id	sid	tid	token	lemma	pos
ejpam-628	1	1	14_628_behboodi.dvi	14_628_behboodi.dvi	PROPN
ejpam-628	1	2	european	european	PROPN
ejpam-628	1	3	journal	journal	PROPN
ejpam-628	1	4	of	of	ADP
ejpam-628	1	5	pure	pure	ADJ
ejpam-628	1	6	and	and	CCONJ
ejpam-628	1	7	applied	apply	VERB
ejpam-628	1	8	mathematics	mathematic	NOUN
ejpam-628	1	9	vol	vol	NOUN
ejpam-628	1	10	.	.	PUNCT
ejpam-628	2	1	3	3	NUM
ejpam-628	2	2	,	,	PUNCT
ejpam-628	2	3	no	no	INTJ
ejpam-628	2	4	.	.	NOUN
ejpam-628	2	5	2	2	NUM
ejpam-628	2	6	,	,	PUNCT
ejpam-628	2	7	2010	2010	NUM
ejpam-628	2	8	,	,	PUNCT
ejpam-628	2	9	303	303	NUM
ejpam-628	2	10	-	-	SYM
ejpam-628	2	11	316	316	NUM
ejpam-628	2	12	issn	issn	PROPN
ejpam-628	2	13	1307	1307	NUM
ejpam-628	2	14	-	-	SYM
ejpam-628	2	15	5543	5543	NUM
ejpam-628	2	16	–	–	PUNCT
ejpam-628	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-628	2	18	on	on	ADP
ejpam-628	2	19	the	the	DET
ejpam-628	2	20	structure	structure	NOUN
ejpam-628	2	21	of	of	ADP
ejpam-628	2	22	commutative	commutative	ADJ
ejpam-628	2	23	rings	ring	NOUN
ejpam-628	2	24	with	with	ADP
ejpam-628	2	25	p1	p1	PROPN
ejpam-628	2	26	k1	k1	X
ejpam-628	2	27	·	·	PUNCT
ejpam-628	2	28	·	·	PUNCT
ejpam-628	2	29	·	·	PUNCT
ejpam-628	3	1	pn	pn	PROPN
ejpam-628	3	2	kn	kn	PROPN
ejpam-628	3	3	(	(	PUNCT
ejpam-628	3	4	1≤	1≤	INTJ
ejpam-628	3	5	ki	ki	PROPN
ejpam-628	3	6	≤	≤	PROPN
ejpam-628	3	7	7	7	NUM
ejpam-628	3	8	)	)	PUNCT
ejpam-628	3	9	zero	zero	NUM
ejpam-628	3	10	-	-	PUNCT
ejpam-628	3	11	divisors	divisor	NOUN
ejpam-628	3	12	m.	m.	NOUN
ejpam-628	3	13	behboodi1,2∗	behboodi1,2∗	PROPN
ejpam-628	3	14	and	and	CCONJ
ejpam-628	3	15	r.	r.	PROPN
ejpam-628	3	16	beyranvand1	beyranvand1	PROPN
ejpam-628	3	17	1	1	NUM
ejpam-628	3	18	department	department	NOUN
ejpam-628	3	19	of	of	ADP
ejpam-628	3	20	mathematical	mathematical	ADJ
ejpam-628	3	21	science	science	NOUN
ejpam-628	3	22	,	,	PUNCT
ejpam-628	3	23	isfahan	isfahan	PROPN
ejpam-628	3	24	university	university	PROPN
ejpam-628	3	25	of	of	ADP
ejpam-628	3	26	technology	technology	PROPN
ejpam-628	3	27	,	,	PUNCT
ejpam-628	3	28	isfahan	isfahan	PROPN
ejpam-628	3	29	,	,	PUNCT
ejpam-628	3	30	iran	iran	PROPN
ejpam-628	3	31	2	2	NUM
ejpam-628	3	32	school	school	NOUN
ejpam-628	3	33	of	of	ADP
ejpam-628	3	34	mathematics	mathematic	NOUN
ejpam-628	3	35	,	,	PUNCT
ejpam-628	3	36	institute	institute	NOUN
ejpam-628	3	37	for	for	ADP
ejpam-628	3	38	research	research	NOUN
ejpam-628	3	39	in	in	ADP
ejpam-628	3	40	fundamental	fundamental	ADJ
ejpam-628	3	41	sciences	science	NOUN
ejpam-628	3	42	(	(	PUNCT
ejpam-628	3	43	ipm	ipm	NOUN
ejpam-628	3	44	)	)	PUNCT
ejpam-628	3	45	,	,	PUNCT
ejpam-628	3	46	tehran	tehran	PROPN
ejpam-628	3	47	,	,	PUNCT
ejpam-628	3	48	iran	iran	PROPN
ejpam-628	3	49	abstract	abstract	ADJ
ejpam-628	3	50	.	.	PUNCT
ejpam-628	4	1	let	let	VERB
ejpam-628	4	2	r	r	PRON
ejpam-628	4	3	be	be	AUX
ejpam-628	4	4	a	a	DET
ejpam-628	4	5	finite	finite	ADJ
ejpam-628	4	6	commutative	commutative	ADJ
ejpam-628	4	7	ring	ring	NOUN
ejpam-628	4	8	with	with	ADP
ejpam-628	4	9	identity	identity	NOUN
ejpam-628	4	10	and	and	CCONJ
ejpam-628	4	11	z(r	z(r	NOUN
ejpam-628	4	12	)	)	PUNCT
ejpam-628	4	13	denote	denote	VERB
ejpam-628	4	14	the	the	DET
ejpam-628	4	15	set	set	NOUN
ejpam-628	4	16	of	of	ADP
ejpam-628	4	17	all	all	DET
ejpam-628	4	18	zero	zero	NUM
ejpam-628	4	19	-	-	PUNCT
ejpam-628	4	20	divisors	divisor	NOUN
ejpam-628	4	21	of	of	ADP
ejpam-628	4	22	r.	r.	PROPN
ejpam-628	4	23	note	note	PROPN
ejpam-628	4	24	that	that	SCONJ
ejpam-628	4	25	r	r	NOUN
ejpam-628	4	26	is	be	AUX
ejpam-628	4	27	uniquely	uniquely	ADV
ejpam-628	4	28	expressible	expressible	ADJ
ejpam-628	4	29	as	as	ADP
ejpam-628	4	30	a	a	DET
ejpam-628	4	31	direct	direct	ADJ
ejpam-628	4	32	sum	sum	NOUN
ejpam-628	4	33	of	of	ADP
ejpam-628	4	34	local	local	ADJ
ejpam-628	4	35	rings	ring	NOUN
ejpam-628	4	36	ri	ri	PROPN
ejpam-628	4	37	(	(	PUNCT
ejpam-628	4	38	1	1	NUM
ejpam-628	4	39	≤	≤	NUM
ejpam-628	4	40	i	i	X
ejpam-628	4	41	≤	≤	NOUN
ejpam-628	4	42	m	m	VERB
ejpam-628	4	43	)	)	PUNCT
ejpam-628	4	44	for	for	ADP
ejpam-628	4	45	some	some	DET
ejpam-628	4	46	m	m	NOUN
ejpam-628	4	47	≥	≥	NOUN
ejpam-628	4	48	1	1	NUM
ejpam-628	4	49	.	.	PUNCT
ejpam-628	5	1	in	in	ADP
ejpam-628	5	2	this	this	DET
ejpam-628	5	3	paper	paper	NOUN
ejpam-628	5	4	,	,	PUNCT
ejpam-628	5	5	we	we	PRON
ejpam-628	5	6	investigate	investigate	VERB
ejpam-628	5	7	the	the	DET
ejpam-628	5	8	relationship	relationship	NOUN
ejpam-628	5	9	between	between	ADP
ejpam-628	5	10	the	the	DET
ejpam-628	5	11	prime	prime	ADJ
ejpam-628	5	12	factorizations	factorization	NOUN
ejpam-628	5	13	|z(r)|	|z(r)|	PROPN
ejpam-628	5	14	=	=	SYM
ejpam-628	5	15	p1	p1	PROPN
ejpam-628	5	16	k1	k1	X
ejpam-628	5	17	·	·	PUNCT
ejpam-628	5	18	·	·	PUNCT
ejpam-628	5	19	·	·	PUNCT
ejpam-628	6	1	pn	pn	INTJ
ejpam-628	6	2	kn	kn	PROPN
ejpam-628	6	3	and	and	CCONJ
ejpam-628	6	4	the	the	DET
ejpam-628	6	5	summands	summand	NOUN
ejpam-628	6	6	ri	ri	PROPN
ejpam-628	6	7	.	.	PUNCT
ejpam-628	7	1	it	it	PRON
ejpam-628	7	2	is	be	AUX
ejpam-628	7	3	shown	show	VERB
ejpam-628	7	4	that	that	SCONJ
ejpam-628	7	5	for	for	ADP
ejpam-628	7	6	each	each	DET
ejpam-628	7	7	i	i	PRON
ejpam-628	7	8	,	,	PUNCT
ejpam-628	7	9	|z(ri)|	|z(ri)|	X
ejpam-628	7	10	=	=	SYM
ejpam-628	7	11	p	p	PROPN
ejpam-628	7	12	j	j	PROPN
ejpam-628	7	13	t	t	PROPN
ejpam-628	7	14	j	j	PROPN
ejpam-628	7	15	for	for	ADP
ejpam-628	7	16	some	some	DET
ejpam-628	7	17	1	1	NUM
ejpam-628	7	18	≤	≤	NUM
ejpam-628	7	19	j	j	PROPN
ejpam-628	7	20	≤	≤	PROPN
ejpam-628	7	21	n	n	ADP
ejpam-628	7	22	and	and	CCONJ
ejpam-628	7	23	0	0	NUM
ejpam-628	7	24	≤	≤	NUM
ejpam-628	7	25	t	t	PROPN
ejpam-628	7	26	j	j	PROPN
ejpam-628	7	27	≤	≤	PROPN
ejpam-628	8	1	k	k	PROPN
ejpam-628	8	2	j	j	PROPN
ejpam-628	8	3	.	.	PUNCT
ejpam-628	9	1	in	in	ADP
ejpam-628	9	2	particular	particular	ADJ
ejpam-628	9	3	,	,	PUNCT
ejpam-628	9	4	rings	ring	NOUN
ejpam-628	9	5	r	r	NOUN
ejpam-628	9	6	with	with	ADP
ejpam-628	9	7	|z(r)|	|z(r)|	PROPN
ejpam-628	9	8	=	=	SYM
ejpam-628	9	9	pk	pk	NOUN
ejpam-628	10	1	where	where	SCONJ
ejpam-628	10	2	1	1	NUM
ejpam-628	10	3	≤	≤	NUM
ejpam-628	10	4	k	k	X
ejpam-628	10	5	≤	≤	NUM
ejpam-628	10	6	7	7	NUM
ejpam-628	10	7	,	,	PUNCT
ejpam-628	10	8	are	be	AUX
ejpam-628	10	9	characterized	characterize	VERB
ejpam-628	10	10	.	.	PUNCT
ejpam-628	11	1	moreover	moreover	ADV
ejpam-628	11	2	,	,	PUNCT
ejpam-628	11	3	the	the	DET
ejpam-628	11	4	structure	structure	NOUN
ejpam-628	11	5	and	and	CCONJ
ejpam-628	11	6	classification	classification	NOUN
ejpam-628	11	7	up	up	ADP
ejpam-628	11	8	to	to	PART
ejpam-628	11	9	isomorphism	isomorphism	VERB
ejpam-628	11	10	all	all	DET
ejpam-628	11	11	commutative	commutative	ADJ
ejpam-628	11	12	rings	ring	NOUN
ejpam-628	11	13	r	r	NOUN
ejpam-628	11	14	with	with	ADP
ejpam-628	11	15	|z(r)|	|z(r)|	PROPN
ejpam-628	11	16	=	=	SYM
ejpam-628	11	17	p1	p1	PROPN
ejpam-628	11	18	k1	k1	NOUN
ejpam-628	11	19	.	.	PUNCT
ejpam-628	11	20	.	.	PUNCT
ejpam-628	11	21	.	.	PUNCT
ejpam-628	12	1	pn	pn	PROPN
ejpam-628	12	2	kn	kn	PROPN
ejpam-628	12	3	,	,	PUNCT
ejpam-628	12	4	where	where	SCONJ
ejpam-628	12	5	n	n	X
ejpam-628	12	6	∈	∈	PROPN
ejpam-628	12	7	n	n	CCONJ
ejpam-628	12	8	,	,	PUNCT
ejpam-628	12	9	p	p	X
ejpam-628	12	10	,	,	PUNCT
ejpam-628	12	11	i	i	PRON
ejpam-628	12	12	s	s	AUX
ejpam-628	12	13	are	be	AUX
ejpam-628	12	14	distinct	distinct	ADJ
ejpam-628	12	15	prime	prime	ADJ
ejpam-628	12	16	numbers	number	NOUN
ejpam-628	12	17	,	,	PUNCT
ejpam-628	12	18	1	1	NUM
ejpam-628	12	19	≤	≤	NUM
ejpam-628	12	20	ki	ki	PROPN
ejpam-628	12	21	≤	≤	ADV
ejpam-628	12	22	3	3	NUM
ejpam-628	12	23	and	and	CCONJ
ejpam-628	12	24	nonlocal	nonlocal	ADJ
ejpam-628	12	25	commutative	commutative	ADJ
ejpam-628	12	26	rings	ring	NOUN
ejpam-628	12	27	r	r	NOUN
ejpam-628	12	28	with	with	ADP
ejpam-628	12	29	|z(r)|	|z(r)|	PROPN
ejpam-628	12	30	=	=	SYM
ejpam-628	12	31	pk	pk	NOUN
ejpam-628	12	32	where	where	SCONJ
ejpam-628	12	33	k	k	NOUN
ejpam-628	12	34	=	=	NOUN
ejpam-628	12	35	4	4	NUM
ejpam-628	12	36	or	or	CCONJ
ejpam-628	12	37	5	5	NUM
ejpam-628	12	38	,	,	PUNCT
ejpam-628	12	39	are	be	AUX
ejpam-628	12	40	determined	determine	VERB
ejpam-628	12	41	.	.	PUNCT
ejpam-628	13	1	2000	2000	NUM
ejpam-628	13	2	mathematics	mathematic	NOUN
ejpam-628	13	3	subject	subject	NOUN
ejpam-628	13	4	classifications	classification	NOUN
ejpam-628	13	5	:	:	PUNCT
ejpam-628	13	6	16b99	16b99	NUM
ejpam-628	13	7	;	;	PUNCT
ejpam-628	13	8	13a99	13a99	NUM
ejpam-628	13	9	;	;	PUNCT
ejpam-628	13	10	68r10	68r10	NUM
ejpam-628	13	11	key	key	ADJ
ejpam-628	13	12	words	word	NOUN
ejpam-628	13	13	and	and	CCONJ
ejpam-628	13	14	phrases	phrase	NOUN
ejpam-628	13	15	:	:	PUNCT
ejpam-628	13	16	finite	finite	PROPN
ejpam-628	13	17	ring	ring	NOUN
ejpam-628	13	18	,	,	PUNCT
ejpam-628	13	19	zero	zero	NUM
ejpam-628	13	20	-	-	PUNCT
ejpam-628	13	21	divisor	divisor	NOUN
ejpam-628	13	22	,	,	PUNCT
ejpam-628	13	23	local	local	ADJ
ejpam-628	13	24	rings	ring	NOUN
ejpam-628	13	25	1	1	NUM
ejpam-628	13	26	.	.	PUNCT
ejpam-628	13	27	introduction	introduction	NOUN
ejpam-628	13	28	throughout	throughout	ADP
ejpam-628	13	29	the	the	DET
ejpam-628	13	30	paper	paper	NOUN
ejpam-628	14	1	r	r	NOUN
ejpam-628	14	2	always	always	ADV
ejpam-628	14	3	denotes	denote	VERB
ejpam-628	14	4	a	a	DET
ejpam-628	14	5	commutative	commutative	ADJ
ejpam-628	14	6	ring	ring	NOUN
ejpam-628	14	7	with	with	ADP
ejpam-628	14	8	identity	identity	NOUN
ejpam-628	14	9	,	,	PUNCT
ejpam-628	14	10	j(r	j(r	PROPN
ejpam-628	14	11	)	)	PUNCT
ejpam-628	14	12	is	be	AUX
ejpam-628	14	13	the	the	DET
ejpam-628	14	14	jacobson	jacobson	PROPN
ejpam-628	14	15	radical	radical	PROPN
ejpam-628	14	16	of	of	ADP
ejpam-628	14	17	r	r	NOUN
ejpam-628	14	18	and	and	CCONJ
ejpam-628	14	19	z(r	z(r	PROPN
ejpam-628	14	20	)	)	PUNCT
ejpam-628	14	21	denotes	denote	VERB
ejpam-628	14	22	the	the	DET
ejpam-628	14	23	set	set	NOUN
ejpam-628	14	24	of	of	ADP
ejpam-628	14	25	all	all	DET
ejpam-628	14	26	zero	zero	NUM
ejpam-628	14	27	-	-	PUNCT
ejpam-628	14	28	divisors	divisor	NOUN
ejpam-628	14	29	of	of	ADP
ejpam-628	14	30	r.	r.	PROPN
ejpam-628	14	31	we	we	PRON
ejpam-628	14	32	denote	denote	VERB
ejpam-628	14	33	fq	fq	PROPN
ejpam-628	14	34	for	for	ADP
ejpam-628	14	35	the	the	DET
ejpam-628	14	36	finite	finite	ADJ
ejpam-628	14	37	field	field	NOUN
ejpam-628	14	38	of	of	ADP
ejpam-628	14	39	order	order	NOUN
ejpam-628	14	40	q	q	NOUN
ejpam-628	15	1	and	and	CCONJ
ejpam-628	15	2	for	for	ADP
ejpam-628	15	3	any	any	DET
ejpam-628	15	4	finite	finite	NOUN
ejpam-628	15	5	subset	subset	VERB
ejpam-628	15	6	y	y	PROPN
ejpam-628	15	7	of	of	ADP
ejpam-628	15	8	r	r	PROPN
ejpam-628	15	9	,	,	PUNCT
ejpam-628	15	10	we	we	PRON
ejpam-628	15	11	denote	denote	VERB
ejpam-628	15	12	|y	|y	NOUN
ejpam-628	15	13	|	|	ADV
ejpam-628	15	14	for	for	ADP
ejpam-628	15	15	the	the	DET
ejpam-628	15	16	cardinality	cardinality	NOUN
ejpam-628	15	17	of	of	ADP
ejpam-628	15	18	y	y	PROPN
ejpam-628	15	19	.	.	PUNCT
ejpam-628	16	1	the	the	DET
ejpam-628	16	2	zero	zero	NUM
ejpam-628	16	3	-	-	PUNCT
ejpam-628	16	4	divisor	divisor	NOUN
ejpam-628	16	5	graph	graph	NOUN
ejpam-628	16	6	of	of	ADP
ejpam-628	16	7	r	r	NOUN
ejpam-628	16	8	,	,	PUNCT
ejpam-628	16	9	denoted	denote	VERB
ejpam-628	16	10	by	by	ADP
ejpam-628	16	11	γ(r	γ(r	PROPN
ejpam-628	16	12	)	)	PUNCT
ejpam-628	16	13	,	,	PUNCT
ejpam-628	16	14	is	be	AUX
ejpam-628	16	15	the	the	DET
ejpam-628	16	16	graph	graph	NOUN
ejpam-628	16	17	whose	whose	DET
ejpam-628	16	18	vertices	vertex	NOUN
ejpam-628	16	19	are	be	AUX
ejpam-628	16	20	the	the	DET
ejpam-628	16	21	nonzero	nonzero	ADJ
ejpam-628	16	22	zero	zero	NUM
ejpam-628	16	23	-	-	PUNCT
ejpam-628	16	24	divisors	divisor	NOUN
ejpam-628	16	25	of	of	ADP
ejpam-628	16	26	r	r	NOUN
ejpam-628	16	27	with	with	ADP
ejpam-628	16	28	two	two	NUM
ejpam-628	16	29	distinct	distinct	ADJ
ejpam-628	16	30	vertices	vertex	NOUN
ejpam-628	16	31	a	a	PRON
ejpam-628	16	32	and	and	CCONJ
ejpam-628	16	33	b	b	NOUN
ejpam-628	16	34	joined	join	VERB
ejpam-628	16	35	by	by	ADP
ejpam-628	16	36	an	an	DET
ejpam-628	16	37	edge	edge	NOUN
ejpam-628	16	38	if	if	SCONJ
ejpam-628	16	39	and	and	CCONJ
ejpam-628	16	40	only	only	ADV
ejpam-628	16	41	if	if	SCONJ
ejpam-628	16	42	ab	ab	PROPN
ejpam-628	16	43	=	=	NOUN
ejpam-628	16	44	0	0	PROPN
ejpam-628	16	45	.	.	PUNCT
ejpam-628	17	1	one	one	PRON
ejpam-628	17	2	might	might	AUX
ejpam-628	17	3	ask	ask	VERB
ejpam-628	17	4	which	which	DET
ejpam-628	17	5	graphs	graph	NOUN
ejpam-628	17	6	on	on	ADP
ejpam-628	17	7	n	n	DET
ejpam-628	17	8	vertices	vertex	NOUN
ejpam-628	17	9	can	can	AUX
ejpam-628	17	10	be	be	AUX
ejpam-628	17	11	realized	realize	VERB
ejpam-628	17	12	as	as	ADP
ejpam-628	17	13	the	the	DET
ejpam-628	17	14	zero	zero	NUM
ejpam-628	17	15	-	-	PUNCT
ejpam-628	17	16	divisor	divisor	NOUN
ejpam-628	17	17	graph	graph	NOUN
ejpam-628	17	18	of	of	ADP
ejpam-628	17	19	a	a	DET
ejpam-628	17	20	commutative	commutative	ADJ
ejpam-628	17	21	ring	ring	NOUN
ejpam-628	17	22	?	?	PUNCT
ejpam-628	18	1	this	this	DET
ejpam-628	18	2	question	question	NOUN
ejpam-628	18	3	has	have	AUX
ejpam-628	18	4	been	be	AUX
ejpam-628	18	5	partially	partially	ADV
ejpam-628	18	6	answered	answer	VERB
ejpam-628	18	7	.	.	PUNCT
ejpam-628	19	1	[	[	X
ejpam-628	19	2	1	1	NUM
