-5709	434	21	a	a	DET
ejpam-5709	434	22	prime	prime	ADJ
ejpam-5709	434	23	element	element	NOUN
ejpam-5709	434	24	because	because	SCONJ
ejpam-5709	434	25	n(π	n(π	NOUN
ejpam-5709	434	26	)	)	PUNCT
ejpam-5709	435	1	=	=	SYM
ejpam-5709	435	2	10457059819	10457059819	NUM
ejpam-5709	435	3	is	be	AUX
ejpam-5709	435	4	a	a	DET
ejpam-5709	435	5	rational	rational	ADJ
ejpam-5709	435	6	prime	prime	NOUN
ejpam-5709	435	7	.	.	PUNCT
ejpam-5709	436	1	now	now	ADV
ejpam-5709	436	2	,	,	PUNCT
ejpam-5709	436	3	we	we	PRON
ejpam-5709	436	4	have	have	VERB
ejpam-5709	436	5	|π|	|π|	NOUN
ejpam-5709	436	6	>	>	X
ejpam-5709	436	7	√	√	PROPN
ejpam-5709	436	8	(	(	PUNCT
ejpam-5709	436	9	9−m)/4	9−m)/4	NUM
ejpam-5709	436	10	(	(	PUNCT
ejpam-5709	436	11	|β|	|β|	NOUN
ejpam-5709	436	12	−	−	PROPN
ejpam-5709	436	13	1	1	NUM
ejpam-5709	436	14	)	)	PUNCT
ejpam-5709	436	15	and	and	CCONJ
ejpam-5709	436	16	ωπ	ωπ	NUM
ejpam-5709	436	17	=	=	NOUN
ejpam-5709	436	18	108686	108686	NUM
ejpam-5709	436	19	+	+	SYM
ejpam-5709	436	20	127230σ−3	127230σ−3	NUM
ejpam-5709	436	21	=	=	SYM
ejpam-5709	436	22	8β5	8β5	NUM
ejpam-5709	437	1	+	+	CCONJ
ejpam-5709	437	2	2β4	2β4	NUM
ejpam-5709	437	3	+	+	NUM
ejpam-5709	437	4	4β3	4β3	NUM
ejpam-5709	437	5	+	+	CCONJ
ejpam-5709	437	6	2β2	2β2	NUM
ejpam-5709	437	7	+	+	CCONJ
ejpam-5709	437	8	6β	6β	NOUN
ejpam-5709	437	9	+	+	CCONJ
ejpam-5709	437	10	2	2	NUM
ejpam-5709	437	11	,	,	PUNCT
ejpam-5709	437	12	which	which	PRON
ejpam-5709	437	13	is	be	AUX
ejpam-5709	437	14	a	a	DET
ejpam-5709	437	15	base	base	NOUN
ejpam-5709	437	16	-	-	PUNCT
ejpam-5709	437	17	β(c′	β(c′	NUM
ejpam-5709	437	18	)	)	PUNCT
ejpam-5709	437	19	representation	representation	NOUN
ejpam-5709	437	20	of	of	ADP
ejpam-5709	437	21	ωπ	ωπ	PROPN
ejpam-5709	437	22	.	.	PUNCT
ejpam-5709	438	1	moreover	moreover	ADV
ejpam-5709	438	2	,	,	PUNCT
ejpam-5709	438	3	re(α5	re(α5	NUM
ejpam-5709	438	4	)	)	PUNCT
ejpam-5709	438	5	=	=	SYM
ejpam-5709	438	6	8	8	NUM
ejpam-5709	438	7	>	>	SYM
ejpam-5709	438	8	1	1	NUM
ejpam-5709	438	9	and	and	CCONJ
ejpam-5709	438	10	re(α4	re(α4	NUM
ejpam-5709	438	11	)	)	PUNCT
ejpam-5709	438	12	im(α5	im(α5	NUM
ejpam-5709	438	13	)	)	PUNCT
ejpam-5709	438	14	=	=	SYM
ejpam-5709	439	1	(	(	PUNCT
ejpam-5709	439	2	2)(0	2)(0	NUM
ejpam-5709	439	3	)	)	PUNCT
ejpam-5709	439	4	≥	≥	NOUN
ejpam-5709	439	5	(	(	PUNCT
ejpam-5709	439	6	8)(0	8)(0	NUM
ejpam-5709	439	7	)	)	PUNCT
ejpam-5709	439	8	=	=	SYM
ejpam-5709	439	9	re(α5	re(α5	NUM
ejpam-5709	439	10	)	)	PUNCT
ejpam-5709	439	11	im(α4	im(α4	NUM
ejpam-5709	439	12	)	)	PUNCT
ejpam-5709	439	13	.	.	PUNCT
ejpam-5709	440	1	by	by	ADP
ejpam-5709	440	2	theorem	theorem	NOUN
ejpam-5709	440	3	4	4	NUM
ejpam-5709	440	4	,	,	PUNCT
ejpam-5709	440	5	we	we	PRON
ejpam-5709	440	6	obtain	obtain	VERB
ejpam-5709	440	7	that	that	SCONJ
ejpam-5709	440	8	f(x	f(x	NOUN
ejpam-5709	440	9	)	)	PUNCT
ejpam-5709	440	10	=	=	PUNCT
ejpam-5709	441	1	8x5	8x5	NUM
ejpam-5709	441	2	+	+	NOUN
ejpam-5709	441	3	2x4	2x4	NUM
ejpam-5709	441	4	+	+	NOUN
ejpam-5709	441	5	4x3	4x3	NUM
ejpam-5709	441	6	+	+	SYM
ejpam-5709	441	7	2x2	2x2	NUM
ejpam-5709	441	8	+	+	SYM
ejpam-5709	441	9	6x+2	6x+2	PROPN
ejpam-5709	441	10	has	have	VERB
ejpam-5709	441	11	no	no	DET
ejpam-5709	441	12	proper	proper	ADJ
ejpam-5709	441	13	factorization	factorization	NOUN
ejpam-5709	441	14	in	in	ADP
ejpam-5709	441	15	ok	ok	ADJ
ejpam-5709	441	16	[	[	X
ejpam-5709	441	17	x	x	X
ejpam-5709	441	18	]	]	X
ejpam-5709	441	19	and	and	CCONJ
ejpam-5709	441	20	so	so	ADV
ejpam-5709	441	21	is	be	AUX
ejpam-5709	441	22	irreducible	irreducible	ADJ
ejpam-5709	441	23	over	over	ADP
ejpam-5709	441	24	k.	k.	PROPN
ejpam-5709	442	1	it	it	PRON
ejpam-5709	442	2	is	be	AUX
ejpam-5709	442	3	noticed	notice	VERB
ejpam-5709	442	4	that	that	SCONJ
ejpam-5709	442	5	f(x	f(x	PROPN
ejpam-5709	442	6	)	)	PUNCT
ejpam-5709	442	7	is	be	AUX
ejpam-5709	442	8	reducible	reducible	ADJ
ejpam-5709	442	9	in	in	ADP
ejpam-5709	442	10	ok	ok	ADJ