ejpam-628	19	3	]	]	PUNCT
ejpam-628	19	4	determines	determine	NOUN
ejpam-628	19	5	,	,	PUNCT
ejpam-628	19	6	up	up	ADP
ejpam-628	19	7	to	to	ADP
ejpam-628	19	8	isomorphism	isomorphism	NOUN
ejpam-628	19	9	,	,	PUNCT
ejpam-628	19	10	all	all	DET
ejpam-628	19	11	such	such	ADJ
ejpam-628	19	12	rings	ring	NOUN
ejpam-628	19	13	for	for	ADP
ejpam-628	19	14	which	which	PRON
ejpam-628	19	15	γ(r	γ(r	PROPN
ejpam-628	19	16	)	)	PUNCT
ejpam-628	19	17	is	be	AUX
ejpam-628	19	18	a	a	DET
ejpam-628	19	19	graphs	graph	NOUN
ejpam-628	19	20	on	on	ADP
ejpam-628	19	21	n	n	NOUN
ejpam-628	19	22	=	=	SYM
ejpam-628	19	23	1,2,3	1,2,3	NUM
ejpam-628	19	24	,	,	PUNCT
ejpam-628	19	25	or	or	CCONJ
ejpam-628	19	26	4	4	NUM
ejpam-628	19	27	vertices	vertex	NOUN
ejpam-628	19	28	.	.	PUNCT
ejpam-628	20	1	this	this	DET
ejpam-628	20	2	list	list	NOUN
ejpam-628	20	3	was	be	AUX
ejpam-628	20	4	extended	extend	VERB
ejpam-628	20	5	to	to	ADP
ejpam-628	20	6	n	n	NOUN
ejpam-628	20	7	=	=	SYM
ejpam-628	20	8	5	5	NUM
ejpam-628	20	9	vertices	vertex	NOUN
ejpam-628	20	10	in	in	ADP
ejpam-628	20	11	[	[	X
ejpam-628	20	12	10	10	NUM
ejpam-628	20	13	]	]	PUNCT
ejpam-628	20	14	,	,	PUNCT
ejpam-628	20	15	and	and	CCONJ
ejpam-628	20	16	to	to	ADP
ejpam-628	20	17	n	n	NOUN
ejpam-628	20	18	=	=	SYM
ejpam-628	20	19	6	6	NUM
ejpam-628	20	20	,	,	PUNCT
ejpam-628	20	21	7	7	NUM
ejpam-628	20	22	,	,	PUNCT
ejpam-628	20	23	.	.	PUNCT
ejpam-628	20	24	.	.	PUNCT
ejpam-628	21	1	.	.	PUNCT
ejpam-628	22	1	,	,	PUNCT
ejpam-628	22	2	14	14	NUM
ejpam-628	22	3	vertices	vertex	NOUN
ejpam-628	22	4	in	in	ADP
ejpam-628	22	5	[	[	X
ejpam-628	22	6	11	11	NUM
ejpam-628	22	7	]	]	PUNCT
ejpam-628	22	8	.	.	PUNCT
ejpam-628	23	1	the	the	DET
ejpam-628	23	2	aim	aim	NOUN
ejpam-628	23	3	of	of	ADP
ejpam-628	23	4	the	the	DET
ejpam-628	23	5	paper	paper	NOUN
ejpam-628	23	6	is	be	AUX
ejpam-628	23	7	to	to	PART
ejpam-628	23	8	develop	develop	VERB
ejpam-628	23	9	this	this	DET
ejpam-628	23	10	list	list	NOUN
ejpam-628	23	11	to	to	ADP
ejpam-628	23	12	a	a	DET
ejpam-628	23	13	wider	wide	ADJ
ejpam-628	23	14	class	class	NOUN
ejpam-628	23	15	of	of	ADP
ejpam-628	23	16	numbers	number	NOUN
ejpam-628	23	17	n.	n.	VERB
ejpam-628	23	18	in	in	ADP
ejpam-628	23	19	fact	fact	NOUN
ejpam-628	23	20	,	,	PUNCT
ejpam-628	23	21	this	this	DET
ejpam-628	23	22	observation	observation	NOUN
ejpam-628	23	23	motivates	motivate	VERB
ejpam-628	23	24	us	we	PRON
ejpam-628	23	25	the	the	DET
ejpam-628	23	26	∗corresponding	∗corresponding	NOUN
ejpam-628	23	27	author	author	NOUN
ejpam-628	23	28	.	.	PUNCT
ejpam-628	24	1	email	email	NOUN
ejpam-628	24	2	addresses	address	NOUN
ejpam-628	24	3	:	:	PUNCT
ejpam-628	24	4	mbehbood	mbehbood	PROPN
ejpam-628	24	5	�	�	PROPN
ejpam-628	24	6	.iut.a	.iut.a	PROPN
ejpam-628	24	7	.ir	.ir	PUNCT
ejpam-628	25	1	(	(	PUNCT
ejpam-628	25	2	m.	m.	NOUN
ejpam-628	25	3	behboodi	behboodi	PROPN
ejpam-628	25	4	)	)	PUNCT
ejpam-628	25	5	,	,	PUNCT
ejpam-628	25	6	r_beyranvand	r_beyranvand	PROPN
ejpam-628	25	7	�	�	PROPN
ejpam-628	25	8	math.iut.a	math.iut.a	NOUN
ejpam-628	25	9	.ir	.ir	PUNCT
ejpam-628	25	10	(	(	PUNCT
ejpam-628	25	11	r.	r.	PROPN
ejpam-628	25	12	beyranvand	beyranvand	PROPN
ejpam-628	25	13	)	)	PUNCT
ejpam-628	25	14	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-628	26	1	303	303	NUM
ejpam-628	26	2	c	c	X
ejpam-628	26	3	©	©	PROPN
ejpam-628	26	4	2010	2010	NUM
ejpam-628	26	5	ejpam	ejpam	NOUN
ejpam-628	26	6	all	all	DET
ejpam-628	26	7	rights	right	NOUN
ejpam-628	26	8	reserved	reserve	VERB
ejpam-628	26	9	.	.	PUNCT
ejpam-628	27	1	m.	m.	NOUN
ejpam-628	27	2	behboodi	behboodi	PROPN
ejpam-628	27	3	,	,	PUNCT
ejpam-628	27	4	r.	r.	PROPN
ejpam-628	27	5	beyranvand	beyranvand	PROPN
ejpam-628	27	6	/	/	SYM
ejpam-628	27	7	eur	eur	PROPN
ejpam-628	27	8	.	.	PUNCT
ejpam-628	28	1	j.	j.	PROPN
ejpam-628	28	2	pure	pure	PROPN
ejpam-628	28	3	appl	appl	PROPN
ejpam-628	28	4	.	.	PROPN
ejpam-628	28	5	math	math	PROPN
ejpam-628	28	6	,	,	PUNCT
ejpam-628	28	7	3	3	NUM
ejpam-628	28	8	(	(	PUNCT
ejpam-628	28	9	2010	2010	NUM
ejpam-628	28	10	)	)	PUNCT
ejpam-628	28	11	,	,	PUNCT
ejpam-628	28	12	303	303	NUM
ejpam-628	28	13	-	-	SYM
ejpam-628	28	14	316	316	NUM
ejpam-628	28	15	304	304	NUM
ejpam-628	28	16	following	follow	VERB
ejpam-628	28	17	fundamental	fundamental	ADJ
ejpam-628	28	18	question	question	NOUN
ejpam-628	28	19	:	:	PUNCT
ejpam-628	28	20	question	question	NOUN
ejpam-628	28	21	.	.	PUNCT
ejpam-628	29	1	if	if	SCONJ
ejpam-628	29	2	r	r	NOUN
ejpam-628	29	3	is	be	AUX
ejpam-628	29	4	a	a	DET
ejpam-628	29	5	finite	finite	ADJ
ejpam-628	29	6	commutative	commutative	ADJ
ejpam-628	29	7	ring	ring	NOUN
ejpam-628	29	8	,	,	PUNCT
ejpam-628	29	9	can	can	AUX
ejpam-628	29	10	we	we	PRON
ejpam-628	29	11	find	find	VERB
ejpam-628	29	12	the	the	DET
ejpam-628	29	13	relationship	relationship	NOUN
ejpam-628	29	14	between	between	ADP
ejpam-628	29	15	the	the	DET
ejpam-628	29	16	prime	prime	ADJ
ejpam-628	29	17	factorizations	factorization	NOUN
ejpam-628	29	18	|z(r)|=	|z(r)|=	VERB
ejpam-628	29	19	p1	p1	NOUN
ejpam-628	29	20	k1	k1	NOUN
ejpam-628	29	21	·	·	PUNCT
ejpam-628	29	22	·	·	PUNCT
ejpam-628	29	23	·	·	PUNCT
ejpam-628	30	1	pn	pn	INTJ
ejpam-628	30	2	kn	kn	PROPN
ejpam-628	30	3	and	and	CCONJ
ejpam-628	30	4	the	the	DET
ejpam-628	30	5	summands	summand	NOUN
ejpam-628	30	6	ri	ri	PROPN
ejpam-628	30	7	,	,	PUNCT
ejpam-628	30	8	where	where	SCONJ
ejpam-628	30	9	r=	r=	ADJ
ejpam-628	30	10	r1×r2×	r1×r2×	ADV
ejpam-628	30	11	.	.	PUNCT
ejpam-628	30	12	.	.	PUNCT
ejpam-628	31	1	.×rm	.×rm	PUNCT
ejpam-628	31	2	(	(	PUNCT
ejpam-628	31	3	m	m	NOUN
ejpam-628	31	4	≥	≥	NOUN
ejpam-628	31	5	1	1	NUM
ejpam-628	31	6	)	)	PUNCT
ejpam-628	31	7	and	and	CCONJ
ejpam-628	31	8	r	r	NOUN
ejpam-628	31	9	,	,	PUNCT
ejpam-628	31	10	i	i	PRON
ejpam-628	31	11	s	s	VERB
ejpam-628	31	12	are	be	AUX
ejpam-628	31	13	local	local	ADJ
ejpam-628	31	14	rings	ring	NOUN
ejpam-628	31	15	?	?	PUNCT
ejpam-628	32	1	then	then	ADV
ejpam-628	32	2	we	we	PRON
ejpam-628	32	3	will	will	AUX
ejpam-628	32	4	give	give	VERB
ejpam-628	32	5	an	an	DET
ejpam-628	32	6	answer	answer	NOUN
ejpam-628	32	7	to	to	ADP
ejpam-628	32	8	this	this	DET
ejpam-628	32	9	question	question	NOUN
ejpam-628	32	10	.	.	PUNCT
ejpam-628	33	1	we	we	PRON
ejpam-628	33	2	show	show	VERB
ejpam-628	33	3	that	that	SCONJ
ejpam-628	33	4	the	the	DET
ejpam-628	33	5	answer	answer	NOUN
ejpam-628	33	6	is	be	AUX
ejpam-628	33	7	“	"	PUNCT
ejpam-628	33	8	yes	yes	INTJ
ejpam-628	33	9	”	"	PUNCT
ejpam-628	33	10	and	and	CCONJ
ejpam-628	33	11	a	a	DET
ejpam-628	33	12	preliminary	preliminary	ADJ
ejpam-628	33	13	answer	answer	NOUN
ejpam-628	33	14	is	be	AUX
ejpam-628	33	15	given	give	VERB
ejpam-628	33	16	in	in	ADP
ejpam-628	33	17	theorem	theorem	NOUN
ejpam-628	33	18	1	1	NUM
ejpam-628	33	19	of	of	ADP
ejpam-628	33	20	section	section	NOUN
ejpam-628	33	21	2	2	NUM
ejpam-628	33	22	;	;	PUNCT
ejpam-628	33	23	which	which	PRON
ejpam-628	33	24	shows	show	VERB
ejpam-628	33	25	that	that	SCONJ
ejpam-628	33	26	if	if	SCONJ
ejpam-628	33	27	r	r	NOUN
ejpam-628	33	28	is	be	AUX
ejpam-628	33	29	a	a	DET
ejpam-628	33	30	finite	finite	ADJ
ejpam-628	33	31	commutative	commutative	ADJ
ejpam-628	33	32	ring	ring	NOUN
ejpam-628	33	33	,	,	PUNCT
ejpam-628	33	34	then	then	ADV
ejpam-628	33	35	either	either	CCONJ
ejpam-628	33	36	r	r	NOUN
ejpam-628	33	37	is	be	AUX
ejpam-628	33	38	a	a	DET
ejpam-628	33	39	reduced	reduce	VERB
ejpam-628	33	40	ring	ring	NOUN
ejpam-628	33	41	or	or	CCONJ
ejpam-628	33	42	there	there	PRON
ejpam-628	33	43	are	be	VERB
ejpam-628	33	44	positive	positive	ADJ
ejpam-628	33	45	integers	integer	NOUN
ejpam-628	33	46	s	s	NOUN
ejpam-628	33	47	,	,	PUNCT
ejpam-628	33	48	m	m	PROPN
ejpam-628	33	49	,	,	PUNCT
ejpam-628	33	50	t1	t1	NOUN
ejpam-628	33	51	,	,	PUNCT
ejpam-628	33	52	.	.	PUNCT
ejpam-628	33	53	.	.	PUNCT
ejpam-628	34	1	.	.	PUNCT
ejpam-628	35	1	,	,	PUNCT
ejpam-628	35	2	ts	ts	NOUN
ejpam-628	35	3	,	,	PUNCT
ejpam-628	35	4	prime	prime	ADJ
ejpam-628	35	5	numbers	number	NOUN
ejpam-628	35	6	p1	p1	NOUN
ejpam-628	35	7	,	,	PUNCT
ejpam-628	35	8	p2	p2	NOUN
ejpam-628	35	9	,	,	PUNCT
ejpam-628	35	10	.	.	PUNCT
ejpam-628	35	11	.	.	PUNCT
ejpam-628	35	12	.	.	PUNCT
ejpam-628	36	1	,	,	PUNCT
ejpam-628	36	2	ps	ps	PROPN
ejpam-628	36	3	and	and	CCONJ
ejpam-628	36	4	a	a	DET
ejpam-628	36	5	non	non	ADJ
ejpam-628	36	6	-	-	ADJ
ejpam-628	36	7	negative	negative	ADJ
ejpam-628	36	8	integer	integer	NOUN
ejpam-628	36	9	t	t	PROPN
ejpam-628	36	10	such	such	ADJ
ejpam-628	36	11	that	that	SCONJ
ejpam-628	36	12	|z(r)|=	|z(r)|=	VERB
ejpam-628	36	13	p1	p1	PROPN
ejpam-628	36	14	t1	t1	NOUN
ejpam-628	36	15	p2	p2	PROPN
ejpam-628	36	16	t2	t2	PROPN
ejpam-628	36	17	.	.	PUNCT
ejpam-628	36	18	.	.	PUNCT
ejpam-628	36	19	.	.	PUNCT
ejpam-628	37	1	ps	ps	INTJ
ejpam-628	37	2	ts	ts	ADP
ejpam-628	37	3	m	m	PROPN
ejpam-628	37	4	and	and	CCONJ
ejpam-628	37	5	r∼=	r∼=	NUM
ejpam-628	37	6	r1×	r1×	NOUN
ejpam-628	37	7	.	.	PUNCT
ejpam-628	37	8	.	.	PUNCT
ejpam-628	38	1	.×rs×	.×rs×	PUNCT
ejpam-628	39	1	fq1	fq1	ADV
ejpam-628	39	2	×	×	NOUN
ejpam-628	39	3	.	.	PUNCT
ejpam-628	39	4	.	.	PUNCT
ejpam-628	40	1	.×	.×	PROPN
ejpam-628	40	2	fqt	fqt	PROPN
ejpam-628	40	3	with	with	ADP
ejpam-628	40	4	|z(ri)|	|z(ri)|	NOUN
ejpam-628	40	5	=	=	SYM
ejpam-628	41	1	p	p	X
ejpam-628	41	2	ti	ti	X
ejpam-628	41	3	i	i	PRON
ejpam-628	41	4	.	.	PUNCT
ejpam-628	42	1	therefore	therefore	ADV
ejpam-628	42	2	,	,	PUNCT
ejpam-628	42	3	in	in	ADP
ejpam-628	42	4	classifying	classify	VERB
ejpam-628	42	5	commutative	commutative	ADJ
ejpam-628	42	6	rings	ring	NOUN
ejpam-628	42	7	with	with	ADP
ejpam-628	42	8	p1	p1	PROPN
ejpam-628	42	9	k1	k1	NOUN
ejpam-628	42	10	.	.	PUNCT
ejpam-628	42	11	.	.	PUNCT
ejpam-628	42	12	.	.	PUNCT
ejpam-628	43	1	pn	pn	PROPN
ejpam-628	43	2	kn	kn	PROPN
ejpam-628	43	3	zero	zero	NUM
ejpam-628	43	4	-	-	PUNCT
ejpam-628	43	5	divisors	divisor	NOUN
ejpam-628	43	6	it	it	PRON
ejpam-628	43	7	suffices	suffice	VERB
ejpam-628	43	8	to	to	PART
ejpam-628	43	9	deal	deal	VERB
ejpam-628	43	10	with	with	ADP
ejpam-628	43	11	local	local	ADJ
ejpam-628	43	12	rings	ring	NOUN
ejpam-628	43	13	with	with	ADP
ejpam-628	43	14	pi	pi	NOUN
ejpam-628	43	15	ti	ti	NOUN
ejpam-628	43	16	zero	zero	NUM
ejpam-628	43	17	-	-	PUNCT
ejpam-628	43	18	divisors	divisor	NOUN
ejpam-628	43	19	where	where	SCONJ
ejpam-628	43	20	that	that	SCONJ
ejpam-628	43	21	1	1	NUM
ejpam-628	43	22	≤	≤	PUNCT
ejpam-628	43	23	i	i	PRON
ejpam-628	43	24	≤	≤	ADJ
ejpam-628	43	25	n	n	CCONJ
ejpam-628	43	26	and	and	CCONJ
ejpam-628	43	27	0	0	NUM
ejpam-628	43	28	≤	≤	NUM
ejpam-628	43	29	t	t	NOUN
ejpam-628	44	1	i	i	PRON
ejpam-628	44	2	≤	≤	ADV
ejpam-628	44	3	ki	ki	ADV
ejpam-628	44	4	,	,	PUNCT
ejpam-628	44	5	and	and	CCONJ
ejpam-628	44	6	henceforth	henceforth	ADV
ejpam-628	44	7	we	we	PRON
ejpam-628	44	8	focus	focus	VERB
ejpam-628	44	9	on	on	ADP
ejpam-628	44	10	rings	ring	NOUN
ejpam-628	44	11	r	r	NOUN
ejpam-628	44	12	with	with	ADP
ejpam-628	44	13	|z(r)|	|z(r)|	PROPN
ejpam-628	44	14	=	=	SYM
ejpam-628	44	15	pk	pk	NOUN
ejpam-628	44	16	where	where	SCONJ
ejpam-628	44	17	p	p	NOUN
ejpam-628	44	18	is	be	AUX
ejpam-628	44	19	a	a	DET
ejpam-628	44	20	prime	prime	ADJ
ejpam-628	44	21	number	number	NOUN
ejpam-628	44	22	and	and	CCONJ
ejpam-628	44	23	k	k	PROPN
ejpam-628	44	24	≥	≥	NUM
ejpam-628	44	25	1	1	NUM
ejpam-628	44	26	.	.	PUNCT
ejpam-628	45	1	it	it	PRON
ejpam-628	45	2	is	be	AUX
ejpam-628	45	3	shown	show	VERB
ejpam-628	45	4	that	that	SCONJ
ejpam-628	45	5	a	a	DET
ejpam-628	45	6	finite	finite	ADJ
ejpam-628	45	7	commutative	commutative	ADJ
ejpam-628	45	8	ring	ring	NOUN
ejpam-628	45	9	r	r	NOUN
ejpam-628	45	10	is	be	AUX
ejpam-628	45	11	local	local	ADJ
ejpam-628	45	12	if	if	SCONJ
ejpam-628	45	13	and	and	CCONJ
ejpam-628	45	14	only	only	ADV
ejpam-628	45	15	if	if	SCONJ
ejpam-628	45	16	|z(r)|	|z(r)|	PROPN
ejpam-628	45	17	=	=	SYM
ejpam-628	45	18	pk	pk	NOUN
ejpam-628	45	19	and	and	CCONJ
ejpam-628	45	20	|r|	|r|	NOUN
ejpam-628	45	21	=	=	NOUN
ejpam-628	45	22	pn	pn	NOUN
ejpam-628	45	23	for	for	ADP
ejpam-628	45	24	some	some	DET
ejpam-628	45	25	prime	prime	ADJ
ejpam-628	45	26	number	number	NOUN
ejpam-628	45	27	p	p	NOUN
ejpam-628	45	28	and	and	CCONJ
ejpam-628	45	29	n	n	PROPN
ejpam-628	45	30	>	>	X
ejpam-628	45	31	k	k	X
ejpam-628	45	32	≥	≥	PROPN
ejpam-628	45	33	0	0	NUM
ejpam-628	45	34	(	(	PUNCT
ejpam-628	45	35	theorem	theorem	NOUN
ejpam-628	45	36	3	3	NUM
ejpam-628	45	37	)	)	PUNCT
ejpam-628	45	38	.	.	PUNCT
ejpam-628	46	1	in	in	ADP
ejpam-628	46	2	section	section	NOUN
ejpam-628	46	3	3	3	NUM
ejpam-628	46	4	,	,	PUNCT
ejpam-628	46	5	first	first	ADV
ejpam-628	46	6	we	we	PRON
ejpam-628	46	7	characterize	characterize	VERB
ejpam-628	46	8	commutative	commutative	ADJ
ejpam-628	46	9	rings	ring	NOUN
ejpam-628	46	10	r	r	NOUN
ejpam-628	46	11	with	with	ADP
ejpam-628	46	12	|z(r)|=	|z(r)|=	ADJ
ejpam-628	46	13	pk	pk	NOUN
ejpam-628	46	14	where	where	SCONJ
ejpam-628	46	15	1≤	1≤	PROPN
ejpam-628	46	16	k	k	PROPN
ejpam-628	47	1	≤	≤	ADV
ejpam-628	47	2	7	7	NUM
ejpam-628	47	3	.	.	PUNCT
ejpam-628	48	1	then	then	ADV
ejpam-628	48	2	the	the	DET
ejpam-628	48	3	structure	structure	NOUN
ejpam-628	48	4	and	and	CCONJ
ejpam-628	48	5	classification	classification	NOUN
ejpam-628	48	6	up	up	ADP
ejpam-628	48	7	to	to	PART
ejpam-628	48	8	isomorphism	isomorphism	VERB