ejpam-5709	442	11	[	[	X
ejpam-5709	442	12	x	x	X
ejpam-5709	442	13	]	]	X
ejpam-5709	442	14	because	because	SCONJ
ejpam-5709	442	15	f(x	f(x	PROPN
ejpam-5709	442	16	)	)	PUNCT
ejpam-5709	442	17	=	=	SYM
ejpam-5709	443	1	2(4x5	2(4x5	NOUN
ejpam-5709	444	1	+	+	CCONJ
ejpam-5709	444	2	x4	x4	PROPN
ejpam-5709	445	1	+	+	CCONJ
ejpam-5709	445	2	2x3	2x3	NUM
ejpam-5709	445	3	+	+	CCONJ
ejpam-5709	445	4	x2	x2	PROPN
ejpam-5709	445	5	+	+	CCONJ
ejpam-5709	445	6	3x+	3x+	NUM
ejpam-5709	445	7	1	1	NUM
ejpam-5709	445	8	)	)	PUNCT
ejpam-5709	445	9	.	.	PUNCT
ejpam-5709	446	1	example	example	NOUN
ejpam-5709	447	1	7	7	NUM
ejpam-5709	447	2	.	.	PUNCT
ejpam-5709	447	3	let	let	VERB
ejpam-5709	447	4	k	k	NOUN
ejpam-5709	447	5	=	=	PUNCT
ejpam-5709	447	6	q	q	ADJ
ejpam-5709	447	7	(	(	PUNCT
ejpam-5709	447	8	√	√	NUM
ejpam-5709	447	9	−7	−7	NOUN
ejpam-5709	447	10	)	)	PUNCT
ejpam-5709	447	11	,	,	PUNCT
ejpam-5709	447	12	β	β	X
ejpam-5709	447	13	=	=	PUNCT
ejpam-5709	447	14	11	11	NUM
ejpam-5709	447	15	+	+	NOUN
ejpam-5709	447	16	3σ−7	3σ−7	NUM
ejpam-5709	447	17	,	,	PUNCT
ejpam-5709	447	18	ω	ω	NOUN
ejpam-5709	447	19	=	=	SYM
ejpam-5709	447	20	2+σ−7	2+σ−7	NUM
ejpam-5709	447	21	,	,	PUNCT
ejpam-5709	447	22	and	and	CCONJ
ejpam-5709	447	23	π	π	X
ejpam-5709	447	24	=	=	PUNCT
ejpam-5709	447	25	158481	158481	NUM
ejpam-5709	447	26	+	+	SYM
ejpam-5709	447	27	166844σ−7	166844σ−7	NUM
ejpam-5709	447	28	.	.	PUNCT
ejpam-5709	448	1	then	then	ADV
ejpam-5709	448	2	d	d	X
ejpam-5709	448	3	=	=	SYM
ejpam-5709	448	4	1	1	NUM
ejpam-5709	448	5	and	and	CCONJ
ejpam-5709	448	6	thus	thus	ADV
ejpam-5709	448	7	c′	c′	VERB
ejpam-5709	448	8	=	=	SYM
ejpam-5709	448	9	{	{	PUNCT
ejpam-5709	448	10	0	0	NUM
ejpam-5709	448	11	,	,	PUNCT
ejpam-5709	448	12	1	1	NUM
ejpam-5709	448	13	,	,	PUNCT
ejpam-5709	448	14	.	.	PUNCT
ejpam-5709	448	15	.	.	PUNCT
ejpam-5709	449	1	.	.	PUNCT
ejpam-5709	450	1	,	,	PUNCT
ejpam-5709	450	2	10	10	NUM
ejpam-5709	450	3	}	}	PUNCT
ejpam-5709	450	4	.	.	PUNCT
ejpam-5709	451	1	one	one	PRON
ejpam-5709	451	2	can	can	AUX
ejpam-5709	451	3	see	see	VERB
ejpam-5709	451	4	that	that	PRON
ejpam-5709	451	5	|β|	|β|	NOUN
ejpam-5709	451	6	=	=	SYM
ejpam-5709	451	7	√	√	PROPN
ejpam-5709	451	8	172	172	NUM
ejpam-5709	451	9	>	>	SYM
ejpam-5709	451	10	2	2	NUM
ejpam-5709	451	11	√	√	NUM
ejpam-5709	451	12	8	8	NUM
ejpam-5709	451	13	+	+	CCONJ
ejpam-5709	451	14	2	2	NUM
ejpam-5709	451	15	=	=	SYM
ejpam-5709	451	16	2|ω|+	2|ω|+	NUM
ejpam-5709	451	17	√	√	NUM
ejpam-5709	451	18	(	(	PUNCT
ejpam-5709	451	19	9−m)/4	9−m)/4	NUM
ejpam-5709	451	20	,	,	PUNCT
ejpam-5709	451	21	a	a	DET
ejpam-5709	451	22	=	=	SYM
ejpam-5709	451	23	11	11	NUM
ejpam-5709	451	24	>	>	SYM
ejpam-5709	451	25	1	1	NUM
ejpam-5709	451	26	,	,	PUNCT
ejpam-5709	451	27	and	and	CCONJ
ejpam-5709	451	28	a+	a+	X
ejpam-5709	451	29	(	(	PUNCT
ejpam-5709	451	30	b/2	b/2	NUM
ejpam-5709	451	31	)	)	PUNCT
ejpam-5709	451	32	=	=	SYM
ejpam-5709	452	1	11	11	NUM
ejpam-5709	452	2	+	+	CCONJ
ejpam-5709	452	3	(	(	PUNCT
ejpam-5709	452	4	3/2	3/2	NUM
ejpam-5709	452	5	)	)	PUNCT
ejpam-5709	452	6	>	>	X
ejpam-5709	453	1	√	√	NUM
ejpam-5709	453	2	8	8	NUM
ejpam-5709	453	3	=	=	SYM
ejpam-5709	453	4	|ω|	|ω|	PROPN
ejpam-5709	453	5	.	.	PROPN
ejpam-5709	453	6	note	note	NOUN
ejpam-5709	453	7	that	that	SCONJ
ejpam-5709	453	8	ok	ok	INTJ
ejpam-5709	453	9	is	be	AUX
ejpam-5709	453	10	a	a	DET
ejpam-5709	453	11	unique	unique	ADJ
ejpam-5709	453	12	factorization	factorization	NOUN
ejpam-5709	453	13	domain	domain	NOUN
ejpam-5709	453	14	and	and	CCONJ
ejpam-5709	453	15	π	π	PROPN
ejpam-5709	453	16	is	be	AUX
ejpam-5709	453	17	a	a	DET
ejpam-5709	453	18	prime	prime	ADJ
ejpam-5709	453	19	element	element	NOUN
ejpam-5709	453	20	because	because	SCONJ
ejpam-5709	453	21	n(π	n(π	NOUN
ejpam-5709	453	22	)	)	PUNCT
ejpam-5709	453	23	=	=	PUNCT
ejpam-5709	453	24	107231671997	107231671997	NUM
ejpam-5709	453	25	is	be	AUX
ejpam-5709	453	26	a	a	DET
ejpam-5709	453	27	rational	rational	ADJ