ejpam-628	48	9	all	all	DET
ejpam-628	48	10	commutative	commutative	ADJ
ejpam-628	48	11	rings	ring	NOUN
ejpam-628	48	12	r	r	NOUN
ejpam-628	48	13	with	with	ADP
ejpam-628	48	14	|z(r)|=	|z(r)|=	VERB
ejpam-628	48	15	p1	p1	NOUN
ejpam-628	48	16	k1	k1	NOUN
ejpam-628	48	17	.	.	PUNCT
ejpam-628	48	18	.	.	PUNCT
ejpam-628	48	19	.	.	PUNCT
ejpam-628	49	1	pn	pn	PROPN
ejpam-628	49	2	kn	kn	PROPN
ejpam-628	49	3	,	,	PUNCT
ejpam-628	49	4	where	where	SCONJ
ejpam-628	49	5	n	n	X
ejpam-628	49	6	∈	∈	PROPN
ejpam-628	49	7	n	n	CCONJ
ejpam-628	49	8	,	,	PUNCT
ejpam-628	49	9	p	p	X
ejpam-628	49	10	,	,	PUNCT
ejpam-628	49	11	i	i	PRON
ejpam-628	49	12	s	s	AUX
ejpam-628	49	13	are	be	AUX
ejpam-628	49	14	distinct	distinct	ADJ
ejpam-628	49	15	prime	prime	ADJ
ejpam-628	49	16	numbers	number	NOUN
ejpam-628	49	17	and	and	CCONJ
ejpam-628	49	18	1	1	NUM
ejpam-628	49	19	≤	≤	NUM
ejpam-628	49	20	ki	ki	PROPN
ejpam-628	49	21	≤	≤	ADJ
ejpam-628	49	22	3	3	NUM
ejpam-628	49	23	,	,	PUNCT
ejpam-628	49	24	are	be	AUX
ejpam-628	49	25	determined	determine	VERB
ejpam-628	49	26	.	.	PUNCT
ejpam-628	50	1	finally	finally	ADV
ejpam-628	50	2	,	,	PUNCT
ejpam-628	50	3	we	we	PRON
ejpam-628	50	4	determine	determine	VERB
ejpam-628	50	5	the	the	DET
ejpam-628	50	6	structure	structure	NOUN
ejpam-628	50	7	of	of	ADP
ejpam-628	50	8	nonlocal	nonlocal	ADJ
ejpam-628	50	9	rings	ring	NOUN
ejpam-628	50	10	r	r	NOUN
ejpam-628	50	11	with	with	ADP
ejpam-628	50	12	|z(r)|=	|z(r)|=	ADJ
ejpam-628	50	13	pk	pk	NOUN
ejpam-628	50	14	where	where	SCONJ
ejpam-628	50	15	k	k	PROPN
ejpam-628	50	16	=	=	NOUN
ejpam-628	50	17	4	4	NUM
ejpam-628	50	18	or	or	CCONJ
ejpam-628	50	19	5	5	NUM
ejpam-628	50	20	.	.	NOUN
ejpam-628	50	21	2	2	NUM
ejpam-628	50	22	.	.	X
ejpam-628	50	23	on	on	ADP
ejpam-628	50	24	rings	ring	NOUN
ejpam-628	50	25	with	with	ADP
ejpam-628	50	26	pk	pk	NOUN
ejpam-628	50	27	zero	zero	NUM
ejpam-628	50	28	-	-	PUNCT
ejpam-628	50	29	divisors	divisor	NOUN
ejpam-628	50	30	recall	recall	VERB
ejpam-628	50	31	that	that	SCONJ
ejpam-628	50	32	an	an	DET
ejpam-628	50	33	artinian	artinian	ADJ
ejpam-628	50	34	commutative	commutative	ADJ
ejpam-628	50	35	ring	ring	NOUN
ejpam-628	50	36	r	r	NOUN
ejpam-628	50	37	is	be	AUX
ejpam-628	50	38	called	call	VERB
ejpam-628	50	39	completely	completely	ADV
ejpam-628	50	40	primary	primary	ADJ
ejpam-628	50	41	if	if	SCONJ
ejpam-628	50	42	r	r	NOUN
ejpam-628	50	43	/	/	SYM
ejpam-628	50	44	j(r	j(r	PROPN
ejpam-628	50	45	)	)	PUNCT
ejpam-628	50	46	is	be	AUX
ejpam-628	50	47	a	a	DET
ejpam-628	50	48	field	field	NOUN
ejpam-628	50	49	.	.	PUNCT
ejpam-628	51	1	one	one	PRON
ejpam-628	51	2	can	can	AUX
ejpam-628	51	3	easily	easily	ADV
ejpam-628	51	4	see	see	VERB
ejpam-628	51	5	that	that	SCONJ
ejpam-628	51	6	an	an	DET
ejpam-628	51	7	artinian	artinian	ADJ
ejpam-628	51	8	commutative	commutative	ADJ
ejpam-628	51	9	ring	ring	NOUN
ejpam-628	51	10	r	r	NOUN
ejpam-628	51	11	is	be	AUX
ejpam-628	51	12	completely	completely	ADV
ejpam-628	51	13	primary	primary	ADJ
ejpam-628	51	14	if	if	SCONJ
ejpam-628	51	15	and	and	CCONJ
ejpam-628	51	16	only	only	ADV
ejpam-628	51	17	if	if	SCONJ
ejpam-628	51	18	z(r	z(r	NOUN
ejpam-628	51	19	)	)	PUNCT
ejpam-628	51	20	is	be	AUX
ejpam-628	51	21	an	an	DET
ejpam-628	51	22	ideal	ideal	NOUN
ejpam-628	51	23	of	of	ADP
ejpam-628	51	24	r	r	NOUN
ejpam-628	51	25	,	,	PUNCT
ejpam-628	51	26	if	if	SCONJ
ejpam-628	51	27	and	and	CCONJ
ejpam-628	51	28	only	only	ADV
ejpam-628	51	29	if	if	SCONJ
ejpam-628	51	30	r	r	NOUN
ejpam-628	51	31	is	be	AUX
ejpam-628	51	32	a	a	DET
ejpam-628	51	33	local	local	ADJ
ejpam-628	51	34	ring	ring	NOUN
ejpam-628	51	35	.	.	PUNCT
ejpam-628	52	1	moreover	moreover	ADV
ejpam-628	52	2	,	,	PUNCT
ejpam-628	52	3	we	we	PRON
ejpam-628	52	4	have	have	VERB
ejpam-628	52	5	the	the	DET
ejpam-628	52	6	following	follow	VERB
ejpam-628	52	7	lemma	lemma	PROPN
ejpam-628	52	8	which	which	PRON
ejpam-628	52	9	is	be	AUX
ejpam-628	52	10	essentially	essentially	ADV
ejpam-628	52	11	theorem	theorem	VERB
ejpam-628	52	12	2	2	NUM
ejpam-628	52	13	of	of	ADP
ejpam-628	52	14	[	[	X
ejpam-628	52	15	9	9	NUM
ejpam-628	52	16	]	]	PUNCT
ejpam-628	52	17	.	.	PUNCT
ejpam-628	53	1	lemma	lemma	PROPN
ejpam-628	53	2	1	1	NUM
ejpam-628	53	3	.	.	PUNCT
ejpam-628	54	1	[	[	X
ejpam-628	54	2	9	9	NUM
ejpam-628	54	3	,	,	PUNCT
ejpam-628	54	4	theorem	theorem	ADJ
ejpam-628	54	5	2	2	NUM
ejpam-628	54	6	]	]	PUNCT
ejpam-628	54	7	let	let	VERB
ejpam-628	54	8	r	r	PRON
ejpam-628	54	9	be	be	AUX
ejpam-628	54	10	a	a	DET
ejpam-628	54	11	finite	finite	NOUN
ejpam-628	54	12	completely	completely	ADV
ejpam-628	54	13	primary	primary	ADJ
ejpam-628	54	14	ring	ring	NOUN
ejpam-628	54	15	.	.	PUNCT
ejpam-628	55	1	then	then	ADV
ejpam-628	55	2	(	(	PUNCT
ejpam-628	55	3	i	i	NOUN
ejpam-628	55	4	)	)	PUNCT
ejpam-628	55	5	z(r	z(r	PROPN
ejpam-628	55	6	)	)	PUNCT
ejpam-628	55	7	=	=	SYM
ejpam-628	55	8	j(r	j(r	PROPN
ejpam-628	55	9	)	)	PUNCT
ejpam-628	55	10	;	;	PUNCT
ejpam-628	55	11	(	(	PUNCT
ejpam-628	55	12	ii	ii	NOUN
ejpam-628	55	13	)	)	PUNCT
ejpam-628	55	14	|z(r)|=	|z(r)|=	VERB
ejpam-628	55	15	p(n−1)r	p(n−1)r	NOUN
ejpam-628	55	16	and	and	CCONJ
ejpam-628	55	17	|r|=	|r|=	NOUN
ejpam-628	55	18	pnr	pnr	NOUN
ejpam-628	55	19	for	for	ADP
ejpam-628	55	20	some	some	DET
ejpam-628	55	21	prime	prime	ADJ
ejpam-628	55	22	number	number	NOUN
ejpam-628	55	23	p	p	NOUN
ejpam-628	55	24	,	,	PUNCT
ejpam-628	55	25	and	and	CCONJ
ejpam-628	55	26	some	some	DET
ejpam-628	55	27	positive	positive	ADJ
ejpam-628	55	28	integers	integer	NOUN
ejpam-628	55	29	n	n	CCONJ
ejpam-628	55	30	,	,	PUNCT
ejpam-628	55	31	r	r	NOUN
ejpam-628	55	32	;	;	PUNCT
ejpam-628	55	33	(	(	PUNCT
ejpam-628	55	34	iii	iii	X
ejpam-628	55	35	)	)	PUNCT
ejpam-628	55	36	z(r)n	z(r)n	NOUN
ejpam-628	55	37	=	=	SYM
ejpam-628	55	38	(	(	PUNCT
ejpam-628	55	39	0	0	NUM
ejpam-628	55	40	)	)	PUNCT
ejpam-628	55	41	;	;	PUNCT
ejpam-628	55	42	(	(	PUNCT
ejpam-628	55	43	iv	iv	X
ejpam-628	55	44	)	)	PUNCT
ejpam-628	55	45	char(r	char(r	NOUN
ejpam-628	55	46	)	)	PUNCT
ejpam-628	55	47	=	=	SYM
ejpam-628	55	48	pk	pk	NOUN
ejpam-628	55	49	for	for	ADP
ejpam-628	55	50	some	some	DET
ejpam-628	55	51	integer	integer	NOUN
ejpam-628	55	52	k	k	PROPN
ejpam-628	55	53	with	with	ADP
ejpam-628	55	54	1≤	1≤	PROPN
ejpam-628	55	55	k	k	PROPN
ejpam-628	55	56	≤	≤	PROPN
ejpam-628	55	57	n	n	CCONJ
ejpam-628	55	58	;	;	PUNCT
ejpam-628	55	59	(	(	PUNCT
ejpam-628	55	60	v	v	NOUN
ejpam-628	55	61	)	)	PUNCT
ejpam-628	55	62	r	r	NOUN
ejpam-628	55	63	/	/	SYM
ejpam-628	55	64	j(r)∼=	j(r)∼=	NOUN
ejpam-628	55	65	fq	fq	NOUN
ejpam-628	55	66	,	,	PUNCT
ejpam-628	55	67	where	where	SCONJ
ejpam-628	55	68	q	q	NOUN
ejpam-628	55	69	=	=	NOUN
ejpam-628	55	70	pr	pr	X
ejpam-628	55	71	.	.	PUNCT
ejpam-628	56	1	let	let	VERB
ejpam-628	56	2	ri	ri	PROPN
ejpam-628	56	3	(	(	PUNCT
ejpam-628	56	4	1	1	NUM
ejpam-628	56	5	≤	≤	NUM
ejpam-628	56	6	i	i	X
ejpam-628	56	7	≤	≤	PROPN
ejpam-628	56	8	s	s	AUX
ejpam-628	56	9	)	)	PUNCT
ejpam-628	56	10	be	be	AUX
ejpam-628	56	11	a	a	DET
ejpam-628	56	12	finite	finite	ADJ
ejpam-628	56	13	commutative	commutative	ADJ
ejpam-628	56	14	ring	ring	NOUN
ejpam-628	56	15	with	with	ADP
ejpam-628	56	16	mi	mi	NOUN
ejpam-628	56	17	elements	element	NOUN
ejpam-628	56	18	and	and	CCONJ
ejpam-628	56	19	ni	ni	PROPN
ejpam-628	56	20	zero	zero	NUM
ejpam-628	56	21	-	-	PUNCT
ejpam-628	56	22	divisors	divisor	NOUN
ejpam-628	56	23	.	.	PUNCT
ejpam-628	57	1	let	let	VERB
ejpam-628	57	2	r=	r=	ADJ
ejpam-628	57	3	r1×	r1×	VERB
ejpam-628	57	4	.	.	PUNCT
ejpam-628	57	5	.	.	PUNCT
ejpam-628	58	1	.×rs	.×rs	PROPN
ejpam-628	58	2	.	.	PUNCT
ejpam-628	59	1	then	then	ADV
ejpam-628	59	2	by	by	ADP
ejpam-628	59	3	[	[	X
ejpam-628	59	4	6	6	NUM
ejpam-628	59	5	,	,	PUNCT
ejpam-628	59	6	theorem	theorem	VERB
ejpam-628	59	7	2	2	NUM
ejpam-628	59	8	]	]	PUNCT
ejpam-628	59	9	,	,	PUNCT
ejpam-628	59	10	|z(r)|=	|z(r)|=	PROPN
ejpam-628	59	11	m1m2	m1m2	X
ejpam-628	59	12	.	.	PUNCT
ejpam-628	59	13	.	.	PUNCT
ejpam-628	59	14	.	.	PUNCT
ejpam-628	60	1	ms−(m1−n1)(m2−n2	ms−(m1−n1)(m2−n2	X
ejpam-628	60	2	)	)	PUNCT
ejpam-628	60	3	.	.	PUNCT
ejpam-628	60	4	.	.	PUNCT
ejpam-628	61	1	.	.	PUNCT
ejpam-628	62	1	(	(	PUNCT
ejpam-628	62	2	ms−	ms−	NOUN
ejpam-628	62	3	ns	ns	NUM
ejpam-628	62	4	)	)	PUNCT
ejpam-628	62	5	.	.	PUNCT
ejpam-628	63	1	thus	thus	ADV
ejpam-628	63	2	by	by	ADP
ejpam-628	63	3	using	use	VERB
ejpam-628	63	4	this	this	DET
ejpam-628	63	5	fact	fact	NOUN
ejpam-628	63	6	,	,	PUNCT
ejpam-628	63	7	lemma	lemma	PROPN
ejpam-628	63	8	1	1	NUM
ejpam-628	63	9	and	and	CCONJ
ejpam-628	63	10	the	the	DET
ejpam-628	63	11	fact	fact	NOUN
ejpam-628	63	12	that	that	SCONJ
ejpam-628	63	13	every	every	DET
ejpam-628	63	14	finite	finite	PROPN
ejpam-628	63	15	commutative	commutative	ADJ
ejpam-628	63	16	ring	ring	NOUN
ejpam-628	63	17	is	be	AUX
ejpam-628	63	18	uniquely	uniquely	ADV
ejpam-628	63	19	expressible	expressible	ADJ
ejpam-628	63	20	as	as	ADP
ejpam-628	63	21	a	a	DET
ejpam-628	63	22	direct	direct	ADJ
ejpam-628	63	23	sum	sum	NOUN
ejpam-628	63	24	of	of	ADP
ejpam-628	63	25	completely	completely	ADV
ejpam-628	63	26	primary	primary	ADJ
ejpam-628	63	27	(	(	PUNCT
ejpam-628	63	28	local	local	ADJ
ejpam-628	63	29	)	)	PUNCT
ejpam-628	63	30	rings	ring	NOUN
ejpam-628	63	31	(	(	PUNCT
ejpam-628	63	32	see	see	VERB
ejpam-628	63	33	for	for	ADP
ejpam-628	63	34	example	example	NOUN
ejpam-628	63	35	[	[	X
ejpam-628	63	36	8	8	NUM
ejpam-628	63	37	,	,	PUNCT
ejpam-628	63	38	p.95	p.95	NOUN
ejpam-628	63	39	]	]	X
ejpam-628	63	40	)	)	PUNCT
ejpam-628	63	41	,	,	PUNCT
ejpam-628	63	42	we	we	PRON
ejpam-628	63	43	have	have	VERB
ejpam-628	63	44	the	the	DET
ejpam-628	63	45	following	follow	VERB
ejpam-628	63	46	evident	evident	ADJ
ejpam-628	63	47	result	result	NOUN
ejpam-628	63	48	.	.	PUNCT
ejpam-628	64	1	m.	m.	NOUN
ejpam-628	64	2	behboodi	behboodi	PROPN
ejpam-628	64	3	,	,	PUNCT
ejpam-628	64	4	r.	r.	PROPN
ejpam-628	64	5	beyranvand	beyranvand	PROPN
ejpam-628	64	6	/	/	SYM
ejpam-628	64	7	eur	eur	PROPN
ejpam-628	64	8	.	.	PUNCT
ejpam-628	65	1	j.	j.	PROPN
ejpam-628	65	2	pure	pure	PROPN
ejpam-628	65	3	appl	appl	PROPN
ejpam-628	65	4	.	.	PROPN
ejpam-628	65	5	math	math	PROPN
ejpam-628	65	6	,	,	PUNCT
ejpam-628	65	7	3	3	NUM
ejpam-628	65	8	(	(	PUNCT
ejpam-628	65	9	2010	2010	NUM
ejpam-628	65	10	)	)	PUNCT
ejpam-628	65	11	,	,	PUNCT
ejpam-628	65	12	303	303	NUM
ejpam-628	65	13	-	-	SYM
ejpam-628	65	14	316	316	NUM
ejpam-628	65	15	305	305	NUM
ejpam-628	65	16	lemma	lemma	PROPN
ejpam-628	65	17	2	2	NUM
ejpam-628	65	18	.	.	PUNCT
ejpam-628	66	1	let	let	VERB
ejpam-628	66	2	r	r	PRON
ejpam-628	66	3	be	be	AUX
ejpam-628	66	4	a	a	DET
ejpam-628	66	5	finite	finite	ADJ
ejpam-628	66	6	commutative	commutative	ADJ
ejpam-628	66	7	ring	ring	NOUN
ejpam-628	66	8	.	.	PUNCT
ejpam-628	67	1	then	then	ADV
ejpam-628	67	2	r	r	NOUN
ejpam-628	67	3	∼=	∼=	PROPN
ejpam-628	67	4	r1	r1	NOUN
ejpam-628	67	5	×	×	NOUN
ejpam-628	67	6	.	.	PUNCT
ejpam-628	67	7	.	.	PUNCT
ejpam-628	67	8	.	.	PUNCT
ejpam-628	68	1	×	×	NOUN
ejpam-628	68	2	rs	rs	INTJ
ejpam-628	68	3	where	where	SCONJ
ejpam-628	68	4	s	s	VERB
ejpam-628	68	5	∈	∈	PROPN
ejpam-628	68	6	n	n	NOUN
ejpam-628	68	7	and	and	CCONJ
ejpam-628	68	8	ri	ri	PROPN
ejpam-628	68	9	,	,	PUNCT
ejpam-628	68	10	s	s	VERB
ejpam-628	68	11	are	be	AUX
ejpam-628	68	12	local	local	ADJ
ejpam-628	68	13	rings	ring	NOUN
ejpam-628	68	14	with	with	ADP
ejpam-628	68	15	|ri|	|ri|	NOUN
ejpam-628	68	16	=	=	PUNCT
ejpam-628	69	1	p	p	X
ejpam-628	69	2	ki	ki	PROPN
ejpam-628	70	1	i	i	PRON
ejpam-628	70	2	,	,	PUNCT
ejpam-628	70	3	|z(ri)|	|z(ri)|	X
ejpam-628	70	4	=	=	SYM
ejpam-628	70	5	p	p	X
ejpam-628	70	6	ti	ti	VERB
ejpam-628	70	7	i	i	PRON
ejpam-628	70	8	for	for	ADP
ejpam-628	70	9	some	some	DET
ejpam-628	70	10	prime	prime	ADJ
ejpam-628	70	11	numbers	number	NOUN
ejpam-628	70	12	p1	p1	NOUN
ejpam-628	70	13	,	,	PUNCT
ejpam-628	70	14	p2	p2	NOUN
ejpam-628	70	15	,	,	PUNCT
ejpam-628	70	16	.	.	PUNCT
ejpam-628	70	17	.	.	PUNCT
ejpam-628	70	18	.	.	PUNCT
ejpam-628	71	1	,	,	PUNCT
ejpam-628	71	2	ps	ps	PROPN
ejpam-628	71	3	and	and	CCONJ
ejpam-628	71	4	ki	ki	PROPN
ejpam-628	71	5	≥	≥	PROPN
ejpam-628	71	6	1	1	NUM
ejpam-628	71	7	,	,	PUNCT
ejpam-628	71	8	t	t	PROPN
ejpam-628	71	9	i	i	PRON
ejpam-628	71	10	≥	≥	PROPN
ejpam-628	71	11	0	0	NUM
ejpam-628	71	12	.	.	PUNCT
ejpam-628	72	1	consequently	consequently	ADV
ejpam-628	72	2	,	,	PUNCT
ejpam-628	72	3	|z(r)|=	|z(r)|=	PROPN
ejpam-628	72	4	s	s	PART
ejpam-628	72	5	∏	∏	NUM
ejpam-628	72	6	i=1	i=1	X
ejpam-628	72	7	p	p	X
ejpam-628	72	8	ti	ti	NOUN
ejpam-628	72	9	i	i	PRON
ejpam-628	72	10	(	(	PUNCT
ejpam-628	72	11	s	s	VERB
ejpam-628	72	12	∏	∏	NUM
ejpam-628	72	13	i=1	i=1	PROPN
ejpam-628	73	1	p	p	NOUN
ejpam-628	73	2	ki−ti	ki−ti	NOUN
ejpam-628	74	1	i	i	PRON
ejpam-628	74	2	−	−	NOUN
ejpam-628	74	3	s	s	PART
ejpam-628	74	4	∏	∏	PROPN
ejpam-628	74	5	i=1	i=1	PROPN
ejpam-628	74	6	(	(	PUNCT
ejpam-628	75	1	p	p	NOUN
ejpam-628	75	2	ki−ti	ki−ti	NOUN
ejpam-628	76	1	i	i	PRON
ejpam-628	76	2	−	−	NOUN
ejpam-628	76	3	1	1	NUM
ejpam-628	76	4	)	)	PUNCT
ejpam-628	76	5	)	)	PUNCT
ejpam-628	76	6	.	.	PUNCT
ejpam-628	77	1	now	now	ADV
ejpam-628	77	2	we	we	PRON
ejpam-628	77	3	are	be	AUX
ejpam-628	77	4	in	in	ADP
ejpam-628	77	5	position	position	NOUN
ejpam-628	77	6	to	to	PART
ejpam-628	77	7	prove	prove	VERB
ejpam-628	77	8	the	the	DET
ejpam-628	77	9	following	follow	VERB
ejpam-628	77	10	two	two	NUM
ejpam-628	77	11	theorems	theorem	NOUN
ejpam-628	77	12	which	which	PRON
ejpam-628	77	13	are	be	AUX
ejpam-628	77	14	crucial	crucial	ADJ
ejpam-628	77	15	in	in	ADP
ejpam-628	77	16	our	our	PRON
ejpam-628	77	17	investigation	investigation	NOUN
ejpam-628	77	18	.	.	PUNCT
ejpam-628	78	1	theorem	theorem	NOUN
ejpam-628	78	2	1	1	NUM
ejpam-628	78	3	.	.	PUNCT
ejpam-628	79	1	let	let	VERB
ejpam-628	79	2	r	r	PRON
ejpam-628	79	3	be	be	AUX
ejpam-628	79	4	a	a	DET
ejpam-628	79	5	finite	finite	ADJ
ejpam-628	79	6	commutative	commutative	ADJ
ejpam-628	79	7	ring	ring	NOUN
ejpam-628	79	8	.	.	PUNCT
ejpam-628	80	1	then	then	ADV
ejpam-628	80	2	(	(	PUNCT
ejpam-628	80	3	i	i	NOUN
ejpam-628	80	4	)	)	PUNCT
ejpam-628	80	5	if	if	SCONJ
ejpam-628	80	6	r	r	NOUN
ejpam-628	80	7	is	be	AUX
ejpam-628	80	8	reduced	reduce	VERB
ejpam-628	80	9	,	,	PUNCT
ejpam-628	80	10	then	then	ADV
ejpam-628	80	11	there	there	PRON
ejpam-628	80	12	are	be	VERB
ejpam-628	80	13	finite	finite	ADJ
ejpam-628	80	14	fields	field	NOUN
ejpam-628	80	15	fq1	fq1	ADV
ejpam-628	80	16	,	,	PUNCT
ejpam-628	80	17	.	.	PUNCT
ejpam-628	80	18	.	.	PUNCT
ejpam-628	81	1	.	.	PUNCT
ejpam-628	82	1	,	,	PUNCT
ejpam-628	82	2	fqt	fqt	PROPN
ejpam-628	82	3	(	(	PUNCT
ejpam-628	82	4	t	t	X
ejpam-628	82	5	≥	≥	NUM
ejpam-628	82	6	1	1	NUM
ejpam-628	82	7	)	)	PUNCT
ejpam-628	82	8	such	such	ADJ
ejpam-628	82	9	that	that	SCONJ
ejpam-628	82	10	r	r	NOUN
ejpam-628	82	11	∼=	∼=	PROPN
ejpam-628	82	12	fq1	fq1	NUM
ejpam-628	82	13	×	×	NOUN
ejpam-628	82	14	.	.	PUNCT
ejpam-628	82	15	.	.	PUNCT
ejpam-628	82	16	.	.	PUNCT
ejpam-628	83	1	×	×	NOUN
ejpam-628	83	2	fqt	fqt	NOUN
ejpam-628	83	3	with	with	ADP
ejpam-628	83	4	|z(r)|=	|z(r)|=	ADJ
ejpam-628	83	5	q1q2	q1q2	PROPN
ejpam-628	83	6	.	.	PUNCT
ejpam-628	83	7	.	.	PUNCT
ejpam-628	83	8	.	.	PUNCT
ejpam-628	84	1	qt	qt	INTJ
ejpam-628	84	2	−	−	PROPN
ejpam-628	84	3	(	(	PUNCT
ejpam-628	84	4	q1−	q1−	VERB
ejpam-628	84	5	1)(q2−	1)(q2−	NUM
ejpam-628	84	6	1	1	NUM
ejpam-628	84	7	)	)	PUNCT
ejpam-628	84	8	.	.	PUNCT
ejpam-628	84	9	.	.	PUNCT
ejpam-628	84	10	.	.	PUNCT
ejpam-628	85	1	(	(	PUNCT
ejpam-628	85	2	qt	qt	INTJ
ejpam-628	85	3	−	−	NOUN
ejpam-628	85	4	1	1	NUM
ejpam-628	85	5	)	)	PUNCT
ejpam-628	85	6	.	.	PUNCT
ejpam-628	86	1	(	(	PUNCT
ejpam-628	86	2	ii	ii	NOUN
ejpam-628	86	3	)	)	PUNCT
ejpam-628	86	4	if	if	SCONJ
ejpam-628	86	5	r	r	NOUN
ejpam-628	86	6	is	be	AUX
ejpam-628	86	7	not	not	PART
ejpam-628	86	8	reduced	reduce	VERB
ejpam-628	86	9	,	,	PUNCT
ejpam-628	86	10	then	then	ADV
ejpam-628	86	11	there	there	PRON
ejpam-628	86	12	are	be	VERB
ejpam-628	86	13	a	a	DET
ejpam-628	86	14	positive	positive	ADJ
ejpam-628	86	15	integer	integer	NOUN
ejpam-628	86	16	s	s	PROPN
ejpam-628	86	17	,	,	PUNCT
ejpam-628	86	18	a	a	DET
ejpam-628	86	19	non	non	ADJ
ejpam-628	86	20	-	-	ADJ
ejpam-628	86	21	negative	negative	ADJ
ejpam-628	86	22	integer	integer	PROPN
ejpam-628	86	23	t	t	PROPN
ejpam-628	86	24	,	,	PUNCT
ejpam-628	86	25	prime	prime	ADJ
ejpam-628	86	26	numbers	number	NOUN
ejpam-628	86	27	p1	p1	NOUN
ejpam-628	86	28	,	,	PUNCT
ejpam-628	86	29	p2	p2	NOUN
ejpam-628	86	30	,	,	PUNCT
ejpam-628	86	31	.	.	PUNCT
ejpam-628	86	32	.	.	PUNCT
ejpam-628	87	1	.	.	PUNCT
ejpam-628	88	1	,	,	PUNCT
ejpam-628	88	2	ps	ps	NOUN
ejpam-628	88	3	and	and	CCONJ
ejpam-628	88	4	positive	positive	ADJ
ejpam-628	88	5	integers	integer	NOUN
ejpam-628	88	6	k1	k1	NOUN
ejpam-628	88	7	,	,	PUNCT
ejpam-628	88	8	.	.	PUNCT
ejpam-628	88	9	.	.	PUNCT
ejpam-628	89	1	.	.	PUNCT
ejpam-628	90	1	,	,	PUNCT
ejpam-628	91	1	ks	k	NOUN
ejpam-628	91	2	such	such	ADJ
ejpam-628	91	3	that	that	SCONJ
ejpam-628	91	4	|z(r)|=	|z(r)|=	PROPN
ejpam-628	91	5	s	s	PART
ejpam-628	91	6	∏	∏	NUM
ejpam-628	91	7	i=1	i=1	X
ejpam-628	91	8	p	p	X
ejpam-628	91	9	ti	ti	NOUN
ejpam-628	91	10	i	i	PROPN
ejpam-628	91	11	[	[	X
ejpam-628	91	12	q1	q1	NOUN
ejpam-628	91	13	.	.	PUNCT
ejpam-628	91	14	.	.	PUNCT
ejpam-628	91	15	.	.	PUNCT
ejpam-628	92	1	qt	qt	ADP
ejpam-628	92	2	s	s	X
ejpam-628	92	3	∏	∏	NUM
ejpam-628	92	4	i=1	i=1	PROPN
ejpam-628	93	1	p	p	NOUN
ejpam-628	93	2	ki−ti	ki−ti	NOUN
ejpam-628	94	1	i	i	PRON
ejpam-628	94	2	−	−	PROPN
ejpam-628	94	3	(	(	PUNCT
ejpam-628	94	4	q1−	q1−	NOUN
ejpam-628	94	5	1	1	NUM
ejpam-628	94	6	)	)	PUNCT
ejpam-628	94	7	.	.	PUNCT
ejpam-628	94	8	.	.	PUNCT
ejpam-628	94	9	.	.	PUNCT
ejpam-628	95	1	(	(	PUNCT
ejpam-628	95	2	qt	qt	INTJ
ejpam-628	95	3	−	−	NOUN
ejpam-628	95	4	1	1	NUM
ejpam-628	95	5	)	)	PUNCT
ejpam-628	95	6	s	s	PART
ejpam-628	95	7	∏	∏	PROPN
ejpam-628	95	8	i=1	i=1	X
ejpam-628	95	9	(	(	PUNCT
ejpam-628	95	10	p	p	NOUN
ejpam-628	95	11	ki−ti	ki−ti	NOUN
ejpam-628	95	12	i	i	PRON
ejpam-628	95	13	−	−	NOUN
ejpam-628	95	14	1	1	NUM
ejpam-628	95	15	)	)	PUNCT
ejpam-628	95	16	]	]	PUNCT
ejpam-628	96	1	(	(	PUNCT
ejpam-628	96	2	1	1	X
ejpam-628	96	3	)	)	PUNCT
ejpam-628	96	4	and	and	CCONJ
ejpam-628	96	5	r∼=	r∼=	NUM
ejpam-628	96	6	r1×	r1×	NOUN
ejpam-628	96	7	.	.	PUNCT
ejpam-628	97	1	.	.	PUNCT
ejpam-628	98	1	.×rs×	.×rs×	PUNCT
ejpam-628	99	1	fq1	fq1	ADV
ejpam-628	99	2	×	×	NOUN
ejpam-628	99	3	.	.	PUNCT
ejpam-628	99	4	.	.	PUNCT
ejpam-628	100	1	.×	.×	PROPN
ejpam-628	100	2	fqt	fqt	PROPN
ejpam-628	100	3	where	where	SCONJ
ejpam-628	100	4	each	each	DET
ejpam-628	100	5	fqi	fqi	NOUN
ejpam-628	100	6	is	be	AUX
ejpam-628	100	7	a	a	DET
ejpam-628	100	8	finite	finite	ADJ
ejpam-628	100	9	field	field	NOUN
ejpam-628	100	10	and	and	CCONJ
ejpam-628	100	11	each	each	DET
ejpam-628	100	12	ri	ri	PROPN
ejpam-628	100	13	is	be	AUX
ejpam-628	100	14	a	a	DET
ejpam-628	100	15	finite	finite	ADJ
ejpam-628	100	16	local	local	ADJ
ejpam-628	100	17	ring	ring	NOUN
ejpam-628	100	18	such	such	ADJ
ejpam-628	100	19	that	that	PRON
ejpam-628	100	20	|z(ri)|	|z(ri)|	NOUN
ejpam-628	100	21	=	=	SYM
ejpam-628	101	1	p	p	X
ejpam-628	101	2	ti	ti	X
ejpam-628	101	3	i	i	PRON
ejpam-628	101	4	for	for	ADP
ejpam-628	101	5	some	some	DET
ejpam-628	101	6	1≤	1≤	NUM
ejpam-628	101	7	t	t	NOUN
ejpam-628	102	1	i	i	PRON
ejpam-628	102	2	≤	≤	PROPN
ejpam-628	102	3	ki	ki	PROPN
ejpam-628	102	4	.	.	PUNCT
ejpam-628	103	1	consequently	consequently	ADV
ejpam-628	103	2	,	,	PUNCT
ejpam-628	103	3	for	for	ADP
ejpam-628	103	4	each	each	DET
ejpam-628	103	5	i	i	NOUN
ejpam-628	103	6	=	=	NOUN
ejpam-628	103	7	1	1	NUM
ejpam-628	103	8	,	,	PUNCT
ejpam-628	103	9	.	.	PUNCT
ejpam-628	103	10	.	.	PUNCT
ejpam-628	103	11	.	.	PUNCT
ejpam-628	104	1	,	,	PUNCT
ejpam-628	104	2	s	s	X
ejpam-628	104	3	,	,	PUNCT
ejpam-628	104	4	|z(ri)|	|z(ri)|	NUM
ejpam-628	104	5	is	be	AUX
ejpam-628	104	6	a	a	DET
ejpam-628	104	7	divisor	divisor	NOUN
ejpam-628	104	8	of	of	ADP
ejpam-628	104	9	|z(r)|	|z(r)|	PROPN
ejpam-628	104	10	.	.	PUNCT
ejpam-628	105	1	proof	proof	NOUN
ejpam-628	105	2	.	.	PUNCT
ejpam-628	106	1	the	the	DET
ejpam-628	106	2	proof	proof	NOUN
ejpam-628	106	3	is	be	AUX
ejpam-628	106	4	clear	clear	ADJ
ejpam-628	106	5	by	by	ADP
ejpam-628	106	6	lemma	lemma	PROPN
ejpam-628	106	7	1	1	NUM
ejpam-628	106	8	and	and	CCONJ
ejpam-628	106	9	lemma	lemma	PROPN
ejpam-628	106	10	2	2	NUM
ejpam-628	106	11	.	.	PUNCT
ejpam-628	106	12	theorem	theorem	NOUN
ejpam-628	106	13	2	2	NUM
ejpam-628	106	14	.	.	PUNCT
ejpam-628	107	1	let	let	VERB
ejpam-628	107	2	r	r	PRON
ejpam-628	107	3	be	be	AUX
ejpam-628	107	4	a	a	DET
ejpam-628	107	5	commutative	commutative	ADJ
ejpam-628	107	6	ring	ring	NOUN
ejpam-628	107	7	such	such	ADJ
ejpam-628	107	8	that	that	SCONJ
ejpam-628	107	9	|z(r)|=	|z(r)|=	VERB
ejpam-628	107	10	pk	pk	NOUN
ejpam-628	107	11	for	for	ADP
ejpam-628	107	12	some	some	DET
ejpam-628	107	13	prime	prime	ADJ
ejpam-628	107	14	number	number	NOUN
ejpam-628	107	15	p	p	NOUN
ejpam-628	107	16	and	and	CCONJ
ejpam-628	107	17	a	a	DET
ejpam-628	107	18	positive	positive	ADJ
ejpam-628	107	19	number	number	NOUN
ejpam-628	107	20	k.	k.	PROPN
ejpam-628	108	1	then	then	ADV
ejpam-628	108	2	either	either	CCONJ
ejpam-628	108	3	(	(	PUNCT
ejpam-628	108	4	i	i	NOUN
ejpam-628	108	5	)	)	PUNCT
ejpam-628	108	6	r	r	NOUN
ejpam-628	108	7	is	be	AUX
ejpam-628	108	8	local	local	ADJ
ejpam-628	108	9	,	,	PUNCT
ejpam-628	108	10	(	(	PUNCT
ejpam-628	108	11	ii	ii	NOUN
ejpam-628	108	12	)	)	PUNCT
ejpam-628	108	13	r	r	NOUN
ejpam-628	108	14	is	be	AUX
ejpam-628	108	15	reduced	reduce	VERB
ejpam-628	108	16	,	,	PUNCT
ejpam-628	108	17	or	or	CCONJ
ejpam-628	108	18	(	(	PUNCT
ejpam-628	108	19	iii	iii	X
ejpam-628	108	20	)	)	PUNCT
ejpam-628	109	1	k	k	PROPN
ejpam-628	109	2	≥	≥	NUM
ejpam-628	109	3	3	3	NUM
ejpam-628	109	4	and	and	CCONJ
ejpam-628	109	5	r∼=	r∼=	NUM
ejpam-628	109	6	r1×	r1×	NOUN
ejpam-628	109	7	.	.	PUNCT
ejpam-628	109	8	.	.	PUNCT
ejpam-628	110	1	.×rs×	.×rs×	PUNCT
ejpam-628	111	1	fq1	fq1	ADV
ejpam-628	111	2	×	×	NOUN
ejpam-628	111	3	.	.	PUNCT
ejpam-628	111	4	.	.	PUNCT
ejpam-628	112	1	.×	.×	PROPN
ejpam-628	112	2	fqt	fqt	PROPN
ejpam-628	112	3	where	where	SCONJ
ejpam-628	112	4	s	s	PRON
ejpam-628	112	5	and	and	CCONJ
ejpam-628	112	6	t	t	PROPN
ejpam-628	112	7	are	be	AUX
ejpam-628	112	8	positive	positive	ADJ
ejpam-628	112	9	integers	integer	NOUN
ejpam-628	112	10	,	,	PUNCT
ejpam-628	112	11	each	each	DET
ejpam-628	112	12	fqi	fqi	VERB
ejpam-628	112	13	is	be	AUX
ejpam-628	112	14	a	a	DET
ejpam-628	112	15	field	field	NOUN
ejpam-628	112	16	,	,	PUNCT
ejpam-628	112	17	and	and	CCONJ
ejpam-628	112	18	where	where	SCONJ
ejpam-628	112	19	each	each	DET
ejpam-628	112	20	ri	ri	PROPN
ejpam-628	112	21	is	be	AUX
ejpam-628	112	22	a	a	DET
ejpam-628	112	23	commutative	commutative	ADJ
ejpam-628	112	24	finite	finite	ADJ
ejpam-628	112	25	local	local	ADJ
ejpam-628	112	26	ring	ring	NOUN
ejpam-628	112	27	with	with	ADP
ejpam-628	112	28	|z(ri)|	|z(ri)|	NOUN
ejpam-628	112	29	=	=	SYM
ejpam-628	112	30	pti	pti	PROPN
ejpam-628	112	31	,	,	PUNCT
ejpam-628	112	32	|ri|	|ri|	NOUN
ejpam-628	112	33	=	=	SYM
ejpam-628	112	34	pki	pki	NOUN
ejpam-628	112	35	for	for	ADP
ejpam-628	112	36	some	some	DET
ejpam-628	112	37	positive	positive	ADJ
ejpam-628	112	38	integers	integer	NOUN
ejpam-628	112	39	ki	ki	PROPN
ejpam-628	112	40	and	and	CCONJ
ejpam-628	112	41	t	t	PROPN
ejpam-628	112	42	i	i	PRON
ejpam-628	112	43	with	with	ADP
ejpam-628	112	44	1≤	1≤	NUM
ejpam-628	113	1	∑s	∑s	PROPN
ejpam-628	113	2	i=1	i=1	PROPN
ejpam-628	113	3	t	t	PROPN
ejpam-628	113	4	i	i	NOUN
ejpam-628	113	5	≤	≤	NUM
ejpam-628	114	1	∑s	∑s	PROPN
ejpam-628	115	1	i=1	i=1	PROPN
ejpam-628	116	1	ki	ki	PROPN
ejpam-628	117	1	−	−	PROPN
ejpam-628	117	2	s	s	PART
ejpam-628	117	3	≤	≤	PROPN
ejpam-628	117	4	k−	k−	NOUN
ejpam-628	117	5	s−	s−	PROPN
ejpam-628	117	6	1	1	NUM
ejpam-628	117	7	such	such	ADJ
ejpam-628	117	8	that	that	SCONJ
ejpam-628	117	9	pk−σs	pk−σs	PROPN
ejpam-628	117	10	i=1	i=1	PROPN
ejpam-628	117	11	ti	ti	PROPN
ejpam-628	117	12	=	=	PROPN
ejpam-628	117	13	q1	q1	PROPN
ejpam-628	117	14	.	.	PUNCT
ejpam-628	117	15	.	.	PUNCT
ejpam-628	117	16	.	.	PUNCT
ejpam-628	118	1	qt	qt	ADP
ejpam-628	118	2	p	p	NOUN
ejpam-628	118	3	σs	σs	ADP
ejpam-628	118	4	i=1(ki−ti	i=1(ki−ti	NOUN
ejpam-628	118	5	)	)	PUNCT
ejpam-628	118	6	−	−	PROPN
ejpam-628	119	1	(	(	PUNCT
ejpam-628	119	2	q1−	q1−	NOUN
ejpam-628	119	3	1	1	NUM
ejpam-628	119	4	)	)	PUNCT
ejpam-628	119	5	.	.	PUNCT
ejpam-628	119	6	.	.	PUNCT
ejpam-628	119	7	.	.	PUNCT
ejpam-628	120	1	(	(	PUNCT
ejpam-628	120	2	qt	qt	INTJ
ejpam-628	120	3	−	−	PROPN
ejpam-628	121	1	1)πs	1)πs	NUM
ejpam-628	122	1	i=1(p	i=1(p	NOUN
ejpam-628	122	2	ki−ti	ki−ti	X
ejpam-628	122	3	−	−	NOUN
ejpam-628	122	4	1	1	NUM
ejpam-628	122	5	)	)	PUNCT
ejpam-628	122	6	.	.	PUNCT
ejpam-628	123	1	(	(	PUNCT
ejpam-628	123	2	2	2	X
ejpam-628	123	3	)	)	PUNCT
ejpam-628	123	4	consequently	consequently	ADV
ejpam-628	123	5	,	,	PUNCT
ejpam-628	123	6	in	in	ADP
ejpam-628	123	7	the	the	DET
ejpam-628	123	8	latter	latter	ADJ
ejpam-628	123	9	case	case	NOUN
ejpam-628	123	10	,	,	PUNCT
ejpam-628	123	11	t	t	PROPN
ejpam-628	123	12	i	i	NOUN
ejpam-628	123	13	≤	≤	NUM
ejpam-628	124	1	k	k	PRON
ejpam-628	125	1	−	−	PROPN
ejpam-628	125	2	2	2	NUM
ejpam-628	125	3	for	for	ADP
ejpam-628	125	4	each	each	DET
ejpam-628	125	5	i	i	NOUN
ejpam-628	125	6	=	=	NOUN
ejpam-628	125	7	1	1	NUM
ejpam-628	125	8	,	,	PUNCT
ejpam-628	125	9	.	.	PUNCT
ejpam-628	125	10	.	.	PUNCT
ejpam-628	126	1	.	.	PUNCT
ejpam-628	127	1	,	,	PUNCT
ejpam-628	127	2	s	s	VERB
ejpam-628	127	3	and	and	CCONJ
ejpam-628	127	4	q	q	PROPN
ejpam-628	127	5	j	j	PROPN
ejpam-628	127	6	≡	≡	PROPN
ejpam-628	127	7	1	1	NUM
ejpam-628	127	8	(	(	PUNCT
ejpam-628	127	9	p	p	NOUN
ejpam-628	127	10	)	)	PUNCT
ejpam-628	127	11	for	for	ADP
ejpam-628	127	12	some	some	DET
ejpam-628	127	13	j.	j.	PROPN
ejpam-628	127	14	moreover	moreover	ADV
ejpam-628	127	15	,	,	PUNCT
ejpam-628	127	16	if	if	SCONJ
ejpam-628	127	17	t	t	PROPN
ejpam-628	127	18	i	i	NOUN
ejpam-628	127	19	=	=	SYM
ejpam-628	127	20	k−	k−	PROPN
ejpam-628	127	21	2	2	NUM
ejpam-628	127	22	for	for	ADP
ejpam-628	127	23	some	some	DET
ejpam-628	127	24	i	i	PROPN
ejpam-628	127	25	,	,	PUNCT
ejpam-628	127	26	then	then	ADV
ejpam-628	127	27	s	s	VERB
ejpam-628	127	28	=	=	X
ejpam-628	127	29	t	t	X
ejpam-628	127	30	=	=	SYM
ejpam-628	127	31	1	1	NUM
ejpam-628	127	32	,	,	PUNCT
ejpam-628	127	33	i.e.	i.e.	X
ejpam-628	127	34	,	,	PUNCT
ejpam-628	127	35	r∼=	r∼=	NUM
ejpam-628	127	36	r1×	r1×	VERB
ejpam-628	127	37	fq	fq	PROPN