ejpam-5709	453	28	prime	prime	NOUN
ejpam-5709	453	29	.	.	PUNCT
ejpam-5709	454	1	now	now	ADV
ejpam-5709	454	2	,	,	PUNCT
ejpam-5709	454	3	we	we	PRON
ejpam-5709	454	4	have	have	VERB
ejpam-5709	454	5	|π|	|π|	NOUN
ejpam-5709	454	6	>	>	X
ejpam-5709	454	7	√	√	PROPN
ejpam-5709	454	8	(	(	PUNCT
ejpam-5709	454	9	9−m)/4	9−m)/4	NUM
ejpam-5709	454	10	(	(	PUNCT
ejpam-5709	454	11	|β|	|β|	NOUN
ejpam-5709	454	12	−	−	PROPN
ejpam-5709	454	13	1	1	NUM
ejpam-5709	454	14	)	)	PUNCT
ejpam-5709	454	15	and	and	CCONJ
ejpam-5709	454	16	ωπ	ωπ	NUM
ejpam-5709	454	17	=	=	SYM
ejpam-5709	454	18	−16726	−16726	PROPN
ejpam-5709	454	19	+	+	CCONJ
ejpam-5709	454	20	659013σ−7	659013σ−7	NUM
ejpam-5709	454	21	=	=	SYM
ejpam-5709	455	1	31β4	31β4	NUM
ejpam-5709	456	1	+	+	NUM
ejpam-5709	456	2	4β3	4β3	NUM
ejpam-5709	456	3	+	+	CCONJ
ejpam-5709	456	4	3β2	3β2	NUM
ejpam-5709	457	1	+	+	NUM
ejpam-5709	457	2	9β	9β	PRON
ejpam-5709	458	1	+	+	CCONJ
ejpam-5709	458	2	5	5	NUM
ejpam-5709	458	3	,	,	PUNCT
ejpam-5709	458	4	which	which	PRON
ejpam-5709	458	5	is	be	AUX
ejpam-5709	458	6	a	a	DET
ejpam-5709	458	7	base	base	NOUN
ejpam-5709	458	8	-	-	PUNCT
ejpam-5709	458	9	β(c′	β(c′	NUM
ejpam-5709	458	10	)	)	PUNCT
ejpam-5709	458	11	representation	representation	NOUN
ejpam-5709	458	12	of	of	ADP
ejpam-5709	458	13	ωπ	ωπ	PROPN
ejpam-5709	458	14	.	.	PUNCT
ejpam-5709	459	1	moreover	moreover	ADV
ejpam-5709	459	2	,	,	PUNCT
ejpam-5709	459	3	re(α4	re(α4	X
ejpam-5709	459	4	)	)	PUNCT
ejpam-5709	459	5	=	=	NOUN
ejpam-5709	459	6	31	31	NUM
ejpam-5709	459	7	>	>	SYM
ejpam-5709	459	8	1	1	NUM
ejpam-5709	459	9	and	and	CCONJ
ejpam-5709	459	10	re(α3	re(α3	NOUN
ejpam-5709	459	11	)	)	PUNCT
ejpam-5709	459	12	im(α4	im(α4	NUM
ejpam-5709	459	13	)	)	PUNCT
ejpam-5709	460	1	=	=	SYM
ejpam-5709	460	2	(	(	PUNCT
ejpam-5709	460	3	4)(0	4)(0	NUM
ejpam-5709	460	4	)	)	PUNCT
ejpam-5709	460	5	≥	≥	NOUN
ejpam-5709	460	6	(	(	PUNCT
ejpam-5709	460	7	31)(0	31)(0	NUM
ejpam-5709	460	8	)	)	PUNCT
ejpam-5709	460	9	=	=	PUNCT
ejpam-5709	460	10	re(α4	re(α4	X
ejpam-5709	460	11	)	)	PUNCT
ejpam-5709	460	12	im(α3	im(α3	NOUN
ejpam-5709	460	13	)	)	PUNCT
ejpam-5709	460	14	.	.	PUNCT
ejpam-5709	461	1	by	by	ADP
ejpam-5709	461	2	theorem	theorem	NOUN
ejpam-5709	461	3	4	4	NUM
ejpam-5709	461	4	,	,	PUNCT
ejpam-5709	461	5	we	we	PRON
ejpam-5709	461	6	obtain	obtain	VERB
ejpam-5709	461	7	that	that	SCONJ
ejpam-5709	461	8	f(x	f(x	NOUN
ejpam-5709	461	9	)	)	PUNCT
ejpam-5709	462	1	=	=	PUNCT
ejpam-5709	463	1	31x4	31x4	NUM
ejpam-5709	463	2	+	+	NUM
ejpam-5709	463	3	4x3	4x3	NUM
ejpam-5709	463	4	+	+	CCONJ
ejpam-5709	463	5	3x2	3x2	NUM
ejpam-5709	463	6	+	+	NOUN
ejpam-5709	463	7	9x+5	9x+5	NOUN
ejpam-5709	463	8	has	have	VERB
ejpam-5709	463	9	no	no	DET
ejpam-5709	463	10	proper	proper	ADJ
ejpam-5709	463	11	factorization	factorization	NOUN
ejpam-5709	463	12	in	in	ADP
ejpam-5709	463	13	ok	ok	ADJ
ejpam-5709	463	14	[	[	X
ejpam-5709	463	15	x	x	X
ejpam-5709	463	16	]	]	X
ejpam-5709	463	17	and	and	CCONJ
ejpam-5709	463	18	so	so	ADV
ejpam-5709	463	19	is	be	AUX
ejpam-5709	463	20	irreducible	irreducible	ADJ
ejpam-5709	463	21	over	over	ADP
ejpam-5709	463	22	k.	k.	PROPN
ejpam-5709	463	23	by	by	ADP
ejpam-5709	463	24	theorem	theorem	NOUN
ejpam-5709	463	25	9.29(3	9.29(3	NUM
ejpam-5709	463	26	)	)	PUNCT
ejpam-5709	463	27	in	in	ADP
ejpam-5709	463	28	[	[	X
ejpam-5709	463	29	6	6	NUM
ejpam-5709	463	30	]	]	PUNCT
ejpam-5709	463	31	,	,	PUNCT
ejpam-5709	463	32	one	one	PRON
ejpam-5709	463	33	can	can	AUX
ejpam-5709	463	34	verify	verify	VERB
ejpam-5709	463	35	that	that	SCONJ
ejpam-5709	463	36	the	the	DET
ejpam-5709	463	37	rational	rational	ADJ
ejpam-5709	463	38	primes	prime	NOUN
ejpam-5709	463	39	3	3	NUM
ejpam-5709	463	40	,	,	PUNCT
ejpam-5709	463	41	5	5	NUM
ejpam-5709	463	42	,	,	PUNCT
ejpam-5709	463	43	and	and	CCONJ
ejpam-5709	463	44	31	31	NUM
ejpam-5709	463	45	are	be	AUX
ejpam-5709	463	46	prime	prime	ADJ
ejpam-5709	463	47	elements	element	NOUN
ejpam-5709	463	48	of	of	ADP
ejpam-5709	463	49	ok	ok	INTJ
ejpam-5709	463	50	.	.	PUNCT
ejpam-5709	464	1	this	this	PRON
ejpam-5709	464	2	implies	imply	VERB
ejpam-5709	464	3	that	that	SCONJ
ejpam-5709	464	4	f(x	f(x	PROPN
ejpam-5709	464	5	)	)	PUNCT
ejpam-5709	464	6	is	be	AUX
ejpam-5709	464	7	irreducible	irreducible	ADJ
ejpam-5709	464	8	in	in	ADP
ejpam-5709	464	9	ok	ok	ADJ
ejpam-5709	464	10	[	[	X
ejpam-5709	464	11	x	x	X
ejpam-5709	464	12	]	]	X
ejpam-5709	464	13	.	.	PUNCT
ejpam-5709	465	1	by	by	ADP
ejpam-5709	465	2	taking	take	VERB
ejpam-5709	465	3	ω	ω	PROPN
ejpam-5709	465	4	∈	∈	PROPN
ejpam-5709	465	5	u(ok	u(ok	NOUN
ejpam-5709	465	6	)	)	PUNCT
ejpam-5709	465	7	in	in	ADP
ejpam-5709	465	8	theorem	theorem	NOUN
ejpam-5709	465	9	4	4	NUM
ejpam-5709	465	10	,	,	PUNCT
ejpam-5709	465	11	we	we	PRON
ejpam-5709	465	12	obtain	obtain	VERB
ejpam-5709	465	13	that	that	SCONJ
ejpam-5709	465	14	the	the	DET
ejpam-5709	465	15	polynomial	polynomial	ADJ
ejpam-5709	465	16	f(x	f(x	PROPN
ejpam-5709	465	17	)	)	PUNCT
ejpam-5709	465	18	has	have	VERB
ejpam-5709	465	19	no	no	DET
ejpam-5709	465	20	proper	proper	ADJ
ejpam-5709	465	21	factorization	factorization	NOUN
ejpam-5709	465	22	in	in	ADP
ejpam-5709	465	23	ok	ok	ADJ
ejpam-5709	465	24	[	[	X
ejpam-5709	465	25	x	x	X
ejpam-5709	465	26	]	]	X
ejpam-5709	465	27	.	.	PUNCT
ejpam-5709	466	1	since	since	SCONJ
ejpam-5709	466	2	π	π	PROPN
ejpam-5709	466	3	is	be	AUX
ejpam-5709	466	4	also	also	ADV
ejpam-5709	466	5	an	an	DET
ejpam-5709	466	6	irreducible	irreducible	ADJ
ejpam-5709	466	7	element	element	NOUN
ejpam-5709	466	8	,	,	PUNCT
ejpam-5709	466	9	then	then	ADV
ejpam-5709	466	10	δ	δ	PROPN
ejpam-5709	466	11	∤	∤	PROPN
ejpam-5709	466	12	f(x	f(x	PROPN
ejpam-5709	466	13	)	)	PUNCT
ejpam-5709	466	14	for	for	ADP
ejpam-5709	466	15	all	all	DET
ejpam-5709	466	16	δ	δ	PROPN
ejpam-5709	466	17	∈	∈	PROPN
ejpam-5709	466	18	ok\u(ok	ok\u(ok	NOUN
ejpam-5709	466	19	)	)	PUNCT
ejpam-5709	466	20	,	,	PUNCT
ejpam-5709	466	21	by	by	ADP
ejpam-5709	466	22	lemma	lemma	PROPN
ejpam-5709	466	23	6	6	NUM
ejpam-5709	466	24	in	in	ADP
ejpam-5709	466	25	[	[	X
ejpam-5709	466	26	8	8	NUM
ejpam-5709	466	27	]	]	PUNCT
ejpam-5709	466	28	.	.	PUNCT
ejpam-5709	467	1	this	this	PRON
ejpam-5709	467	2	shows	show	VERB
ejpam-5709	467	3	that	that	SCONJ
ejpam-5709	467	4	f(x	f(x	PROPN
ejpam-5709	467	5	)	)	PUNCT
ejpam-5709	467	6	is	be	AUX
ejpam-5709	467	7	irreducible	irreducible	ADJ
ejpam-5709	467	8	in	in	ADP
ejpam-5709	467	9	ok	ok	ADJ
ejpam-5709	467	10	[	[	X
ejpam-5709	467	11	x	x	X
ejpam-5709	467	12	]	]	X
ejpam-5709	467	13	.	.	PUNCT
ejpam-5709	468	1	therefore	therefore	ADV
ejpam-5709	468	2	,	,	PUNCT
ejpam-5709	468	3	theorem	theorem	VERB
ejpam-5709	468	4	4	4	NUM
ejpam-5709	468	5	is	be	AUX
ejpam-5709	468	6	a	a	DET
ejpam-5709	468	7	generalization	generalization	NOUN
ejpam-5709	468	8	of	of	ADP
ejpam-5709	468	9	theorem	theorem	NOUN
ejpam-5709	468	10	d	d	VERB
ejpam-5709	468	11	by	by	ADP
ejpam-5709	468	12	considering	consider	VERB
ejpam-5709	468	13	ωπ	ωπ	INTJ
ejpam-5709	468	14	instead	instead	ADV
ejpam-5709	468	15	of	of	ADP
ejpam-5709	468	16	π	π	PROPN
ejpam-5709	468	17	,	,	PUNCT
ejpam-5709	468	18	where	where	SCONJ
ejpam-5709	468	19	ω	ω	PROPN
ejpam-5709	468	20	∈	∈	PROPN
ejpam-5709	468	21	ok\{0	ok\{0	PROPN
ejpam-5709	468	22	}	}	PUNCT
ejpam-5709	468	23	and	and	CCONJ
ejpam-5709	468	24	π	π	PROPN
ejpam-5709	468	25	is	be	AUX
ejpam-5709	468	26	a	a	DET
ejpam-5709	468	27	prime	prime	ADJ
ejpam-5709	468	28	element	element	NOUN
ejpam-5709	468	29	.	.	PUNCT
ejpam-5709	469	1	4	4	X
ejpam-5709	469	2	.	.	X
ejpam-5709	469	3	conclusion	conclusion	NOUN
ejpam-5709	469	4	for	for	ADP
ejpam-5709	469	5	any	any	DET
ejpam-5709	469	6	imaginary	imaginary	ADJ
ejpam-5709	469	7	quadratic	quadratic	ADJ
ejpam-5709	469	8	field	field	NOUN
ejpam-5709	469	9	k	k	NOUN
ejpam-5709	469	10	,	,	PUNCT
ejpam-5709	469	11	we	we	PRON