ejpam-628	127	38	where	where	SCONJ
ejpam-628	127	39	|z(r1)|=	|z(r1)|=	AUX
ejpam-628	127	40	pk−2	pk−2	ADJ
ejpam-628	127	41	and	and	CCONJ
ejpam-628	127	42	so	so	ADV
ejpam-628	127	43	p2	p2	PROPN
ejpam-628	127	44	=	=	SYM
ejpam-628	127	45	p+	p+	PROPN
ejpam-628	127	46	q−	q−	PROPN
ejpam-628	127	47	1	1	NUM
ejpam-628	127	48	.	.	PUNCT
ejpam-628	128	1	m.	m.	NOUN
ejpam-628	128	2	behboodi	behboodi	PROPN
ejpam-628	128	3	,	,	PUNCT
ejpam-628	128	4	r.	r.	PROPN
ejpam-628	128	5	beyranvand	beyranvand	PROPN
ejpam-628	128	6	/	/	SYM
ejpam-628	128	7	eur	eur	PROPN
ejpam-628	128	8	.	.	PUNCT
ejpam-628	129	1	j.	j.	PROPN
ejpam-628	129	2	pure	pure	PROPN
ejpam-628	129	3	appl	appl	PROPN
ejpam-628	129	4	.	.	PROPN
ejpam-628	129	5	math	math	PROPN
ejpam-628	129	6	,	,	PUNCT
ejpam-628	129	7	3	3	NUM
ejpam-628	129	8	(	(	PUNCT
ejpam-628	129	9	2010	2010	NUM
ejpam-628	129	10	)	)	PUNCT
ejpam-628	129	11	,	,	PUNCT
ejpam-628	129	12	303	303	NUM
ejpam-628	129	13	-	-	SYM
ejpam-628	129	14	316	316	NUM
ejpam-628	129	15	306	306	NUM
ejpam-628	129	16	proof	proof	NOUN
ejpam-628	129	17	.	.	PUNCT
ejpam-628	130	1	suppose	suppose	VERB
ejpam-628	130	2	r	r	NOUN
ejpam-628	130	3	is	be	AUX
ejpam-628	130	4	not	not	PART
ejpam-628	130	5	local	local	ADJ
ejpam-628	130	6	.	.	PUNCT
ejpam-628	131	1	then	then	ADV
ejpam-628	131	2	r	r	NOUN
ejpam-628	131	3	∼=	∼=	PROPN
ejpam-628	131	4	r1	r1	NOUN
ejpam-628	131	5	×	×	NOUN
ejpam-628	131	6	.	.	PUNCT
ejpam-628	131	7	.	.	PUNCT
ejpam-628	131	8	.	.	PUNCT
ejpam-628	132	1	×	×	PROPN
ejpam-628	132	2	rn	rn	PROPN
ejpam-628	132	3	,	,	PUNCT
ejpam-628	132	4	where	where	SCONJ
ejpam-628	132	5	n	n	PRON
ejpam-628	132	6	≥	≥	X
ejpam-628	132	7	2	2	NUM
ejpam-628	132	8	and	and	CCONJ
ejpam-628	132	9	each	each	DET
ejpam-628	132	10	ri	ri	PROPN
ejpam-628	132	11	is	be	AUX
ejpam-628	132	12	a	a	DET
ejpam-628	132	13	local	local	ADJ
ejpam-628	132	14	ring	ring	NOUN
ejpam-628	132	15	.	.	PUNCT
ejpam-628	133	1	if	if	SCONJ
ejpam-628	133	2	for	for	ADP
ejpam-628	133	3	each	each	DET
ejpam-628	133	4	i	i	PRON
ejpam-628	133	5	(	(	PUNCT
ejpam-628	133	6	1	1	NUM
ejpam-628	133	7	≤	≤	NUM
ejpam-628	133	8	i	i	PRON
ejpam-628	133	9	≤	≤	NOUN
ejpam-628	133	10	n	n	CCONJ
ejpam-628	133	11	)	)	PUNCT
ejpam-628	133	12	ri	ri	PROPN
ejpam-628	133	13	is	be	AUX
ejpam-628	133	14	not	not	PART
ejpam-628	133	15	field	field	NOUN
ejpam-628	133	16	,	,	PUNCT
ejpam-628	133	17	then	then	ADV
ejpam-628	133	18	|z(ri)|	|z(ri)|	X
ejpam-628	133	19	=	=	SYM
ejpam-628	133	20	pti	pti	PROPN
ejpam-628	133	21	and	and	CCONJ
ejpam-628	133	22	|ri|	|ri|	NOUN
ejpam-628	133	23	=	=	SYM
ejpam-628	133	24	pki	pki	NOUN
ejpam-628	133	25	for	for	ADP
ejpam-628	133	26	some	some	DET
ejpam-628	133	27	1≤	1≤	NUM
ejpam-628	133	28	t	t	NOUN
ejpam-628	134	1	i	i	PRON
ejpam-628	134	2	<	<	X
ejpam-628	134	3	ki	ki	PROPN
ejpam-628	134	4	≤	≤	PROPN
ejpam-628	134	5	k.	k.	INTJ
ejpam-628	134	6	by	by	ADP
ejpam-628	134	7	the	the	DET
ejpam-628	134	8	relation	relation	NOUN
ejpam-628	134	9	(	(	PUNCT
ejpam-628	134	10	1	1	NUM
ejpam-628	134	11	)	)	PUNCT
ejpam-628	134	12	of	of	ADP
ejpam-628	134	13	theorem	theorem	NOUN
ejpam-628	134	14	1	1	NUM
ejpam-628	134	15	,	,	PUNCT
ejpam-628	134	16	we	we	PRON
ejpam-628	134	17	have	have	VERB
ejpam-628	134	18	pk	pk	NOUN
ejpam-628	134	19	=	=	NOUN
ejpam-628	134	20	p	p	PROPN
ejpam-628	134	21	∑s	∑s	PROPN
ejpam-628	134	22	i=1	i=1	PROPN
ejpam-628	135	1	ti[p	ti[p	PROPN
ejpam-628	135	2	∑s	∑s	PROPN
ejpam-628	135	3	i=1(ki−ti	i=1(ki−ti	PROPN
ejpam-628	135	4	)	)	PUNCT
ejpam-628	136	1	−	−	PROPN
ejpam-628	136	2	s	s	PART
ejpam-628	136	3	∏	∏	PROPN
ejpam-628	136	4	i=1	i=1	PROPN
ejpam-628	136	5	(	(	PUNCT
ejpam-628	136	6	pki−ti	pki−ti	NOUN
ejpam-628	136	7	−	−	NOUN
ejpam-628	136	8	1	1	NUM
ejpam-628	136	9	)	)	PUNCT
ejpam-628	136	10	]	]	PUNCT
ejpam-628	136	11	and	and	CCONJ
ejpam-628	136	12	hence	hence	ADV
ejpam-628	136	13	pk−	pk−	PROPN
ejpam-628	136	14	∑s	∑s	PROPN
ejpam-628	137	1	i=1	i=1	X
ejpam-628	137	2	ti	ti	X
ejpam-628	138	1	=	=	SYM
ejpam-628	138	2	p	p	PROPN
ejpam-628	138	3	∑s	∑s	PROPN
ejpam-628	138	4	i=1(ki−ti)−	i=1(ki−ti)−	PROPN
ejpam-628	138	5	s	s	PART
ejpam-628	138	6	∏	∏	PROPN
ejpam-628	138	7	i=1	i=1	PROPN
ejpam-628	138	8	(	(	PUNCT
ejpam-628	138	9	pki−ti	pki−ti	NOUN
ejpam-628	138	10	−	−	PROPN
ejpam-628	138	11	1	1	NUM
ejpam-628	138	12	)	)	PUNCT
ejpam-628	138	13	.	.	PUNCT
ejpam-628	139	1	this	this	PRON
ejpam-628	139	2	implies	imply	VERB
ejpam-628	139	3	that	that	SCONJ
ejpam-628	139	4	0≡	0≡	NOUN
ejpam-628	139	5	1(p	1(p	NUM
ejpam-628	139	6	)	)	PUNCT
ejpam-628	139	7	or	or	CCONJ
ejpam-628	139	8	0≡	0≡	NUM
ejpam-628	139	9	−1(p	−1(p	NOUN
ejpam-628	139	10	)	)	PUNCT
ejpam-628	139	11	,	,	PUNCT
ejpam-628	139	12	a	a	DET
ejpam-628	139	13	contradiction	contradiction	NOUN
ejpam-628	139	14	.	.	PUNCT
ejpam-628	140	1	thus	thus	ADV
ejpam-628	140	2	r	r	NOUN
ejpam-628	140	3	j	j	PROPN
ejpam-628	140	4	is	be	AUX
ejpam-628	140	5	field	field	NOUN
ejpam-628	140	6	for	for	ADP
ejpam-628	140	7	some	some	DET
ejpam-628	140	8	1≤	1≤	NUM
ejpam-628	140	9	j	j	PROPN
ejpam-628	140	10	≤	≤	PROPN
ejpam-628	140	11	n.	n.	NOUN
ejpam-628	140	12	if	if	SCONJ
ejpam-628	140	13	each	each	DET
ejpam-628	140	14	ri	ri	PROPN
ejpam-628	140	15	is	be	AUX
ejpam-628	140	16	field	field	NOUN
ejpam-628	140	17	,	,	PUNCT
ejpam-628	140	18	then	then	ADV
ejpam-628	140	19	r	r	NOUN
ejpam-628	140	20	is	be	AUX
ejpam-628	140	21	a	a	DET
ejpam-628	140	22	reduced	reduce	VERB
ejpam-628	140	23	ring	ring	NOUN
ejpam-628	140	24	.	.	PUNCT
ejpam-628	141	1	suppose	suppose	VERB
ejpam-628	141	2	r	r	NOUN
ejpam-628	141	3	is	be	AUX
ejpam-628	141	4	non	non	ADJ
ejpam-628	141	5	-	-	ADJ
ejpam-628	141	6	reduced	reduced	ADJ
ejpam-628	141	7	.	.	PUNCT
ejpam-628	142	1	without	without	ADP
ejpam-628	142	2	loss	loss	NOUN
ejpam-628	142	3	of	of	ADP
ejpam-628	142	4	generality	generality	NOUN
ejpam-628	142	5	we	we	PRON
ejpam-628	142	6	can	can	AUX
ejpam-628	142	7	assume	assume	VERB
ejpam-628	142	8	that	that	SCONJ
ejpam-628	142	9	r∼=	r∼=	ADV
ejpam-628	142	10	r1	r1	PROPN
ejpam-628	142	11	×	×	NOUN
ejpam-628	142	12	.	.	PUNCT
ejpam-628	142	13	.	.	PUNCT
ejpam-628	143	1	.×	.×	NOUN
ejpam-628	143	2	rs	r	VERB
ejpam-628	143	3	×	×	NOUN
ejpam-628	144	1	fq1	fq1	INTJ
ejpam-628	144	2	×	×	NOUN
ejpam-628	144	3	.	.	PUNCT
ejpam-628	144	4	.	.	PUNCT
ejpam-628	145	1	.×	.×	PROPN
ejpam-628	145	2	fqt	fqt	VERB
ejpam-628	145	3	where	where	SCONJ
ejpam-628	145	4	s	s	X
ejpam-628	145	5	,	,	PUNCT
ejpam-628	145	6	t	t	PROPN
ejpam-628	145	7	≥	≥	NUM
ejpam-628	145	8	1	1	NUM
ejpam-628	145	9	and	and	CCONJ
ejpam-628	145	10	each	each	DET
ejpam-628	145	11	ri	ri	PROPN
ejpam-628	145	12	is	be	AUX
ejpam-628	145	13	a	a	DET
ejpam-628	145	14	commutative	commutative	ADJ
ejpam-628	145	15	finite	finite	ADJ
ejpam-628	145	16	local	local	ADJ
ejpam-628	145	17	ring	ring	NOUN
ejpam-628	145	18	with	with	ADP
ejpam-628	145	19	|z(ri)|	|z(ri)|	NOUN
ejpam-628	145	20	=	=	SYM
ejpam-628	145	21	pti	pti	PROPN
ejpam-628	145	22	and	and	CCONJ
ejpam-628	145	23	|ri|	|ri|	NOUN
ejpam-628	146	1	=	=	SYM
ejpam-628	146	2	pki	pki	NOUN
ejpam-628	146	3	for	for	ADP
ejpam-628	146	4	some	some	DET
ejpam-628	146	5	1≤	1≤	NUM
ejpam-628	146	6	t	t	NOUN
ejpam-628	147	1	i	i	PRON
ejpam-628	147	2	<	<	X
ejpam-628	147	3	ki	ki	PROPN
ejpam-628	147	4	≤	≤	PROPN
ejpam-628	147	5	k.	k.	PROPN
ejpam-628	148	1	since	since	SCONJ
ejpam-628	148	2	t	t	PROPN
ejpam-628	148	3	≥	≥	NUM
ejpam-628	148	4	1	1	NUM
ejpam-628	148	5	,	,	PUNCT
ejpam-628	148	6	it	it	PRON
ejpam-628	148	7	is	be	AUX
ejpam-628	148	8	easy	easy	ADJ
ejpam-628	148	9	to	to	PART
ejpam-628	148	10	check	check	VERB
ejpam-628	148	11	that	that	PRON
ejpam-628	148	12	pk	pk	NOUN
ejpam-628	148	13	=	=	SYM
ejpam-628	148	14	|z(r)|	|z(r)|	PROPN
ejpam-628	149	1	>	>	X
ejpam-628	149	2	s	s	PART
ejpam-628	149	3	∏	∏	PROPN
ejpam-628	149	4	i=1	i=1	PROPN
ejpam-628	149	5	|ri|	|ri|	NOUN
ejpam-628	149	6	=	=	PUNCT
ejpam-628	150	1	p	p	PROPN
ejpam-628	150	2	∑s	∑s	PROPN
ejpam-628	150	3	i=1	i=1	PROPN
ejpam-628	150	4	ki	ki	PROPN
ejpam-628	150	5	≥	≥	PROPN
ejpam-628	150	6	p	p	PROPN
ejpam-628	150	7	∑s	∑s	PROPN
ejpam-628	150	8	i=1(ti+1	i=1(ti+1	NOUN
ejpam-628	150	9	)	)	PUNCT
ejpam-628	151	1	=	=	SYM
ejpam-628	152	1	p	p	X
ejpam-628	152	2	(	(	PUNCT
ejpam-628	152	3	∑s	∑s	PROPN
ejpam-628	152	4	i=1	i=1	PROPN
ejpam-628	152	5	ti)+s	ti)+s	NOUN
ejpam-628	152	6	.	.	PUNCT
ejpam-628	153	1	consequently	consequently	ADV
ejpam-628	153	2	we	we	PRON
ejpam-628	153	3	have	have	VERB
ejpam-628	153	4	1≤	1≤	NUM
ejpam-628	154	1	∑s	∑s	PROPN
ejpam-628	154	2	i=1	i=1	PROPN
ejpam-628	154	3	t	t	PROPN
ejpam-628	154	4	i	i	NOUN
ejpam-628	154	5	≤	≤	NUM
ejpam-628	154	6	∑s	∑s	PROPN
ejpam-628	155	1	i=1	i=1	PROPN
ejpam-628	155	2	ki−	ki−	PROPN
ejpam-628	155	3	s	s	PART
ejpam-628	155	4	≤	≤	PROPN
ejpam-628	155	5	k−	k−	NOUN
ejpam-628	155	6	s−1	s−1	PROPN
ejpam-628	155	7	and	and	CCONJ
ejpam-628	155	8	hence	hence	ADV
ejpam-628	155	9	we	we	PRON
ejpam-628	155	10	obtain	obtain	VERB
ejpam-628	155	11	relation	relation	NOUN
ejpam-628	155	12	(	(	PUNCT
ejpam-628	155	13	2	2	NUM
ejpam-628	155	14	)	)	PUNCT
ejpam-628	155	15	.	.	PUNCT
ejpam-628	156	1	now	now	ADV
ejpam-628	156	2	since	since	SCONJ
ejpam-628	156	3	k−	k−	PROPN
ejpam-628	156	4	∑s	∑s	PROPN
ejpam-628	157	1	i=1	i=1	PROPN
ejpam-628	158	1	t	t	PROPN
ejpam-628	158	2	i	i	PRON
ejpam-628	158	3	and	and	CCONJ
ejpam-628	158	4	∑s	∑s	PROPN
ejpam-628	158	5	i=1(ki−	i=1(ki−	X
ejpam-628	158	6	t	t	PROPN
ejpam-628	158	7	i	i	PROPN
ejpam-628	158	8	)	)	PUNCT
ejpam-628	158	9	are	be	AUX
ejpam-628	158	10	positive	positive	ADJ
ejpam-628	158	11	,	,	PUNCT
ejpam-628	158	12	the	the	DET
ejpam-628	158	13	relation	relation	NOUN
ejpam-628	158	14	(	(	PUNCT
ejpam-628	158	15	2	2	X
ejpam-628	158	16	)	)	PUNCT
ejpam-628	158	17	shows	show	VERB
ejpam-628	158	18	that	that	SCONJ
ejpam-628	158	19	qi	qi	PROPN
ejpam-628	158	20	≡	≡	PROPN
ejpam-628	158	21	1(p	1(p	NUM
ejpam-628	158	22	)	)	PUNCT
ejpam-628	158	23	for	for	ADP
ejpam-628	158	24	some	some	DET
ejpam-628	158	25	i.	i.	NOUN
ejpam-628	158	26	also	also	ADV
ejpam-628	158	27	,	,	PUNCT
ejpam-628	158	28	since	since	SCONJ
ejpam-628	158	29	s	s	PRON
ejpam-628	158	30	≥	≥	NOUN
ejpam-628	158	31	1	1	NUM
ejpam-628	158	32	,	,	PUNCT
ejpam-628	158	33	t	t	PROPN
ejpam-628	158	34	i	i	PROPN
ejpam-628	158	35	≤	≤	PROPN
ejpam-628	158	36	k−	k−	PROPN
ejpam-628	158	37	2	2	NUM
ejpam-628	158	38	for	for	ADP
ejpam-628	158	39	each	each	DET
ejpam-628	158	40	i	i	NOUN
ejpam-628	158	41	=	=	NOUN
ejpam-628	158	42	1	1	NUM
ejpam-628	158	43	,	,	PUNCT
ejpam-628	158	44	.	.	PUNCT
ejpam-628	158	45	.	.	PUNCT
ejpam-628	159	1	.	.	PUNCT
ejpam-628	160	1	,	,	PUNCT
ejpam-628	160	2	s.	s.	PROPN
ejpam-628	160	3	if	if	SCONJ
ejpam-628	160	4	t	t	PROPN
ejpam-628	160	5	j	j	PROPN
ejpam-628	160	6	=	=	SYM
ejpam-628	160	7	k−	k−	PROPN
ejpam-628	160	8	2	2	NUM
ejpam-628	160	9	for	for	ADP
ejpam-628	160	10	some	some	DET
ejpam-628	160	11	j	j	PROPN
ejpam-628	160	12	∈	∈	PROPN
ejpam-628	160	13	{	{	PUNCT
ejpam-628	160	14	1	1	NUM
ejpam-628	160	15	,	,	PUNCT
ejpam-628	160	16	.	.	PUNCT
ejpam-628	160	17	.	.	PUNCT
ejpam-628	161	1	.	.	PUNCT
ejpam-628	162	1	,	,	PUNCT
ejpam-628	162	2	s	s	X
ejpam-628	162	3	}	}	PUNCT
ejpam-628	162	4	,	,	PUNCT
ejpam-628	162	5	then	then	ADV
ejpam-628	162	6	s	s	VERB
ejpam-628	162	7	=	=	ADJ
ejpam-628	162	8	1	1	X
ejpam-628	162	9	.	.	PUNCT
ejpam-628	163	1	thus	thus	ADV
ejpam-628	163	2	r	r	NOUN
ejpam-628	163	3	∼=	∼=	NOUN
ejpam-628	163	4	r1	r1	NOUN
ejpam-628	163	5	×	×	NOUN
ejpam-628	163	6	fq1	fq1	CCONJ
ejpam-628	163	7	×	×	NOUN
ejpam-628	163	8	.	.	PUNCT
ejpam-628	163	9	.	.	PUNCT
ejpam-628	164	1	.×	.×	PROPN
ejpam-628	164	2	fqt	fqt	PROPN
ejpam-628	164	3	,	,	PUNCT
ejpam-628	164	4	where	where	SCONJ
ejpam-628	164	5	r1	r1	PROPN
ejpam-628	164	6	is	be	AUX
ejpam-628	164	7	a	a	DET
ejpam-628	164	8	local	local	ADJ
ejpam-628	164	9	ring	ring	NOUN
ejpam-628	164	10	with	with	ADP
ejpam-628	164	11	|z(r1)|	|z(r1)|	PROPN
ejpam-628	164	12	=	=	SYM
ejpam-628	164	13	pk−2	pk−2	PROPN
ejpam-628	164	14	.	.	PUNCT
ejpam-628	165	1	since	since	SCONJ
ejpam-628	165	2	|z(r)|=	|z(r)|=	VERB
ejpam-628	165	3	pk	pk	NOUN
ejpam-628	165	4	and	and	CCONJ
ejpam-628	165	5	t	t	PROPN
ejpam-628	165	6	≥	≥	NUM
ejpam-628	165	7	1	1	NUM
ejpam-628	165	8	,	,	PUNCT
ejpam-628	165	9	by	by	ADP
ejpam-628	165	10	theorem	theorem	NOUN
ejpam-628	165	11	1	1	NUM
ejpam-628	165	12	,	,	PUNCT
ejpam-628	165	13	|r1|	|r1|	PROPN
ejpam-628	165	14	=	=	PROPN
ejpam-628	165	15	pk−1	pk−1	PROPN
ejpam-628	165	16	.	.	PUNCT
ejpam-628	165	17	also	also	ADV
ejpam-628	165	18	by	by	ADP
ejpam-628	165	19	the	the	DET
ejpam-628	165	20	relation	relation	NOUN
ejpam-628	165	21	(	(	PUNCT
ejpam-628	165	22	2	2	X
ejpam-628	165	23	)	)	PUNCT
ejpam-628	165	24	we	we	PRON
ejpam-628	165	25	have	have	VERB
ejpam-628	165	26	p2	p2	NOUN
ejpam-628	165	27	=	=	SYM
ejpam-628	165	28	pq1q2	pq1q2	PROPN
ejpam-628	165	29	.	.	PUNCT
ejpam-628	165	30	.	.	PUNCT
ejpam-628	165	31	.	.	PUNCT
ejpam-628	166	1	qt	qt	INTJ
ejpam-628	166	2	−	−	PROPN
ejpam-628	166	3	(	(	PUNCT
ejpam-628	166	4	p−	p−	NOUN
ejpam-628	166	5	1)(q1−	1)(q1−	NUM
ejpam-628	166	6	1)(q2−	1)(q2−	NUM
ejpam-628	166	7	1	1	NUM
ejpam-628	166	8	)	)	PUNCT
ejpam-628	166	9	.	.	PUNCT
ejpam-628	166	10	.	.	PUNCT
ejpam-628	166	11	.	.	PUNCT
ejpam-628	167	1	(	(	PUNCT