ejpam-5709	469	12	provide	provide	VERB
ejpam-5709	469	13	the	the	DET
ejpam-5709	469	14	explicit	explicit	ADJ
ejpam-5709	469	15	shapes	shape	NOUN
ejpam-5709	469	16	of	of	ADP
ejpam-5709	469	17	all	all	DET
ejpam-5709	469	18	base	base	NOUN
ejpam-5709	469	19	-	-	PUNCT
ejpam-5709	469	20	β(c	β(c	NUM
ejpam-5709	469	21	)	)	PUNCT
ejpam-5709	469	22	representations	representation	NOUN
ejpam-5709	469	23	for	for	ADP
ejpam-5709	469	24	nonzero	nonzero	ADJ
ejpam-5709	469	25	elements	element	NOUN
ejpam-5709	469	26	of	of	ADP
ejpam-5709	469	27	ok	ok	INTJ
ejpam-5709	469	28	.	.	PUNCT
ejpam-5709	470	1	using	use	VERB
ejpam-5709	470	2	such	such	DET
ejpam-5709	470	3	a	a	DET
ejpam-5709	470	4	representation	representation	NOUN
ejpam-5709	470	5	,	,	PUNCT
ejpam-5709	470	6	irreducibility	irreducibility	NOUN
ejpam-5709	470	7	criteria	criterion	NOUN
ejpam-5709	470	8	for	for	ADP
ejpam-5709	470	9	polynomials	polynomial	NOUN
ejpam-5709	470	10	in	in	ADP
ejpam-5709	470	11	ok	ok	ADJ
ejpam-5709	470	12	[	[	X
ejpam-5709	470	13	x	x	X
ejpam-5709	470	14	]	]	X
ejpam-5709	470	15	are	be	AUX
ejpam-5709	470	16	established	establish	VERB
ejpam-5709	470	17	,	,	PUNCT
ejpam-5709	470	18	which	which	PRON
ejpam-5709	470	19	extend	extend	VERB
ejpam-5709	470	20	and	and	CCONJ
ejpam-5709	470	21	generalize	generalize	VERB
ejpam-5709	470	22	the	the	DET
ejpam-5709	470	23	authors	author	NOUN
ejpam-5709	470	24	’	'	PUNCT
ejpam-5709	470	25	earlier	early	ADJ
ejpam-5709	470	26	work	work	NOUN
ejpam-5709	470	27	.	.	PUNCT
ejpam-5709	471	1	acknowledgements	acknowledgement	NOUN
ejpam-5709	471	2	this	this	DET
ejpam-5709	471	3	work	work	NOUN
ejpam-5709	471	4	was	be	AUX
ejpam-5709	471	5	supported	support	VERB
ejpam-5709	471	6	by	by	ADP
ejpam-5709	471	7	the	the	DET
ejpam-5709	471	8	science	science	NOUN
ejpam-5709	471	9	achievement	achievement	NOUN
ejpam-5709	471	10	scholarship	scholarship	NOUN
ejpam-5709	471	11	of	of	ADP
ejpam-5709	471	12	thailand	thailand	PROPN
ejpam-5709	471	13	(	(	PUNCT
ejpam-5709	471	14	sast	sast	NOUN
ejpam-5709	471	15	)	)	PUNCT
ejpam-5709	471	16	.	.	PUNCT
ejpam-5709	472	1	p.	p.	NOUN
ejpam-5709	472	2	phetnun	phetnun	PROPN
ejpam-5709	472	3	,	,	PUNCT
ejpam-5709	472	4	n.r	n.r	PROPN
ejpam-5709	472	5	.	.	PROPN
ejpam-5709	472	6	kanasri	kanasri	PROPN
ejpam-5709	472	7	/	/	SYM
ejpam-5709	472	8	eur	eur	PROPN
ejpam-5709	472	9	.	.	PUNCT
ejpam-5709	473	1	j.	j.	PROPN
ejpam-5709	473	2	pure	pure	PROPN
ejpam-5709	473	3	appl	appl	PROPN
ejpam-5709	473	4	.	.	PROPN
ejpam-5709	473	5	math	math	PROPN
ejpam-5709	473	6	,	,	PUNCT
ejpam-5709	473	7	18	18	NUM
ejpam-5709	473	8	(	(	PUNCT
ejpam-5709	473	9	1	1	NUM
ejpam-5709	473	10	)	)	PUNCT
ejpam-5709	473	11	(	(	PUNCT
ejpam-5709	473	12	2025	2025	NUM
ejpam-5709	473	13	)	)	PUNCT
ejpam-5709	473	14	,	,	PUNCT
ejpam-5709	473	15	5709	5709	NUM
ejpam-5709	473	16	15	15	NUM
ejpam-5709	473	17	of	of	ADP
ejpam-5709	473	18	15	15	NUM
ejpam-5709	473	19	references	reference	NOUN
ejpam-5709	473	20	[	[	X
ejpam-5709	473	21	1	1	NUM
ejpam-5709	473	22	]	]	X
ejpam-5709	473	23	s	s	VERB
ejpam-5709	473	24	alaca	alaca	PROPN
ejpam-5709	473	25	and	and	CCONJ
ejpam-5709	473	26	k	k	PROPN
ejpam-5709	473	27	s	s	PROPN
ejpam-5709	473	28	williams	williams	PROPN
ejpam-5709	473	29	.	.	PUNCT
ejpam-5709	474	1	introductory	introductory	ADJ
ejpam-5709	474	2	algebraic	algebraic	ADJ
ejpam-5709	474	3	number	number	NOUN
ejpam-5709	474	4	theory	theory	NOUN
ejpam-5709	474	5	.	.	PUNCT
ejpam-5709	475	1	cambridge	cambridge	PROPN
ejpam-5709	475	2	university	university	PROPN
ejpam-5709	475	3	press	press	PROPN
ejpam-5709	475	4	,	,	PUNCT
ejpam-5709	475	5	cambridge	cambridge	PROPN
ejpam-5709	475	6	,	,	PUNCT
ejpam-5709	475	7	2004	2004	NUM
ejpam-5709	475	8	.	.	PUNCT
ejpam-5709	476	1	[	[	X
ejpam-5709	476	2	2	2	NUM
ejpam-5709	476	3	]	]	PUNCT
ejpam-5709	476	4	j	j	PROPN
ejpam-5709	476	5	brillhart	brillhart	NOUN
ejpam-5709	476	6	,	,	PUNCT
ejpam-5709	476	7	m	m	VERB
ejpam-5709	476	8	filaseta	filaseta	ADJ
ejpam-5709	476	9	,	,	PUNCT
ejpam-5709	476	10	and	and	CCONJ
ejpam-5709	476	11	a	a	DET
ejpam-5709	476	12	odlyzko	odlyzko	NOUN
ejpam-5709	476	13	.	.	PUNCT
ejpam-5709	477	1	on	on	ADP
ejpam-5709	477	2	an	an	DET
ejpam-5709	477	3	irreducibility	irreducibility	NOUN
ejpam-5709	477	4	theorem	theorem	NOUN
ejpam-5709	477	5	of	of	ADP
ejpam-5709	477	6	a	a	DET
ejpam-5709	477	7	cohn	cohn	PROPN
ejpam-5709	477	8	.	.	PUNCT
ejpam-5709	478	1	canadian	canadian	PROPN
ejpam-5709	478	2	journal	journal	PROPN
ejpam-5709	478	3	of	of	ADP
ejpam-5709	478	4	mathematics	mathematic	NOUN
ejpam-5709	478	5	,	,	PUNCT
ejpam-5709	478	6	33(5):1055–1059	33(5):1055–1059	NUM
ejpam-5709	478	7	,	,	PUNCT
ejpam-5709	478	8	1981	1981	NUM
ejpam-5709	478	9	.	.	PUNCT
ejpam-5709	479	1	[	[	X
ejpam-5709	479	2	3	3	X
ejpam-5709	479	3	]	]	X
ejpam-5709	479	4	m	m	VERB
ejpam-5709	479	5	filaseta	filaseta	PROPN
ejpam-5709	479	6	.	.	PUNCT
ejpam-5709	480	1	a	a	DET
ejpam-5709	480	2	further	further	ADJ
ejpam-5709	480	3	generalization	generalization	NOUN
ejpam-5709	480	4	of	of	ADP
ejpam-5709	480	5	an	an	DET
ejpam-5709	480	6	irreducibility	irreducibility	NOUN
ejpam-5709	480	7	theorem	theorem	NOUN
ejpam-5709	480	8	of	of	ADP
ejpam-5709	480	9	a	a	DET
ejpam-5709	480	10	cohn	cohn	PROPN
ejpam-5709	480	11	.	.	PUNCT
ejpam-5709	481	1	canadian	canadian	PROPN
ejpam-5709	481	2	journal	journal	PROPN
ejpam-5709	481	3	of	of	ADP
ejpam-5709	481	4	mathematics	mathematic	NOUN
ejpam-5709	481	5	,	,	PUNCT
ejpam-5709	481	6	34(6):1390–1395	34(6):1390–1395	NUM
ejpam-5709	481	7	,	,	PUNCT
ejpam-5709	481	8	1982	1982	NUM
ejpam-5709	481	9	.	.	PUNCT
ejpam-5709	482	1	[	[	X
ejpam-5709	482	2	4	4	X
ejpam-5709	482	3	]	]	SYM
ejpam-5709	482	4	n	n	PRON
ejpam-5709	482	5	r	r	NOUN
ejpam-5709	482	6	kanasri	kanasri	NOUN
ejpam-5709	482	7	,	,	PUNCT
ejpam-5709	482	8	p	p	PROPN
ejpam-5709	482	9	singthongla	singthongla	PROPN
ejpam-5709	482	10	,	,	PUNCT
ejpam-5709	482	11	and	and	CCONJ
ejpam-5709	482	12	v	v	ADP
ejpam-5709	482	13	laohakosol	laohakosol	NOUN
ejpam-5709	482	14	.	.	PUNCT
ejpam-5709	483	1	irreducibility	irreducibility	NOUN
ejpam-5709	483	2	criteria	criterion	NOUN
ejpam-5709	483	3	for	for	ADP
ejpam-5709	483	4	polynomials	polynomial	NOUN
ejpam-5709	483	5	over	over	ADP
ejpam-5709	483	6	some	some	DET
ejpam-5709	483	7	imaginary	imaginary	ADJ
ejpam-5709	483	8	quadratic	quadratic	ADJ
ejpam-5709	483	9	fields	field	NOUN
ejpam-5709	483	10	.	.	PUNCT
ejpam-5709	484	1	southeast	southeast	ADJ
ejpam-5709	484	2	asian	asian	ADJ
ejpam-5709	484	3	bulletin	bulletin	NOUN
ejpam-5709	484	4	of	of	ADP
ejpam-5709	484	5	mathematics	mathematic	NOUN
ejpam-5709	484	6	,	,	PUNCT
ejpam-5709	484	7	43(1):367–376	43(1):367–376	PROPN
ejpam-5709	484	8	,	,	PUNCT
ejpam-5709	484	9	2019	2019	NUM
ejpam-5709	484	10	.	.	PUNCT
ejpam-5709	485	1	[	[	X
ejpam-5709	485	2	5	5	NUM
ejpam-5709	485	3	]	]	PUNCT
ejpam-5709	485	4	m	m	VERB
ejpam-5709	485	5	rmurty	rmurty	NOUN
ejpam-5709	485	6	.	.	PUNCT
ejpam-5709	486	1	prime	prime	ADJ
ejpam-5709	486	2	numbers	number	NOUN
ejpam-5709	486	3	and	and	CCONJ
ejpam-5709	486	4	irreducible	irreducible	ADJ
ejpam-5709	486	5	polynomials	polynomial	NOUN
ejpam-5709	486	6	.	.	PUNCT
ejpam-5709	487	1	the	the	DET
ejpam-5709	487	2	american	american	PROPN
ejpam-5709	487	3	mathematical	mathematical	PROPN
ejpam-5709	487	4	monthly	monthly	ADV
ejpam-5709	487	5	,	,	PUNCT
ejpam-5709	487	6	109(5):452–458	109(5):452–458	NUM
ejpam-5709	487	7	,	,	PUNCT
ejpam-5709	487	8	2002	2002	NUM
ejpam-5709	487	9	.	.	PUNCT
ejpam-5709	488	1	[	[	X
ejpam-5709	488	2	6	6	NUM
ejpam-5709	488	3	]	]	X
ejpam-5709	488	4	i	i	PRON
ejpam-5709	488	5	niven	niven	VERB
ejpam-5709	488	6	,	,	PUNCT
ejpam-5709	488	7	h	h	PROPN
ejpam-5709	488	8	s	s	PROPN
ejpam-5709	488	9	zuckerman	zuckerman	NOUN
ejpam-5709	488	10	,	,	PUNCT
ejpam-5709	488	11	and	and	CCONJ
ejpam-5709	488	12	h	h	PROPN
ejpam-5709	488	13	l	l	PROPN