ejpam-628	167	2	qt	qt	INTJ
ejpam-628	167	3	−	−	NOUN
ejpam-628	167	4	1	1	NUM
ejpam-628	167	5	)	)	PUNCT
ejpam-628	167	6	.	.	PUNCT
ejpam-628	168	1	since	since	SCONJ
ejpam-628	168	2	qi	qi	PROPN
ejpam-628	168	3	≡	≡	PROPN
ejpam-628	168	4	1	1	NUM
ejpam-628	168	5	(	(	PUNCT
ejpam-628	168	6	p	p	NOUN
ejpam-628	168	7	)	)	PUNCT
ejpam-628	168	8	for	for	ADP
ejpam-628	168	9	some	some	DET
ejpam-628	168	10	i	i	PRON
ejpam-628	168	11	,	,	PUNCT
ejpam-628	168	12	we	we	PRON
ejpam-628	168	13	can	can	AUX
ejpam-628	168	14	assume	assume	VERB
ejpam-628	168	15	that	that	SCONJ
ejpam-628	168	16	q1	q1	PROPN
ejpam-628	168	17	≡	≡	PROPN
ejpam-628	168	18	1	1	NUM
ejpam-628	168	19	(	(	PUNCT
ejpam-628	168	20	p	p	NOUN
ejpam-628	168	21	)	)	PUNCT
ejpam-628	168	22	and	and	CCONJ
ejpam-628	168	23	so	so	ADV
ejpam-628	168	24	q1	q1	PROPN
ejpam-628	168	25	>	>	PUNCT
ejpam-628	169	1	p.	p.	NOUN
ejpam-628	170	1	now	now	ADV
ejpam-628	170	2	if	if	SCONJ
ejpam-628	170	3	t	t	PROPN
ejpam-628	170	4	≥	≥	NOUN
ejpam-628	170	5	2	2	NUM
ejpam-628	170	6	,	,	PUNCT
ejpam-628	170	7	then	then	ADV
ejpam-628	170	8	|z(r)|	|z(r)|	PROPN
ejpam-628	170	9	≥	≥	PRON
ejpam-628	170	10	|r1|q1	|r1|q1	NOUN
ejpam-628	170	11	>	>	X
ejpam-628	170	12	pk−1p	pk−1p	PROPN
ejpam-628	170	13	=	=	SYM
ejpam-628	170	14	pk	pk	PROPN
ejpam-628	170	15	,	,	PUNCT
ejpam-628	170	16	a	a	DET
ejpam-628	170	17	contradiction	contradiction	NOUN
ejpam-628	170	18	.	.	PUNCT
ejpam-628	171	1	thus	thus	ADV
ejpam-628	171	2	t	t	X
ejpam-628	171	3	=	=	SYM
ejpam-628	171	4	1	1	NUM
ejpam-628	171	5	and	and	CCONJ
ejpam-628	171	6	p2	p2	PROPN
ejpam-628	171	7	=	=	NOUN
ejpam-628	171	8	pq1−	pq1−	NOUN
ejpam-628	171	9	(	(	PUNCT
ejpam-628	171	10	p−1)(q1−1	p−1)(q1−1	NOUN
ejpam-628	171	11	)	)	PUNCT
ejpam-628	171	12	,	,	PUNCT
ejpam-628	171	13	i.e.	i.e.	X
ejpam-628	171	14	,	,	PUNCT
ejpam-628	171	15	p2	p2	PROPN
ejpam-628	171	16	=	=	PROPN
ejpam-628	171	17	p+	p+	PROPN
ejpam-628	171	18	q1	q1	PROPN
ejpam-628	171	19	−	−	PROPN
ejpam-628	171	20	1	1	X
ejpam-628	171	21	.	.	PUNCT
ejpam-628	172	1	obviously	obviously	ADV
ejpam-628	172	2	for	for	ADP
ejpam-628	172	3	every	every	DET
ejpam-628	172	4	finite	finite	ADJ
ejpam-628	172	5	local	local	ADJ
ejpam-628	172	6	ring	ring	NOUN
ejpam-628	172	7	r	r	NOUN
ejpam-628	172	8	we	we	PRON
ejpam-628	172	9	have	have	VERB
ejpam-628	172	10	|r|	|r|	NOUN
ejpam-628	172	11	=	=	NOUN
ejpam-628	172	12	pn	pn	NOUN
ejpam-628	172	13	for	for	ADP
ejpam-628	172	14	some	some	DET
ejpam-628	172	15	prime	prime	ADJ
ejpam-628	172	16	number	number	NOUN
ejpam-628	172	17	p	p	NOUN
ejpam-628	172	18	and	and	CCONJ
ejpam-628	172	19	n	n	PRON
ejpam-628	172	20	≥	≥	NOUN
ejpam-628	172	21	0	0	NUM
ejpam-628	172	22	.	.	PUNCT
ejpam-628	173	1	in	in	ADP
ejpam-628	173	2	general	general	ADJ
ejpam-628	173	3	,	,	PUNCT
ejpam-628	173	4	the	the	DET
ejpam-628	173	5	converse	converse	NOUN
ejpam-628	173	6	is	be	AUX
ejpam-628	173	7	not	not	PART
ejpam-628	173	8	true	true	ADJ
ejpam-628	173	9	(	(	PUNCT
ejpam-628	173	10	the	the	DET
ejpam-628	173	11	nonlocal	nonlocal	ADJ
ejpam-628	173	12	ring	ring	NOUN
ejpam-628	173	13	f2	f2	PROPN
ejpam-628	173	14	×	×	NOUN
ejpam-628	173	15	f2	f2	PROPN
ejpam-628	173	16	has	have	VERB
ejpam-628	173	17	4	4	NUM
ejpam-628	173	18	elements	element	NOUN
ejpam-628	173	19	)	)	PUNCT
ejpam-628	173	20	.	.	PUNCT
ejpam-628	174	1	here	here	ADV
ejpam-628	174	2	we	we	PRON
ejpam-628	174	3	show	show	VERB
ejpam-628	174	4	that	that	SCONJ
ejpam-628	174	5	a	a	DET
ejpam-628	174	6	finite	finite	NOUN
ejpam-628	174	7	ring	ring	NOUN
ejpam-628	174	8	r	r	NOUN
ejpam-628	174	9	is	be	AUX
ejpam-628	174	10	local	local	ADJ
ejpam-628	174	11	if	if	SCONJ
ejpam-628	174	12	and	and	CCONJ
ejpam-628	174	13	only	only	ADV
ejpam-628	174	14	if	if	SCONJ
ejpam-628	174	15	|z(r)|	|z(r)|	PROPN
ejpam-628	174	16	=	=	SYM
ejpam-628	174	17	pm	pm	NOUN
ejpam-628	174	18	and	and	CCONJ
ejpam-628	174	19	|r|	|r|	NOUN
ejpam-628	175	1	=	=	NOUN
ejpam-628	175	2	pn	pn	NOUN
ejpam-628	175	3	for	for	ADP
ejpam-628	175	4	some	some	DET
ejpam-628	175	5	prime	prime	ADJ
ejpam-628	175	6	number	number	NOUN
ejpam-628	175	7	p	p	NOUN
ejpam-628	175	8	and	and	CCONJ
ejpam-628	175	9	n	n	CCONJ
ejpam-628	175	10	>	>	X
ejpam-628	175	11	m≥	m≥	PROPN
ejpam-628	175	12	0	0	PROPN
ejpam-628	175	13	.	.	PUNCT
ejpam-628	175	14	theorem	theorem	NOUN
ejpam-628	175	15	3	3	X
ejpam-628	175	16	.	.	PUNCT
ejpam-628	176	1	let	let	VERB
ejpam-628	176	2	r	r	PRON
ejpam-628	176	3	be	be	AUX
ejpam-628	176	4	a	a	DET
ejpam-628	176	5	commutative	commutative	ADJ
ejpam-628	176	6	ring	ring	NOUN
ejpam-628	176	7	.	.	PUNCT
ejpam-628	177	1	then	then	ADV
ejpam-628	177	2	r	r	NOUN
ejpam-628	177	3	is	be	AUX
ejpam-628	177	4	a	a	DET
ejpam-628	177	5	finite	finite	ADJ
ejpam-628	177	6	local	local	ADJ
ejpam-628	177	7	ring	ring	NOUN
ejpam-628	177	8	if	if	SCONJ
ejpam-628	177	9	and	and	CCONJ
ejpam-628	177	10	only	only	ADV
ejpam-628	177	11	if	if	SCONJ
ejpam-628	177	12	|z(r)|=	|z(r)|=	VERB
ejpam-628	177	13	pk	pk	NOUN
ejpam-628	177	14	and	and	CCONJ
ejpam-628	177	15	|r|=	|r|=	PRON
ejpam-628	177	16	pn	pn	NOUN
ejpam-628	177	17	for	for	ADP
ejpam-628	177	18	some	some	DET
ejpam-628	177	19	prime	prime	ADJ
ejpam-628	177	20	number	number	NOUN
ejpam-628	177	21	p	p	NOUN
ejpam-628	177	22	and	and	CCONJ
ejpam-628	177	23	n	n	CCONJ
ejpam-628	177	24	>	>	X
ejpam-628	177	25	k	k	X
ejpam-628	177	26	≥	≥	PROPN
ejpam-628	177	27	0	0	NUM
ejpam-628	177	28	.	.	PUNCT
ejpam-628	178	1	proof	proof	NOUN
ejpam-628	178	2	.	.	PUNCT
ejpam-628	179	1	for	for	ADP
ejpam-628	179	2	one	one	NUM
ejpam-628	179	3	direction	direction	NOUN
ejpam-628	179	4	,	,	PUNCT
ejpam-628	179	5	the	the	DET
ejpam-628	179	6	proof	proof	NOUN
ejpam-628	179	7	is	be	AUX
ejpam-628	179	8	clear	clear	ADJ
ejpam-628	179	9	by	by	ADP
ejpam-628	179	10	lemma	lemma	PROPN
ejpam-628	179	11	1	1	NUM
ejpam-628	179	12	.	.	PUNCT
ejpam-628	180	1	for	for	ADP
ejpam-628	180	2	the	the	DET
ejpam-628	180	3	other	other	ADJ
ejpam-628	180	4	direction	direction	NOUN
ejpam-628	180	5	,	,	PUNCT
ejpam-628	180	6	suppose	suppose	VERB
ejpam-628	180	7	that	that	SCONJ
ejpam-628	180	8	|z(r)|	|z(r)|	PROPN
ejpam-628	180	9	=	=	SYM
ejpam-628	180	10	pk	pk	NOUN
ejpam-628	180	11	and	and	CCONJ
ejpam-628	180	12	|r|	|r|	NOUN
ejpam-628	180	13	=	=	NOUN
ejpam-628	180	14	pn	pn	NOUN
ejpam-628	180	15	for	for	ADP
ejpam-628	180	16	some	some	DET
ejpam-628	180	17	prime	prime	ADJ
ejpam-628	180	18	number	number	NOUN
ejpam-628	180	19	p	p	NOUN
ejpam-628	180	20	and	and	CCONJ
ejpam-628	180	21	n	n	PROPN
ejpam-628	180	22	>	>	X
ejpam-628	180	23	k	k	X
ejpam-628	180	24	≥	≥	PROPN
ejpam-628	180	25	0	0	NUM
ejpam-628	180	26	.	.	PUNCT
ejpam-628	181	1	if	if	SCONJ
ejpam-628	181	2	r	r	NOUN
ejpam-628	181	3	is	be	AUX
ejpam-628	181	4	not	not	PART
ejpam-628	181	5	a	a	DET
ejpam-628	181	6	local	local	ADJ
ejpam-628	181	7	ring	ring	NOUN
ejpam-628	181	8	,	,	PUNCT
ejpam-628	181	9	then	then	ADV
ejpam-628	181	10	by	by	ADP
ejpam-628	181	11	theorem	theorem	NOUN
ejpam-628	181	12	2	2	NUM
ejpam-628	181	13	,	,	PUNCT
ejpam-628	181	14	either	either	CCONJ
ejpam-628	181	15	r∼=	r∼=	ADV
ejpam-628	181	16	fq1	fq1	ADV
ejpam-628	181	17	×	×	NOUN
ejpam-628	181	18	.	.	PUNCT
ejpam-628	181	19	.	.	PUNCT
ejpam-628	182	1	.×	.×	PROPN
ejpam-628	182	2	fqt	fqt	PROPN
ejpam-628	183	1	(	(	PUNCT
ejpam-628	183	2	when	when	SCONJ
ejpam-628	183	3	r	r	NOUN
ejpam-628	183	4	is	be	AUX
ejpam-628	183	5	reduced	reduce	VERB
ejpam-628	183	6	)	)	PUNCT
ejpam-628	183	7	or	or	CCONJ
ejpam-628	183	8	r∼=	r∼=	NUM
ejpam-628	183	9	r1×	r1×	NOUN
ejpam-628	183	10	.	.	PUNCT
ejpam-628	183	11	.	.	PUNCT
ejpam-628	184	1	.×rs×	.×rs×	PUNCT
ejpam-628	185	1	fq1	fq1	ADV
ejpam-628	185	2	×	×	NOUN
ejpam-628	185	3	.	.	PUNCT
ejpam-628	185	4	.	.	PUNCT
ejpam-628	185	5	.	.	PUNCT
ejpam-628	186	1	×	×	NOUN
ejpam-628	186	2	fqt	fqt	NOUN
ejpam-628	186	3	where	where	SCONJ
ejpam-628	186	4	s	s	PRON
ejpam-628	186	5	and	and	CCONJ
ejpam-628	186	6	t	t	PROPN
ejpam-628	186	7	are	be	AUX
ejpam-628	186	8	positive	positive	ADJ
ejpam-628	186	9	integers	integer	NOUN
ejpam-628	186	10	and	and	CCONJ
ejpam-628	186	11	each	each	DET
ejpam-628	186	12	ri	ri	PROPN
ejpam-628	186	13	is	be	AUX
ejpam-628	186	14	a	a	DET
ejpam-628	186	15	commutative	commutative	ADJ
ejpam-628	186	16	finite	finite	ADJ
ejpam-628	186	17	local	local	ADJ
ejpam-628	186	18	m.	m.	NOUN
ejpam-628	186	19	behboodi	behboodi	NOUN
ejpam-628	186	20	,	,	PUNCT
ejpam-628	186	21	r.	r.	PROPN
ejpam-628	186	22	beyranvand	beyranvand	PROPN
ejpam-628	186	23	/	/	SYM
ejpam-628	186	24	eur	eur	PROPN
ejpam-628	186	25	.	.	PUNCT
ejpam-628	187	1	j.	j.	PROPN
ejpam-628	187	2	pure	pure	PROPN
ejpam-628	187	3	appl	appl	PROPN
ejpam-628	187	4	.	.	PROPN
ejpam-628	187	5	math	math	PROPN
ejpam-628	187	6	,	,	PUNCT
ejpam-628	187	7	3	3	NUM
ejpam-628	187	8	(	(	PUNCT
ejpam-628	187	9	2010	2010	NUM
ejpam-628	187	10	)	)	PUNCT
ejpam-628	187	11	,	,	PUNCT
ejpam-628	187	12	303	303	NUM
ejpam-628	187	13	-	-	SYM
ejpam-628	187	14	316	316	NUM
ejpam-628	187	15	307	307	NUM
ejpam-628	187	16	ring	ring	NOUN
ejpam-628	187	17	that	that	PRON
ejpam-628	187	18	is	be	AUX
ejpam-628	187	19	not	not	PART
ejpam-628	187	20	a	a	DET
ejpam-628	187	21	field	field	NOUN
ejpam-628	187	22	,	,	PUNCT
ejpam-628	187	23	and	and	CCONJ
ejpam-628	187	24	where	where	SCONJ
ejpam-628	187	25	each	each	DET
ejpam-628	187	26	fqi	fqi	VERB
ejpam-628	187	27	is	be	AUX
ejpam-628	187	28	a	a	DET
ejpam-628	187	29	field	field	NOUN
ejpam-628	187	30	.	.	PUNCT
ejpam-628	188	1	since	since	SCONJ
ejpam-628	188	2	|r|	|r|	PROPN
ejpam-628	188	3	=	=	SYM
ejpam-628	188	4	pn	pn	PROPN
ejpam-628	188	5	,	,	PUNCT
ejpam-628	188	6	each	each	DET
ejpam-628	188	7	qi	qi	NOUN
ejpam-628	188	8	is	be	AUX
ejpam-628	188	9	a	a	DET
ejpam-628	188	10	divisor	divisor	NOUN
ejpam-628	188	11	of	of	ADP
ejpam-628	188	12	pn	pn	PROPN
ejpam-628	188	13	and	and	CCONJ
ejpam-628	188	14	since	since	SCONJ
ejpam-628	188	15	qi	qi	PROPN
ejpam-628	188	16	is	be	AUX
ejpam-628	188	17	a	a	DET
ejpam-628	188	18	prime	prime	ADJ
ejpam-628	188	19	power	power	NOUN
ejpam-628	188	20	,	,	PUNCT
ejpam-628	188	21	qi	qi	PROPN
ejpam-628	188	22	≡	≡	PROPN
ejpam-628	188	23	0	0	PUNCT
ejpam-628	189	1	(	(	PUNCT
ejpam-628	189	2	p	p	NOUN
ejpam-628	189	3	)	)	PUNCT
ejpam-628	189	4	for	for	ADP
ejpam-628	189	5	each	each	DET
ejpam-628	189	6	i	i	PRON
ejpam-628	189	7	(	(	PUNCT
ejpam-628	189	8	1	1	NUM
ejpam-628	189	9	≤	≤	NUM
ejpam-628	189	10	i	i	NOUN
ejpam-628	189	11	≤	≤	PROPN
ejpam-628	189	12	t	t	PROPN
ejpam-628	189	13	)	)	PUNCT
ejpam-628	189	14	.	.	PUNCT
ejpam-628	190	1	if	if	SCONJ
ejpam-628	190	2	r	r	NOUN
ejpam-628	190	3	is	be	AUX
ejpam-628	190	4	reduced	reduce	VERB
ejpam-628	190	5	,	,	PUNCT
ejpam-628	190	6	then	then	ADV
ejpam-628	190	7	we	we	PRON
ejpam-628	190	8	have	have	VERB
ejpam-628	190	9	pk	pk	NOUN
ejpam-628	190	10	=	=	PROPN
ejpam-628	190	11	q1	q1	PROPN
ejpam-628	190	12	.	.	PUNCT
ejpam-628	190	13	.	.	PUNCT
ejpam-628	190	14	.	.	PUNCT
ejpam-628	191	1	qt	qt	INTJ
ejpam-628	191	2	−	−	PROPN
ejpam-628	191	3	(	(	PUNCT
ejpam-628	191	4	q1−	q1−	NOUN
ejpam-628	191	5	1	1	NUM
ejpam-628	191	6	)	)	PUNCT
ejpam-628	191	7	.	.	PUNCT
ejpam-628	191	8	.	.	PUNCT
ejpam-628	191	9	.	.	PUNCT
ejpam-628	192	1	(	(	PUNCT
ejpam-628	192	2	qt	qt	INTJ
ejpam-628	192	3	−	−	NOUN
ejpam-628	192	4	1	1	NUM
ejpam-628	192	5	)	)	PUNCT
ejpam-628	192	6	and	and	CCONJ
ejpam-628	192	7	if	if	SCONJ
ejpam-628	192	8	r	r	NOUN
ejpam-628	192	9	is	be	AUX
ejpam-628	192	10	not	not	PART
ejpam-628	192	11	reduced	reduce	VERB
ejpam-628	192	12	,	,	PUNCT
ejpam-628	192	13	then	then	ADV
ejpam-628	192	14	we	we	PRON
ejpam-628	192	15	have	have	VERB
ejpam-628	192	16	pk−σs	pk−σ	VERB
ejpam-628	192	17	i=1	i=1	ADP
ejpam-628	192	18	ti	ti	PROPN
ejpam-628	192	19	=	=	PROPN
ejpam-628	192	20	q1	q1	PROPN
ejpam-628	192	21	.	.	PUNCT
ejpam-628	192	22	.	.	PUNCT
ejpam-628	192	23	.	.	PUNCT
ejpam-628	193	1	qt	qt	ADP
ejpam-628	193	2	p	p	NOUN
ejpam-628	193	3	σs	σs	ADP
ejpam-628	193	4	i=1(ki−ti	i=1(ki−ti	NOUN
ejpam-628	193	5	)	)	PUNCT
ejpam-628	193	6	−	−	PROPN
ejpam-628	194	1	(	(	PUNCT
ejpam-628	194	2	q1−	q1−	NOUN
ejpam-628	194	3	1	1	NUM
ejpam-628	194	4	)	)	PUNCT
ejpam-628	194	5	.	.	PUNCT
ejpam-628	194	6	.	.	PUNCT
ejpam-628	194	7	.	.	PUNCT
ejpam-628	195	1	(	(	PUNCT
ejpam-628	195	2	qt	qt	INTJ
ejpam-628	195	3	−	−	PROPN
ejpam-628	196	1	1)πs	1)πs	NUM
ejpam-628	197	1	i=1(p	i=1(p	NOUN
ejpam-628	197	2	ki−ti	ki−ti	X
ejpam-628	197	3	−	−	NOUN
ejpam-628	197	4	1	1	NUM
ejpam-628	197	5	)	)	PUNCT
ejpam-628	197	6	.	.	PUNCT
ejpam-628	198	1	thus	thus	ADV
ejpam-628	198	2	in	in	ADP
ejpam-628	198	3	any	any	DET
ejpam-628	198	4	case	case	NOUN
ejpam-628	198	5	0≡	0≡	X
ejpam-628	198	6	1(p	1(p	NUM
ejpam-628	198	7	)	)	PUNCT
ejpam-628	198	8	or	or	CCONJ
ejpam-628	198	9	0≡	0≡	NUM
ejpam-628	198	10	−1(p	−1(p	NOUN
ejpam-628	198	11	)	)	PUNCT
ejpam-628	198	12	,	,	PUNCT
ejpam-628	198	13	which	which	PRON
ejpam-628	198	14	is	be	AUX
ejpam-628	198	15	impossible	impossible	ADJ
ejpam-628	198	16	.	.	PUNCT
ejpam-628	199	1	thus	thus	ADV
ejpam-628	199	2	r	r	NOUN
ejpam-628	199	3	is	be	AUX
ejpam-628	199	4	a	a	DET
ejpam-628	199	5	local	local	ADJ
ejpam-628	199	6	ring	ring	NOUN
ejpam-628	199	7	.	.	PUNCT
ejpam-628	200	1	3	3	X
ejpam-628	200	2	.	.	X
ejpam-628	200	3	on	on	ADP
ejpam-628	200	4	commutative	commutative	ADJ