ejpam-5709	488	14	montgomery	montgomery	PROPN
ejpam-5709	488	15	.	.	PUNCT
ejpam-5709	489	1	an	an	DET
ejpam-5709	489	2	introduction	introduction	NOUN
ejpam-5709	489	3	to	to	ADP
ejpam-5709	489	4	the	the	DET
ejpam-5709	489	5	theory	theory	NOUN
ejpam-5709	489	6	of	of	ADP
ejpam-5709	489	7	numbers	number	NOUN
ejpam-5709	489	8	.	.	PUNCT
ejpam-5709	490	1	5th	5th	ADJ
ejpam-5709	490	2	ed	ed	NOUN
ejpam-5709	490	3	.	.	PUNCT
ejpam-5709	490	4	wiley	wiley	PROPN
ejpam-5709	490	5	,	,	PUNCT
ejpam-5709	490	6	new	new	PROPN
ejpam-5709	490	7	york	york	PROPN
ejpam-5709	490	8	,	,	PUNCT
ejpam-5709	490	9	1991	1991	NUM
ejpam-5709	490	10	.	.	PUNCT
ejpam-5709	491	1	[	[	X
ejpam-5709	491	2	7	7	X
ejpam-5709	491	3	]	]	X
ejpam-5709	491	4	p	p	X
ejpam-5709	491	5	phetnun	phetnun	NOUN
ejpam-5709	491	6	and	and	CCONJ
ejpam-5709	491	7	n	n	PRON
ejpam-5709	491	8	r	r	NOUN
ejpam-5709	491	9	kanasri	kanasri	NOUN
ejpam-5709	491	10	.	.	PUNCT
ejpam-5709	492	1	further	further	ADJ
ejpam-5709	492	2	irreducibility	irreducibility	NOUN
ejpam-5709	492	3	criteria	criterion	NOUN
ejpam-5709	492	4	for	for	ADP
ejpam-5709	492	5	polynomials	polynomial	NOUN
ejpam-5709	492	6	associated	associate	VERB
ejpam-5709	492	7	with	with	ADP
ejpam-5709	492	8	the	the	DET
ejpam-5709	492	9	complete	complete	ADJ
ejpam-5709	492	10	residue	residue	NOUN
ejpam-5709	492	11	systems	system	NOUN
ejpam-5709	492	12	in	in	ADP
ejpam-5709	492	13	any	any	DET
ejpam-5709	492	14	imaginary	imaginary	ADJ
ejpam-5709	492	15	quadratic	quadratic	ADJ
ejpam-5709	492	16	field	field	NOUN
ejpam-5709	492	17	.	.	PUNCT
ejpam-5709	493	1	aims	aim	VERB
ejpam-5709	493	2	mathematics	mathematic	NOUN
ejpam-5709	493	3	,	,	PUNCT
ejpam-5709	493	4	7(10):18925–18947	7(10):18925–18947	NUM
ejpam-5709	493	5	,	,	PUNCT
ejpam-5709	493	6	2022	2022	NUM
ejpam-5709	493	7	.	.	PUNCT
ejpam-5709	494	1	[	[	X
ejpam-5709	494	2	8	8	X
ejpam-5709	494	3	]	]	X
ejpam-5709	494	4	p	p	NOUN
ejpam-5709	494	5	phetnun	phetnun	NOUN
ejpam-5709	494	6	,	,	PUNCT
ejpam-5709	494	7	n	n	NOUN
ejpam-5709	494	8	r	r	NOUN
ejpam-5709	494	9	kanasri	kanasri	NOUN
ejpam-5709	494	10	,	,	PUNCT
ejpam-5709	494	11	and	and	CCONJ
ejpam-5709	494	12	p	p	X
ejpam-5709	494	13	singthongla	singthongla	PROPN
ejpam-5709	494	14	.	.	PUNCT
ejpam-5709	495	1	on	on	ADP
ejpam-5709	495	2	the	the	DET
ejpam-5709	495	3	irreducibility	irreducibility	NOUN
ejpam-5709	495	4	of	of	ADP
ejpam-5709	495	5	polynomials	polynomial	NOUN
ejpam-5709	495	6	associated	associate	VERB
ejpam-5709	495	7	with	with	ADP
ejpam-5709	495	8	the	the	DET
ejpam-5709	495	9	complete	complete	ADJ
ejpam-5709	495	10	residue	residue	NOUN
ejpam-5709	495	11	systems	system	NOUN
ejpam-5709	495	12	in	in	ADP
ejpam-5709	495	13	any	any	DET
ejpam-5709	495	14	imaginary	imaginary	ADJ
ejpam-5709	495	15	quadratic	quadratic	ADJ
ejpam-5709	495	16	fields	field	NOUN
ejpam-5709	495	17	.	.	PUNCT
ejpam-5709	496	1	international	international	ADJ
ejpam-5709	496	2	journal	journal	PROPN
ejpam-5709	496	3	of	of	ADP
ejpam-5709	496	4	mathematics	mathematics	PROPN
ejpam-5709	496	5	and	and	CCONJ
ejpam-5709	496	6	mathematical	mathematical	ADJ
ejpam-5709	496	7	sciences	science	NOUN
ejpam-5709	496	8	,	,	PUNCT
ejpam-5709	496	9	2021:17	2021:17	NUM
ejpam-5709	496	10	pages	page	NOUN
ejpam-5709	496	11	,	,	PUNCT
ejpam-5709	496	12	2021	2021	NUM
ejpam-5709	496	13	.	.	PUNCT
ejpam-5709	497	1	[	[	X
ejpam-5709	497	2	9	9	NUM
ejpam-5709	497	3	]	]	SYM
ejpam-5709	497	4	h	h	NOUN
ejpam-5709	497	5	pollard	pollard	NOUN
ejpam-5709	497	6	and	and	CCONJ
ejpam-5709	497	7	h	h	NOUN
ejpam-5709	497	8	g	g	PROPN
ejpam-5709	497	9	diamond	diamond	NOUN
ejpam-5709	497	10	.	.	PUNCT
ejpam-5709	498	1	the	the	DET
ejpam-5709	498	2	theory	theory	NOUN
ejpam-5709	498	3	of	of	ADP
ejpam-5709	498	4	algebraic	algebraic	ADJ
ejpam-5709	498	5	numbers	number	NOUN
ejpam-5709	498	6	.	.	PUNCT
ejpam-5709	499	1	cambridge	cambridge	PROPN