ejpam-628	200	5	rings	ring	NOUN
ejpam-628	200	6	with	with	ADP
ejpam-628	200	7	p1	p1	PROPN
ejpam-628	200	8	k1	k1	NOUN
ejpam-628	200	9	.	.	PUNCT
ejpam-628	200	10	.	.	PUNCT
ejpam-628	200	11	.	.	PUNCT
ejpam-628	201	1	pn	pn	PROPN
ejpam-628	201	2	kn(1	kn(1	PROPN
ejpam-628	201	3	≤	≤	PROPN
ejpam-628	201	4	ki	ki	PROPN
ejpam-628	201	5	≤	≤	ADV
ejpam-628	201	6	7	7	NUM
ejpam-628	201	7	)	)	PUNCT
ejpam-628	201	8	zero	zero	NUM
ejpam-628	201	9	-	-	PUNCT
ejpam-628	201	10	divisors	divisor	NOUN
ejpam-628	201	11	by	by	ADP
ejpam-628	201	12	lemma	lemma	PROPN
ejpam-628	201	13	1	1	NUM
ejpam-628	201	14	,	,	PUNCT
ejpam-628	201	15	for	for	ADP
ejpam-628	201	16	each	each	DET
ejpam-628	201	17	finite	finite	ADJ
ejpam-628	201	18	local	local	ADJ
ejpam-628	201	19	ring	ring	NOUN
ejpam-628	201	20	r	r	NOUN
ejpam-628	201	21	we	we	PRON
ejpam-628	201	22	have	have	VERB
ejpam-628	201	23	|z(r)|	|z(r)|	PROPN
ejpam-628	201	24	=	=	SYM
ejpam-628	201	25	pk	pk	NOUN
ejpam-628	201	26	for	for	ADP
ejpam-628	201	27	some	some	DET
ejpam-628	201	28	prime	prime	ADJ
ejpam-628	201	29	number	number	NOUN
ejpam-628	201	30	p	p	NOUN
ejpam-628	201	31	and	and	CCONJ
ejpam-628	201	32	k	k	PROPN
ejpam-628	201	33	≥	≥	PROPN
ejpam-628	201	34	0	0	NUM
ejpam-628	201	35	,	,	PUNCT
ejpam-628	201	36	but	but	CCONJ
ejpam-628	201	37	the	the	DET
ejpam-628	201	38	converse	converse	NOUN
ejpam-628	201	39	is	be	AUX
ejpam-628	201	40	not	not	PART
ejpam-628	201	41	true	true	ADJ
ejpam-628	201	42	in	in	ADP
ejpam-628	201	43	general	general	ADJ
ejpam-628	201	44	.	.	PUNCT
ejpam-628	202	1	for	for	ADP
ejpam-628	202	2	example	example	NOUN
ejpam-628	202	3	,	,	PUNCT
ejpam-628	202	4	the	the	DET
ejpam-628	202	5	nonlocal	nonlocal	ADJ
ejpam-628	202	6	ring	ring	NOUN
ejpam-628	202	7	z8×	z8×	X
ejpam-628	202	8	f7	f7	PROPN
ejpam-628	202	9	has	have	VERB
ejpam-628	202	10	32	32	NUM
ejpam-628	202	11	zero	zero	NUM
ejpam-628	202	12	-	-	PUNCT
ejpam-628	202	13	divisors	divisor	NOUN
ejpam-628	202	14	.	.	PUNCT
ejpam-628	203	1	in	in	ADP
ejpam-628	203	2	this	this	DET
ejpam-628	203	3	section	section	NOUN
ejpam-628	203	4	,	,	PUNCT
ejpam-628	203	5	we	we	PRON
ejpam-628	203	6	will	will	AUX
ejpam-628	203	7	characterize	characterize	VERB
ejpam-628	203	8	rings	ring	NOUN
ejpam-628	203	9	with	with	ADP
ejpam-628	203	10	pk	pk	NOUN
ejpam-628	203	11	zero	zero	NUM
ejpam-628	203	12	-	-	PUNCT
ejpam-628	203	13	divisors	divisor	NOUN
ejpam-628	203	14	where	where	SCONJ
ejpam-628	203	15	k	k	PROPN
ejpam-628	203	16	is	be	AUX
ejpam-628	203	17	a	a	DET
ejpam-628	203	18	positive	positive	ADJ
ejpam-628	203	19	integer	integer	NOUN
ejpam-628	203	20	1≤	1≤	PROPN
ejpam-628	204	1	k	k	PROPN
ejpam-628	204	2	≤	≤	ADV
ejpam-628	204	3	7	7	NUM
ejpam-628	204	4	.	.	PUNCT
ejpam-628	204	5	theorem	theorem	NOUN
ejpam-628	204	6	4	4	NUM
ejpam-628	204	7	.	.	PUNCT
ejpam-628	205	1	let	let	VERB
ejpam-628	205	2	r	r	PRON
ejpam-628	205	3	be	be	AUX
ejpam-628	205	4	a	a	DET
ejpam-628	205	5	commutative	commutative	ADJ
ejpam-628	205	6	ring	ring	NOUN
ejpam-628	205	7	with	with	ADP
ejpam-628	205	8	|z(r)|	|z(r)|	PROPN
ejpam-628	205	9	=	=	SYM
ejpam-628	205	10	pk	pk	NOUN
ejpam-628	205	11	where	where	SCONJ
ejpam-628	205	12	p	p	NOUN
ejpam-628	205	13	is	be	AUX
ejpam-628	205	14	a	a	DET
ejpam-628	205	15	prime	prime	ADJ
ejpam-628	205	16	number	number	NOUN
ejpam-628	205	17	and	and	CCONJ
ejpam-628	205	18	1≤	1≤	NUM
ejpam-628	206	1	k	k	PROPN
ejpam-628	206	2	≤	≤	ADV
ejpam-628	206	3	6	6	NUM
ejpam-628	206	4	.	.	PUNCT
ejpam-628	207	1	then	then	ADV
ejpam-628	207	2	either	either	CCONJ
ejpam-628	207	3	(	(	PUNCT
ejpam-628	207	4	i	i	NOUN
ejpam-628	207	5	)	)	PUNCT
ejpam-628	207	6	r	r	NOUN
ejpam-628	207	7	is	be	AUX
ejpam-628	207	8	a	a	DET
ejpam-628	207	9	local	local	ADJ
ejpam-628	207	10	ring	ring	NOUN
ejpam-628	207	11	;	;	PUNCT
ejpam-628	207	12	(	(	PUNCT
ejpam-628	207	13	ii	ii	NOUN
ejpam-628	207	14	)	)	PUNCT
ejpam-628	207	15	r	r	NOUN
ejpam-628	207	16	is	be	AUX
ejpam-628	207	17	a	a	DET
ejpam-628	207	18	reduced	reduce	VERB
ejpam-628	207	19	ring	ring	NOUN
ejpam-628	207	20	and	and	CCONJ
ejpam-628	207	21	so	so	ADV
ejpam-628	207	22	r	r	NOUN
ejpam-628	207	23	∼=	∼=	PROPN
ejpam-628	207	24	fq1	fq1	NUM
ejpam-628	207	25	×	×	NOUN
ejpam-628	207	26	.	.	PUNCT
ejpam-628	207	27	.	.	PUNCT
ejpam-628	208	1	.×	.×	PROPN
ejpam-628	208	2	fqt	fqt	PROPN
ejpam-628	208	3	,	,	PUNCT
ejpam-628	208	4	where	where	SCONJ
ejpam-628	208	5	each	each	DET
ejpam-628	208	6	fqi	fqi	NOUN
ejpam-628	208	7	(	(	PUNCT
ejpam-628	208	8	1	1	NUM
ejpam-628	208	9	≤	≤	NUM
ejpam-628	208	10	i	i	NOUN
ejpam-628	208	11	≤	≤	PROPN
ejpam-628	208	12	t	t	PROPN
ejpam-628	208	13	)	)	PUNCT
ejpam-628	208	14	is	be	AUX
ejpam-628	208	15	a	a	DET
ejpam-628	208	16	finite	finite	ADJ
ejpam-628	208	17	field	field	NOUN
ejpam-628	208	18	and	and	CCONJ
ejpam-628	209	1	pk	pk	NOUN
ejpam-628	209	2	=	=	SYM
ejpam-628	209	3	q1q2	q1q2	PROPN
ejpam-628	209	4	.	.	PUNCT
ejpam-628	209	5	.	.	PUNCT
ejpam-628	209	6	.	.	PUNCT
ejpam-628	210	1	qt	qt	INTJ
ejpam-628	210	2	−	−	PROPN
ejpam-628	210	3	(	(	PUNCT
ejpam-628	210	4	q1−	q1−	VERB
ejpam-628	210	5	1)(q2−	1)(q2−	NUM
ejpam-628	210	6	1	1	NUM
ejpam-628	210	7	)	)	PUNCT
ejpam-628	210	8	.	.	PUNCT
ejpam-628	210	9	.	.	PUNCT
ejpam-628	210	10	.	.	PUNCT
ejpam-628	211	1	(	(	PUNCT
ejpam-628	211	2	qt	qt	INTJ
ejpam-628	211	3	−	−	PROPN
ejpam-628	211	4	1	1	NUM
ejpam-628	211	5	)	)	PUNCT
ejpam-628	211	6	;	;	PUNCT
ejpam-628	211	7	(	(	PUNCT
ejpam-628	211	8	iii	iii	X
ejpam-628	211	9	)	)	PUNCT
ejpam-628	211	10	r	r	NOUN
ejpam-628	211	11	∼=	∼=	PROPN
ejpam-628	211	12	r1	r1	NOUN
ejpam-628	211	13	×	×	NOUN
ejpam-628	211	14	fq1	fq1	CCONJ
ejpam-628	211	15	×	×	NOUN
ejpam-628	211	16	.	.	PUNCT
ejpam-628	211	17	.	.	PUNCT
ejpam-628	212	1	.×	.×	PROPN
ejpam-628	212	2	fqt	fqt	PROPN
ejpam-628	212	3	,	,	PUNCT
ejpam-628	212	4	where	where	SCONJ
ejpam-628	212	5	each	each	DET
ejpam-628	212	6	fqi	fqi	NOUN
ejpam-628	212	7	(	(	PUNCT
ejpam-628	212	8	1	1	NUM
ejpam-628	212	9	≤	≤	NUM
ejpam-628	212	10	i	i	NOUN
ejpam-628	212	11	≤	≤	PROPN
ejpam-628	212	12	t	t	PROPN
ejpam-628	212	13	)	)	PUNCT
ejpam-628	212	14	is	be	AUX
ejpam-628	212	15	a	a	DET
ejpam-628	212	16	finite	finite	ADJ
ejpam-628	212	17	field	field	NOUN
ejpam-628	212	18	and	and	CCONJ
ejpam-628	212	19	r1	r1	PROPN
ejpam-628	212	20	is	be	AUX
ejpam-628	212	21	a	a	DET
ejpam-628	212	22	local	local	ADJ
ejpam-628	212	23	ring	ring	NOUN
ejpam-628	212	24	with	with	ADP
ejpam-628	212	25	|z(r1)|=	|z(r1)|=	NOUN
ejpam-628	212	26	pm	pm	NOUN
ejpam-628	212	27	,	,	PUNCT
ejpam-628	212	28	|r1|	|r1|	NOUN
ejpam-628	212	29	=	=	PUNCT
ejpam-628	212	30	pn	pn	PROPN
ejpam-628	213	1	such	such	ADJ
ejpam-628	213	2	that	that	SCONJ
ejpam-628	213	3	0	0	NUM
ejpam-628	213	4	<	<	X
ejpam-628	213	5	m	m	X
ejpam-628	213	6	<	<	X
ejpam-628	213	7	n≤	n≤	PRON
ejpam-628	213	8	k−	k−	PROPN
ejpam-628	213	9	1	1	NUM
ejpam-628	213	10	and	and	CCONJ
ejpam-628	213	11	pk	pk	NOUN
ejpam-628	213	12	=	=	NOUN
ejpam-628	213	13	pnq1q2	pnq1q2	NOUN
ejpam-628	213	14	.	.	PUNCT
ejpam-628	213	15	.	.	PUNCT
ejpam-628	213	16	.	.	PUNCT
ejpam-628	214	1	qt	qt	INTJ
ejpam-628	214	2	−	−	PROPN
ejpam-628	215	1	(	(	PUNCT
ejpam-628	215	2	p	p	NOUN
ejpam-628	215	3	n−	n−	NOUN
ejpam-628	215	4	pm)(q1−	pm)(q1−	VERB
ejpam-628	215	5	1)(q2−	1)(q2−	NUM
ejpam-628	215	6	1	1	NUM
ejpam-628	215	7	)	)	PUNCT
ejpam-628	215	8	.	.	PUNCT
ejpam-628	215	9	.	.	PUNCT
ejpam-628	216	1	.	.	PUNCT
ejpam-628	217	1	(	(	PUNCT
ejpam-628	217	2	qt	qt	AUX
ejpam-628	217	3	−	−	PROPN
ejpam-628	217	4	1	1	NUM
ejpam-628	217	5	)	)	PUNCT
ejpam-628	217	6	;	;	PUNCT
ejpam-628	217	7	or	or	CCONJ
ejpam-628	217	8	(	(	PUNCT
ejpam-628	217	9	iv	iv	X
ejpam-628	217	10	)	)	PUNCT
ejpam-628	217	11	r∼=	r∼=	PUNCT
ejpam-628	217	12	r1	r1	PROPN
ejpam-628	217	13	×	×	PROPN
ejpam-628	217	14	r2×	r2×	NOUN
ejpam-628	217	15	f5	f5	VERB
ejpam-628	217	16	where	where	SCONJ
ejpam-628	217	17	each	each	DET
ejpam-628	217	18	ri	ri	PROPN
ejpam-628	217	19	is	be	AUX
ejpam-628	217	20	isomorphic	isomorphic	ADJ
ejpam-628	217	21	to	to	ADP
ejpam-628	217	22	z4	z4	PROPN
ejpam-628	217	23	or	or	CCONJ
ejpam-628	217	24	z2[x]/(x	z2[x]/(x	NUM
ejpam-628	217	25	2	2	NUM
ejpam-628	217	26	)	)	PUNCT
ejpam-628	217	27	.	.	PUNCT
ejpam-628	218	1	proof	proof	NOUN
ejpam-628	218	2	.	.	PUNCT
ejpam-628	219	1	suppose	suppose	VERB
ejpam-628	219	2	|z(r)|	|z(r)|	PROPN
ejpam-628	219	3	=	=	SYM
ejpam-628	219	4	pk	pk	NOUN
ejpam-628	219	5	and	and	CCONJ
ejpam-628	219	6	r	r	NOUN
ejpam-628	219	7	is	be	AUX
ejpam-628	219	8	not	not	PART
ejpam-628	219	9	a	a	DET
ejpam-628	219	10	local	local	ADJ
ejpam-628	219	11	ring	ring	NOUN
ejpam-628	219	12	.	.	PUNCT
ejpam-628	220	1	if	if	SCONJ
ejpam-628	220	2	r	r	NOUN
ejpam-628	220	3	is	be	AUX
ejpam-628	220	4	reduced	reduce	VERB
ejpam-628	220	5	,	,	PUNCT
ejpam-628	220	6	then	then	ADV
ejpam-628	220	7	we	we	PRON
ejpam-628	220	8	are	be	AUX
ejpam-628	220	9	done	do	VERB
ejpam-628	220	10	.	.	PUNCT
ejpam-628	221	1	now	now	ADV
ejpam-628	221	2	let	let	VERB
ejpam-628	221	3	r	r	NOUN
ejpam-628	221	4	is	be	AUX
ejpam-628	221	5	not	not	PART
ejpam-628	221	6	a	a	DET
ejpam-628	221	7	reduced	reduce	VERB
ejpam-628	221	8	ring	ring	NOUN
ejpam-628	221	9	.	.	PUNCT
ejpam-628	222	1	then	then	ADV
ejpam-628	222	2	by	by	ADP
ejpam-628	222	3	theorem	theorem	NOUN
ejpam-628	222	4	2	2	NUM
ejpam-628	222	5	,	,	PUNCT
ejpam-628	222	6	we	we	PRON
ejpam-628	222	7	can	can	AUX
ejpam-628	222	8	assume	assume	VERB
ejpam-628	222	9	that	that	SCONJ
ejpam-628	222	10	r∼=	r∼=	ADV
ejpam-628	222	11	r1	r1	PROPN
ejpam-628	222	12	×	×	NOUN
ejpam-628	222	13	.	.	PUNCT
ejpam-628	222	14	.	.	PUNCT
ejpam-628	223	1	.×	.×	NOUN
ejpam-628	223	2	rs	r	VERB
ejpam-628	223	3	×	×	NOUN
ejpam-628	224	1	fq1	fq1	INTJ
ejpam-628	224	2	×	×	NOUN
ejpam-628	224	3	.	.	PUNCT
ejpam-628	224	4	.	.	PUNCT
ejpam-628	225	1	.×	.×	PROPN
ejpam-628	225	2	fqt	fqt	PROPN
ejpam-628	225	3	,	,	PUNCT
ejpam-628	225	4	where	where	SCONJ
ejpam-628	225	5	s	s	X
ejpam-628	225	6	,	,	PUNCT
ejpam-628	225	7	t	t	PROPN
ejpam-628	225	8	≥	≥	NUM
ejpam-628	225	9	1	1	NUM
ejpam-628	225	10	and	and	CCONJ
ejpam-628	225	11	each	each	DET
ejpam-628	225	12	ri	ri	PROPN
ejpam-628	225	13	is	be	AUX
ejpam-628	225	14	a	a	DET
ejpam-628	225	15	local	local	ADJ
ejpam-628	225	16	ring	ring	NOUN
ejpam-628	225	17	with	with	ADP
ejpam-628	225	18	|z(ri)|	|z(ri)|	NOUN
ejpam-628	225	19	=	=	SYM
ejpam-628	225	20	pti	pti	PROPN
ejpam-628	225	21	,	,	PUNCT
ejpam-628	225	22	|ri|	|ri|	NOUN
ejpam-628	225	23	=	=	SYM
ejpam-628	225	24	pki	pki	NOUN
ejpam-628	225	25	for	for	ADP
ejpam-628	225	26	some	some	DET
ejpam-628	225	27	t	t	NOUN
ejpam-628	226	1	i	i	PRON
ejpam-628	226	2	,	,	PUNCT
ejpam-628	226	3	ki	ki	PROPN
ejpam-628	226	4	≥	≥	PROPN
ejpam-628	226	5	1	1	NUM
ejpam-628	226	6	such	such	ADJ
ejpam-628	226	7	that	that	SCONJ
ejpam-628	226	8	1≤	1≤	NUM
ejpam-628	226	9	s	s	VERB
ejpam-628	226	10	∑	∑	PROPN
ejpam-628	226	11	i=1	i=1	PROPN
ejpam-628	227	1	t	t	PROPN
ejpam-628	228	1	i	i	PRON
ejpam-628	228	2	≤	≤	PROPN
ejpam-628	228	3	s	s	VERB
ejpam-628	228	4	∑	∑	PROPN
ejpam-628	228	5	i=1	i=1	PROPN
ejpam-628	228	6	ki	ki	PROPN
ejpam-628	229	1	−	−	PROPN
ejpam-628	229	2	s	s	PART
ejpam-628	229	3	≤	≤	PROPN
ejpam-628	229	4	k−	k−	PROPN
ejpam-628	229	5	s−	s−	PROPN
ejpam-628	229	6	1≤	1≤	ADP
ejpam-628	229	7	6−	6−	NUM
ejpam-628	229	8	1−	1−	NUM
ejpam-628	229	9	1=	1=	NUM
ejpam-628	229	10	4	4	NUM
ejpam-628	229	11	.	.	PUNCT
ejpam-628	230	1	it	it	PRON
ejpam-628	230	2	follows	follow	VERB
ejpam-628	230	3	that	that	PRON
ejpam-628	230	4	s	s	VERB
ejpam-628	230	5	≤	≤	ADJ
ejpam-628	230	6	4	4	NUM
ejpam-628	230	7	.	.	PUNCT
ejpam-628	231	1	if	if	SCONJ
ejpam-628	231	2	s	s	PART
ejpam-628	231	3	=	=	SYM
ejpam-628	231	4	3	3	NUM
ejpam-628	231	5	or	or	CCONJ
ejpam-628	231	6	4	4	NUM
ejpam-628	231	7	,	,	PUNCT
ejpam-628	231	8	then	then	ADV
ejpam-628	231	9	since	since	SCONJ
ejpam-628	231	10	t	t	PROPN
ejpam-628	231	11	≥	≥	NUM
ejpam-628	231	12	1	1	NUM
ejpam-628	231	13	,	,	PUNCT
ejpam-628	231	14	pk	pk	NOUN
ejpam-628	231	15	=	=	SYM
ejpam-628	231	16	|z(r)|	|z(r)|	PROPN
ejpam-628	231	17	>	>	X
ejpam-628	231	18	|r1||r2||r3|	|r1||r2||r3|	ADP
ejpam-628	231	19	≥	≥	NOUN
ejpam-628	231	20	p6	p6	PROPN
ejpam-628	231	21	,	,	PUNCT
ejpam-628	231	22	this	this	PRON
ejpam-628	231	23	is	be	AUX
ejpam-628	231	24	a	a	DET
ejpam-628	231	25	contradiction	contradiction	NOUN
ejpam-628	231	26	.	.	PUNCT
ejpam-628	232	1	hence	hence	ADV
ejpam-628	232	2	s	s	VERB
ejpam-628	232	3	≤	≤	ADJ
ejpam-628	232	4	2	2	NUM
ejpam-628	232	5	.	.	PUNCT
ejpam-628	233	1	if	if	SCONJ
ejpam-628	233	2	s	s	PART
ejpam-628	233	3	=	=	NOUN
ejpam-628	233	4	1	1	NUM
ejpam-628	233	5	,	,	PUNCT
ejpam-628	233	6	then	then	ADV
ejpam-628	233	7	by	by	ADP
ejpam-628	233	8	theorem	theorem	NOUN
ejpam-628	233	9	2	2	NUM
ejpam-628	233	10	,	,	PUNCT
ejpam-628	233	11	we	we	PRON
ejpam-628	233	12	are	be	AUX
ejpam-628	233	13	done	do	VERB
ejpam-628	233	14	.	.	PUNCT
ejpam-628	234	1	thus	thus	ADV
ejpam-628	234	2	we	we	PRON
ejpam-628	234	3	can	can	AUX
ejpam-628	234	4	assume	assume	VERB
ejpam-628	234	5	that	that	SCONJ
ejpam-628	234	6	s	s	VERB
ejpam-628	234	7	=	=	SYM
ejpam-628	234	8	2	2	NUM
ejpam-628	234	9	,	,	PUNCT
ejpam-628	234	10	i.e.	i.e.	X
ejpam-628	234	11	,	,	PUNCT
ejpam-628	234	12	r∼=	r∼=	PUNCT
ejpam-628	234	13	r1×r2×	r1×r2×	X