ejpam-5709	499	2	university	university	PROPN
ejpam-5709	499	3	press	press	PROPN
ejpam-5709	499	4	,	,	PUNCT
ejpam-5709	499	5	cambridge	cambridge	PROPN
ejpam-5709	499	6	,	,	PUNCT
ejpam-5709	499	7	1975	1975	NUM
ejpam-5709	499	8	.	.	PUNCT
ejpam-5709	500	1	[	[	X
ejpam-5709	500	2	10	10	NUM
ejpam-5709	500	3	]	]	X
ejpam-5709	500	4	g	g	PROPN
ejpam-5709	500	5	pólya	pólya	PROPN
ejpam-5709	500	6	and	and	CCONJ
ejpam-5709	500	7	g	g	NOUN
ejpam-5709	500	8	szegö.	szegö.	NOUN
ejpam-5709	500	9	problems	problem	NOUN
ejpam-5709	500	10	and	and	CCONJ
ejpam-5709	500	11	theorems	theorem	NOUN
ejpam-5709	500	12	in	in	ADP
ejpam-5709	500	13	analysis	analysis	NOUN
ejpam-5709	500	14	.	.	PUNCT
ejpam-5709	501	1	springer	springer	NOUN
ejpam-5709	501	2	-	-	PUNCT
ejpam-5709	501	3	verlag	verlag	PROPN
ejpam-5709	501	4	,	,	PUNCT
ejpam-5709	501	5	new	new	PROPN
ejpam-5709	501	6	york	york	PROPN
ejpam-5709	501	7	,	,	PUNCT
ejpam-5709	501	8	1976	1976	NUM
ejpam-5709	501	9	.	.	PUNCT
ejpam-5709	502	1	[	[	X
ejpam-5709	502	2	11	11	NUM
ejpam-5709	502	3	]	]	X
ejpam-5709	502	4	k	k	PROPN
ejpam-5709	502	5	h	h	PROPN
ejpam-5709	502	6	rosen	rosen	PROPN
ejpam-5709	502	7	.	.	PUNCT
ejpam-5709	503	1	elementary	elementary	ADJ
ejpam-5709	503	2	number	number	NOUN
ejpam-5709	503	3	theory	theory	NOUN
ejpam-5709	503	4	and	and	CCONJ
ejpam-5709	503	5	its	its	PRON
ejpam-5709	503	6	applications	application	NOUN
ejpam-5709	503	7	.	.	PUNCT
ejpam-5709	504	1	5th	5th	ADJ
ejpam-5709	504	2	ed	ed	NOUN
ejpam-5709	504	3	.	.	PUNCT
ejpam-5709	505	1	addison	addison	PROPN
ejpam-5709	505	2	-	-	PUNCT
ejpam-5709	505	3	wesley	wesley	PROPN
ejpam-5709	505	4	,	,	PUNCT
ejpam-5709	505	5	new	new	PROPN
ejpam-5709	505	6	york	york	PROPN
ejpam-5709	505	7	,	,	PUNCT
ejpam-5709	505	8	2005	2005	NUM
ejpam-5709	505	9	.	.	PUNCT
ejpam-5709	506	1	[	[	X
ejpam-5709	506	2	12	12	NUM
ejpam-5709	506	3	]	]	X
ejpam-5709	507	1	p	p	X
ejpam-5709	507	2	singthongla	singthongla	PROPN
ejpam-5709	507	3	,	,	PUNCT
ejpam-5709	507	4	n	n	NOUN
ejpam-5709	507	5	r	r	NOUN
ejpam-5709	507	6	kanasri	kanasri	NOUN
ejpam-5709	507	7	,	,	PUNCT
ejpam-5709	507	8	and	and	CCONJ
ejpam-5709	507	9	v	v	ADP
ejpam-5709	507	10	laohakosol	laohakosol	NOUN
ejpam-5709	507	11	.	.	PUNCT
ejpam-5709	508	1	prime	prime	ADJ
ejpam-5709	508	2	elements	element	NOUN
ejpam-5709	508	3	and	and	CCONJ
ejpam-5709	508	4	irreducible	irreducible	ADJ
ejpam-5709	508	5	polynomials	polynomial	NOUN
ejpam-5709	508	6	over	over	ADP
ejpam-5709	508	7	some	some	DET
ejpam-5709	508	8	imaginary	imaginary	ADJ
ejpam-5709	508	9	quadratic	quadratic	ADJ
ejpam-5709	508	10	fields	field	NOUN
ejpam-5709	508	11	.	.	PUNCT
ejpam-5709	509	1	kyungpook	kyungpook	PROPN
ejpam-5709	509	2	mathematical	mathematical	PROPN
ejpam-5709	509	3	journal	journal	PROPN
ejpam-5709	509	4	,	,	PUNCT
ejpam-5709	509	5	57(4):581–600	57(4):581–600	PROPN
ejpam-5709	509	6	,	,	PUNCT
ejpam-5709	509	7	2017	2017	NUM
ejpam-5709	509	8	.	.	PUNCT
ejpam-5709	510	1	[	[	X
ejpam-5709	510	2	13	13	NUM
ejpam-5709	510	3	]	]	X
ejpam-5709	510	4	s	s	VERB
ejpam-5709	510	5	tadee	tadee	NOUN
ejpam-5709	510	6	,	,	PUNCT
ejpam-5709	510	7	v	v	NOUN
ejpam-5709	510	8	laohakosol	laohakosol	NOUN
ejpam-5709	510	9	,	,	PUNCT
ejpam-5709	510	10	and	and	CCONJ
ejpam-5709	510	11	s	s	AUX
ejpam-5709	510	12	damkaew	damkaew	VERB
ejpam-5709	510	13	.	.	PUNCT
ejpam-5709	511	1	explicit	explicit	ADJ
ejpam-5709	511	2	complete	complete	ADJ
ejpam-5709	511	3	residue	residue	NOUN
ejpam-5709	511	4	systems	system	NOUN
ejpam-5709	511	5	in	in	ADP
ejpam-5709	511	6	a	a	DET
ejpam-5709	511	7	general	general	ADJ
ejpam-5709	511	8	quadratic	quadratic	ADJ
ejpam-5709	511	9	field	field	NOUN
ejpam-5709	511	10	.	.	PUNCT
ejpam-5709	512	1	divulgaciones	divulgacione	NOUN
ejpam-5709	512	2	matemáticas	matemática	NOUN
ejpam-5709	512	3	,	,	PUNCT
ejpam-5709	512	4	18(2):1–17	18(2):1–17	NUM
ejpam-5709	512	5	,	,	PUNCT
ejpam-5709	512	6	2017	2017	NUM
ejpam-5709	512	7	.	.	PUNCT