ejpam-628	234	14	fq1	fq1	ADV
ejpam-628	234	15	×	×	NOUN
ejpam-628	234	16	.	.	PUNCT
ejpam-628	234	17	.	.	PUNCT
ejpam-628	235	1	.×	.×	PROPN
ejpam-628	235	2	fqt	fqt	PROPN
ejpam-628	235	3	,	,	PUNCT
ejpam-628	235	4	where	where	SCONJ
ejpam-628	235	5	r1	r1	PROPN
ejpam-628	235	6	and	and	CCONJ
ejpam-628	235	7	r2	r2	PROPN
ejpam-628	235	8	are	be	AUX
ejpam-628	235	9	local	local	ADJ
ejpam-628	235	10	rings	ring	NOUN
ejpam-628	235	11	with	with	ADP
ejpam-628	235	12	|z(ri)|	|z(ri)|	NUM
ejpam-628	235	13	=	=	SYM
ejpam-628	235	14	pti	pti	PROPN
ejpam-628	235	15	.	.	PUNCT
ejpam-628	236	1	clearly	clearly	ADV
ejpam-628	236	2	|ri|	|ri|	VERB
ejpam-628	236	3	≥	≥	NUM
ejpam-628	236	4	pti+1	pti+1	NOUN
ejpam-628	236	5	for	for	ADP
ejpam-628	236	6	i	i	PROPN
ejpam-628	236	7	=	=	SYM
ejpam-628	236	8	1	1	NUM
ejpam-628	236	9	,	,	PUNCT
ejpam-628	236	10	2	2	NUM
ejpam-628	236	11	and	and	CCONJ
ejpam-628	236	12	since	since	SCONJ
ejpam-628	236	13	t	t	PROPN
ejpam-628	236	14	≥	≥	NUM
ejpam-628	236	15	1	1	NUM
ejpam-628	236	16	,	,	PUNCT
ejpam-628	236	17	|z(r)|	|z(r)|	PROPN
ejpam-628	236	18	>	>	X
ejpam-628	236	19	|r1||r2|	|r1||r2|	PROPN
ejpam-628	236	20	.	.	PUNCT
ejpam-628	237	1	if	if	SCONJ
ejpam-628	237	2	t	t	PROPN
ejpam-628	237	3	i	i	PRON
ejpam-628	237	4	≥	≥	VERB
ejpam-628	237	5	3	3	NUM
ejpam-628	237	6	for	for	ADP
ejpam-628	237	7	some	some	DET
ejpam-628	237	8	i	i	NOUN
ejpam-628	237	9	or	or	CCONJ
ejpam-628	237	10	t1	t1	NOUN
ejpam-628	237	11	=	=	SYM
ejpam-628	237	12	t2	t2	NOUN
ejpam-628	237	13	=	=	SYM
ejpam-628	237	14	2	2	NUM
ejpam-628	237	15	,	,	PUNCT
ejpam-628	237	16	then	then	ADV
ejpam-628	237	17	|z(r)|	|z(r)|	PROPN
ejpam-628	237	18	>	>	SYM
ejpam-628	237	19	|r1||r2|	|r1||r2|	PROPN
ejpam-628	237	20	=	=	SYM
ejpam-628	238	1	pt1+t2	pt1+t2	PROPN
ejpam-628	238	2	+	+	NOUN
ejpam-628	238	3	2	2	NUM
ejpam-628	238	4	≥	≥	NOUN
ejpam-628	238	5	p6	p6	PROPN
ejpam-628	238	6	,	,	PUNCT
ejpam-628	238	7	a	a	DET
ejpam-628	238	8	contradiction	contradiction	NOUN
ejpam-628	238	9	.	.	PUNCT
ejpam-628	239	1	thus	thus	ADV
ejpam-628	239	2	without	without	ADP
ejpam-628	239	3	loss	loss	NOUN
ejpam-628	239	4	of	of	ADP
ejpam-628	239	5	generality	generality	NOUN
ejpam-628	239	6	we	we	PRON
ejpam-628	239	7	can	can	AUX
ejpam-628	239	8	assume	assume	VERB
ejpam-628	239	9	that	that	SCONJ
ejpam-628	239	10	either	either	ADV
ejpam-628	239	11	t1	t1	NOUN
ejpam-628	239	12	=	=	SYM
ejpam-628	239	13	2	2	NUM
ejpam-628	239	14	,	,	PUNCT
ejpam-628	239	15	t2	t2	NOUN
ejpam-628	239	16	=	=	SYM
ejpam-628	239	17	1	1	NUM
ejpam-628	239	18	or	or	CCONJ
ejpam-628	239	19	t1	t1	NOUN
ejpam-628	239	20	=	=	SYM
ejpam-628	239	21	t2	t2	NOUN
ejpam-628	239	22	=	=	SYM
ejpam-628	239	23	1	1	X
ejpam-628	239	24	.	.	X
ejpam-628	239	25	m.	m.	NOUN
ejpam-628	239	26	behboodi	behboodi	PROPN
ejpam-628	239	27	,	,	PUNCT
ejpam-628	239	28	r.	r.	PROPN
ejpam-628	239	29	beyranvand	beyranvand	PROPN
ejpam-628	239	30	/	/	SYM
ejpam-628	239	31	eur	eur	PROPN
ejpam-628	239	32	.	.	PUNCT
ejpam-628	240	1	j.	j.	PROPN
ejpam-628	240	2	pure	pure	PROPN
ejpam-628	240	3	appl	appl	PROPN
ejpam-628	240	4	.	.	PROPN
ejpam-628	240	5	math	math	PROPN
ejpam-628	240	6	,	,	PUNCT
ejpam-628	240	7	3	3	NUM
ejpam-628	240	8	(	(	PUNCT
ejpam-628	240	9	2010	2010	NUM
ejpam-628	240	10	)	)	PUNCT
ejpam-628	240	11	,	,	PUNCT
ejpam-628	240	12	303	303	NUM
ejpam-628	240	13	-	-	SYM
ejpam-628	240	14	316	316	NUM
ejpam-628	240	15	308	308	NUM
ejpam-628	240	16	•	•	NOUN
ejpam-628	240	17	case	case	NOUN
ejpam-628	240	18	1	1	NUM
ejpam-628	240	19	:	:	PUNCT
ejpam-628	240	20	t1	t1	NOUN
ejpam-628	240	21	=	=	SYM
ejpam-628	240	22	2	2	NUM
ejpam-628	240	23	,	,	PUNCT
ejpam-628	240	24	t2	t2	NOUN
ejpam-628	240	25	=	=	SYM
ejpam-628	240	26	1	1	NUM
ejpam-628	240	27	i.e.	i.e.	X
ejpam-628	240	28	,	,	PUNCT
ejpam-628	240	29	|z(r1)|	|z(r1)|	NOUN
ejpam-628	240	30	=	=	NOUN
ejpam-628	240	31	p2	p2	PROPN
ejpam-628	240	32	and	and	CCONJ
ejpam-628	240	33	|z(r2)|=	|z(r2)|=	NUM
ejpam-628	241	1	p.	p.	NOUN
ejpam-628	241	2	then	then	ADV
ejpam-628	241	3	by	by	ADP
ejpam-628	241	4	lemma	lemma	PROPN
ejpam-628	241	5	1	1	NUM
ejpam-628	241	6	,	,	PUNCT
ejpam-628	241	7	we	we	PRON
ejpam-628	241	8	conclude	conclude	VERB
ejpam-628	241	9	that	that	SCONJ
ejpam-628	241	10	|r1|	|r1|	PROPN
ejpam-628	241	11	=	=	SYM
ejpam-628	241	12	p3	p3	PROPN
ejpam-628	241	13	or	or	CCONJ
ejpam-628	241	14	|r1|	|r1|	NOUN
ejpam-628	241	15	=	=	PUNCT
ejpam-628	241	16	p4	p4	ADJ
ejpam-628	241	17	and	and	CCONJ
ejpam-628	241	18	|r2|	|r2|	ADJ
ejpam-628	241	19	=	=	ADJ
ejpam-628	241	20	p2	p2	NOUN
ejpam-628	241	21	.	.	PUNCT
ejpam-628	242	1	if	if	SCONJ
ejpam-628	242	2	|r1|	|r1|	NOUN
ejpam-628	242	3	=	=	SYM
ejpam-628	242	4	p4	p4	ADJ
ejpam-628	242	5	,	,	PUNCT
ejpam-628	242	6	then	then	ADV
ejpam-628	242	7	|z(r)|	|z(r)|	PROPN
ejpam-628	242	8	>	>	X
ejpam-628	242	9	p6	p6	PROPN
ejpam-628	242	10	,	,	PUNCT
ejpam-628	242	11	a	a	DET
ejpam-628	242	12	contradiction	contradiction	NOUN
ejpam-628	242	13	.	.	PUNCT
ejpam-628	243	1	thus	thus	ADV
ejpam-628	243	2	|r1|	|r1|	PROPN
ejpam-628	243	3	=	=	SYM
ejpam-628	243	4	p3	p3	PROPN
ejpam-628	243	5	and	and	CCONJ
ejpam-628	243	6	|r2|	|r2|	ADJ
ejpam-628	243	7	=	=	NOUN
ejpam-628	243	8	p2	p2	PROPN
ejpam-628	243	9	and	and	CCONJ
ejpam-628	243	10	so	so	ADV
ejpam-628	243	11	|z(r)|	|z(r)|	PROPN
ejpam-628	243	12	>	>	X
ejpam-628	243	13	|r1||r2|	|r1||r2|	PROPN
ejpam-628	243	14	≥	≥	PRON
ejpam-628	243	15	p5	p5	ADJ
ejpam-628	243	16	i.e.	i.e.	X
ejpam-628	243	17	,	,	PUNCT
ejpam-628	243	18	k	k	PROPN
ejpam-628	243	19	=	=	SYM
ejpam-628	243	20	6	6	X
ejpam-628	243	21	.	.	PUNCT
ejpam-628	244	1	we	we	PRON
ejpam-628	244	2	claim	claim	VERB
ejpam-628	244	3	that	that	SCONJ
ejpam-628	244	4	t	t	NOUN
ejpam-628	244	5	=	=	SYM
ejpam-628	244	6	1	1	NUM
ejpam-628	244	7	,	,	PUNCT
ejpam-628	244	8	for	for	ADP
ejpam-628	244	9	if	if	SCONJ
ejpam-628	244	10	not	not	PART
ejpam-628	244	11	,	,	PUNCT
ejpam-628	244	12	since	since	SCONJ
ejpam-628	244	13	qi	qi	PROPN
ejpam-628	244	14	>	>	X
ejpam-628	244	15	p	p	PROPN
ejpam-628	244	16	for	for	ADP
ejpam-628	244	17	some	some	PRON
ejpam-628	244	18	i	i	PRON
ejpam-628	244	19	(	(	PUNCT
ejpam-628	244	20	see	see	VERB
ejpam-628	244	21	theorem	theorem	NOUN
ejpam-628	244	22	2	2	NUM
ejpam-628	244	23	)	)	PUNCT
ejpam-628	244	24	,	,	PUNCT
ejpam-628	244	25	|z(r)|	|z(r)|	PROPN
ejpam-628	244	26	≥	≥	PRON
ejpam-628	244	27	|r1||r2||fqi	|r1||r2||fqi	PROPN
ejpam-628	244	28	|	|	ADV
ejpam-628	244	29	>	>	X
ejpam-628	244	30	p6	p6	PROPN
ejpam-628	244	31	,	,	PUNCT
ejpam-628	244	32	a	a	DET
ejpam-628	244	33	contradiction	contradiction	NOUN
ejpam-628	244	34	.	.	PUNCT
ejpam-628	245	1	thus	thus	ADV
ejpam-628	245	2	t	t	X
ejpam-628	245	3	=	=	SYM
ejpam-628	245	4	1	1	NUM
ejpam-628	245	5	and	and	CCONJ
ejpam-628	245	6	hence	hence	ADV
ejpam-628	245	7	by	by	ADP
ejpam-628	245	8	using	use	VERB
ejpam-628	245	9	the	the	DET
ejpam-628	245	10	relation	relation	NOUN
ejpam-628	245	11	(	(	PUNCT
ejpam-628	245	12	2	2	X
ejpam-628	245	13	)	)	PUNCT
ejpam-628	245	14	we	we	PRON
ejpam-628	245	15	have	have	VERB
ejpam-628	245	16	p3	p3	NOUN
ejpam-628	245	17	=	=	PUNCT
ejpam-628	246	1	p2q1	p2q1	PROPN
ejpam-628	247	1	−	−	PROPN
ejpam-628	247	2	(	(	PUNCT
ejpam-628	247	3	p−	p−	NOUN
ejpam-628	247	4	1)2(q1−	1)2(q1−	NUM
ejpam-628	247	5	1	1	NUM
ejpam-628	247	6	)	)	PUNCT
ejpam-628	247	7	.	.	PUNCT
ejpam-628	248	1	this	this	PRON
ejpam-628	248	2	implies	imply	VERB
ejpam-628	248	3	that	that	SCONJ
ejpam-628	248	4	q1(2p−1	q1(2p−1	NOUN
ejpam-628	248	5	)	)	PUNCT
ejpam-628	249	1	=	=	PUNCT
ejpam-628	249	2	p3−	p3−	NOUN
ejpam-628	249	3	p2	p2	NOUN
ejpam-628	249	4	+	+	NOUN
ejpam-628	249	5	2p−1	2p−1	NUM
ejpam-628	249	6	and	and	CCONJ
ejpam-628	249	7	so	so	ADV
ejpam-628	249	8	(	(	PUNCT
ejpam-628	249	9	2p−1	2p−1	NUM
ejpam-628	249	10	)	)	PUNCT
ejpam-628	249	11	is	be	AUX
ejpam-628	249	12	a	a	DET
ejpam-628	249	13	divisor	divisor	NOUN
ejpam-628	249	14	of	of	ADP
ejpam-628	249	15	p2(p−1	p2(p−1	PROPN
ejpam-628	249	16	)	)	PUNCT
ejpam-628	249	17	.	.	PUNCT
ejpam-628	250	1	but	but	CCONJ
ejpam-628	250	2	since	since	SCONJ
ejpam-628	250	3	(	(	PUNCT
ejpam-628	250	4	2p−	2p−	NOUN
ejpam-628	250	5	1	1	NUM
ejpam-628	250	6	,	,	PUNCT
ejpam-628	250	7	p2	p2	X
ejpam-628	250	8	)	)	PUNCT
ejpam-628	250	9	=	=	SYM
ejpam-628	250	10	1	1	NUM
ejpam-628	250	11	,	,	PUNCT
ejpam-628	250	12	2p−	2p−	NOUN
ejpam-628	250	13	1	1	NUM
ejpam-628	250	14	is	be	AUX
ejpam-628	250	15	a	a	DET
ejpam-628	250	16	divisor	divisor	NOUN
ejpam-628	250	17	of	of	ADP
ejpam-628	250	18	p−	p−	NOUN
ejpam-628	250	19	1	1	NUM
ejpam-628	250	20	,	,	PUNCT
ejpam-628	250	21	a	a	DET
ejpam-628	250	22	contradiction	contradiction	NOUN
ejpam-628	250	23	.	.	PUNCT
ejpam-628	251	1	•	•	NUM
ejpam-628	251	2	case	case	NOUN
ejpam-628	251	3	2	2	NUM
ejpam-628	251	4	:	:	PUNCT
ejpam-628	251	5	t1	t1	NOUN
ejpam-628	251	6	=	=	SYM
ejpam-628	251	7	t2	t2	NOUN
ejpam-628	251	8	=	=	SYM
ejpam-628	251	9	1	1	NUM
ejpam-628	251	10	i.e.	i.e.	X
ejpam-628	251	11	,	,	PUNCT
ejpam-628	251	12	|z(r1)|	|z(r1)|	NOUN
ejpam-628	251	13	=	=	SYM
ejpam-628	251	14	|z(r2)|	|z(r2)|	PROPN
ejpam-628	251	15	=	=	SYM
ejpam-628	252	1	p.	p.	NOUN
ejpam-628	252	2	then	then	ADV
ejpam-628	252	3	|z(r)|	|z(r)|	PROPN
ejpam-628	252	4	>	>	X
ejpam-628	252	5	|r1||r2|	|r1||r2|	PROPN
ejpam-628	252	6	≥	≥	PRON
ejpam-628	252	7	p4	p4	ADJ
ejpam-628	252	8	,	,	PUNCT
ejpam-628	252	9	i.e.	i.e.	X
ejpam-628	252	10	,	,	PUNCT
ejpam-628	252	11	k	k	X
ejpam-628	252	12	≥	≥	NUM
ejpam-628	252	13	5	5	NUM
ejpam-628	252	14	.	.	PUNCT
ejpam-628	253	1	if	if	SCONJ
ejpam-628	253	2	t	t	PROPN
ejpam-628	253	3	≥	≥	NUM
ejpam-628	253	4	3	3	NUM
ejpam-628	253	5	,	,	PUNCT
ejpam-628	253	6	then	then	ADV
ejpam-628	253	7	by	by	ADP
ejpam-628	253	8	the	the	DET
ejpam-628	253	9	relation	relation	NOUN
ejpam-628	253	10	(	(	PUNCT
ejpam-628	253	11	2	2	NUM
ejpam-628	253	12	)	)	PUNCT
ejpam-628	253	13	,	,	PUNCT
ejpam-628	253	14	p2	p2	PROPN
ejpam-628	253	15	is	be	AUX
ejpam-628	253	16	a	a	DET
ejpam-628	253	17	divisor	divisor	NOUN
ejpam-628	253	18	of	of	ADP
ejpam-628	253	19	(	(	PUNCT
ejpam-628	253	20	qi	qi	PROPN
ejpam-628	253	21	−	−	PROPN
ejpam-628	253	22	1)(q	1)(q	NUM
ejpam-628	253	23	j	j	PROPN
ejpam-628	253	24	−	−	NOUN
ejpam-628	253	25	1	1	NUM
ejpam-628	253	26	)	)	PUNCT
ejpam-628	253	27	for	for	ADP
ejpam-628	253	28	some	some	DET
ejpam-628	253	29	1≤	1≤	NOUN
ejpam-628	254	1	i	i	PRON
ejpam-628	254	2	,	,	PUNCT
ejpam-628	254	3	j	j	PROPN
ejpam-628	254	4	≤	≤	PROPN
ejpam-628	254	5	t.	t.	NOUN
ejpam-628	254	6	it	it	PRON
ejpam-628	254	7	follows	follow	VERB
ejpam-628	254	8	that	that	SCONJ
ejpam-628	254	9	qiq	qiq	PROPN
ejpam-628	254	10	j	j	X
ejpam-628	254	11	>	>	X
ejpam-628	254	12	p2	p2	PROPN
ejpam-628	254	13	and	and	CCONJ
ejpam-628	254	14	hence	hence	ADV
ejpam-628	254	15	|z(r)|	|z(r)|	PROPN
ejpam-628	254	16	>	>	PUNCT
ejpam-628	254	17	|r1||r2|qiq	|r1||r2|qiq	PROPN
ejpam-628	254	18	j	j	PROPN
ejpam-628	254	19	>	>	X
ejpam-628	254	20	p4p2	p4p2	PROPN
ejpam-628	254	21	=	=	SYM
ejpam-628	254	22	p6	p6	PROPN
ejpam-628	254	23	,	,	PUNCT
ejpam-628	254	24	a	a	DET
ejpam-628	254	25	contradiction	contradiction	NOUN
ejpam-628	254	26	.	.	PUNCT
ejpam-628	255	1	therefore	therefore	ADV
ejpam-628	255	2	t	t	X
ejpam-628	255	3	≤	≤	NOUN
ejpam-628	255	4	2	2	NUM
ejpam-628	255	5	.	.	PUNCT
ejpam-628	256	1	we	we	PRON
ejpam-628	256	2	claim	claim	VERB
ejpam-628	256	3	that	that	SCONJ
ejpam-628	256	4	t	t	NOUN
ejpam-628	256	5	=	=	SYM
ejpam-628	256	6	1	1	X
ejpam-628	256	7	.	.	PUNCT
ejpam-628	257	1	if	if	SCONJ
ejpam-628	257	2	t	t	NOUN
ejpam-628	257	3	=	=	SYM
ejpam-628	257	4	2	2	NUM
ejpam-628	257	5	,	,	PUNCT
ejpam-628	257	6	then	then	ADV
ejpam-628	257	7	by	by	ADP
ejpam-628	257	8	the	the	DET
ejpam-628	257	9	relation	relation	NOUN
ejpam-628	257	10	(	(	PUNCT
ejpam-628	257	11	2	2	X
ejpam-628	257	12	)	)	PUNCT
ejpam-628	257	13	we	we	PRON
ejpam-628	257	14	have	have	VERB
ejpam-628	257	15	pk−2	pk−2	ADJ
ejpam-628	257	16	=	=	SYM
ejpam-628	257	17	p2q1q2−	p2q1q2−	NOUN
ejpam-628	257	18	(	(	PUNCT
ejpam-628	257	19	p−	p−	NOUN
ejpam-628	257	20	1)2(q1−	1)2(q1−	NUM
ejpam-628	257	21	1)(q2−	1)(q2−	NUM
ejpam-628	257	22	1	1	NUM
ejpam-628	257	23	)	)	PUNCT
ejpam-628	257	24	.	.	PUNCT
ejpam-628	258	1	(	(	PUNCT
ejpam-628	258	2	3	3	X
ejpam-628	258	3	)	)	PUNCT
ejpam-628	258	4	since	since	SCONJ
ejpam-628	258	5	k	k	PROPN
ejpam-628	258	6	≥	≥	NUM
ejpam-628	258	7	5	5	NUM
ejpam-628	258	8	,	,	PUNCT
ejpam-628	258	9	p2	p2	PROPN
ejpam-628	258	10	is	be	AUX
ejpam-628	258	11	a	a	DET
ejpam-628	258	12	divisor	divisor	NOUN
ejpam-628	258	13	of	of	ADP
ejpam-628	258	14	(	(	PUNCT
ejpam-628	258	15	q1	q1	PROPN
ejpam-628	258	16	−	−	PROPN
ejpam-628	258	17	1)(q2−	1)(q2−	NUM
ejpam-628	258	18	1	1	NUM
ejpam-628	258	19	)	)	PUNCT
ejpam-628	258	20	.	.	PUNCT
ejpam-628	259	1	if	if	SCONJ
ejpam-628	259	2	p2	p2	PROPN
ejpam-628	259	3	is	be	AUX
ejpam-628	259	4	a	a	DET
ejpam-628	259	5	divisor	divisor	NOUN
ejpam-628	259	6	of	of	ADP
ejpam-628	259	7	qi	qi	PROPN
ejpam-628	259	8	−	−	PROPN
ejpam-628	259	9	1	1	NUM
ejpam-628	259	10	,	,	PUNCT
ejpam-628	259	11	then	then	ADV
ejpam-628	259	12	qi	qi	X
ejpam-628	259	13	>	>	X
ejpam-628	259	14	p2	p2	PROPN
ejpam-628	259	15	and	and	CCONJ
ejpam-628	259	16	so	so	ADV
ejpam-628	259	17	|z(r)|	|z(r)|	PROPN
ejpam-628	259	18	>	>	VERB
ejpam-628	259	19	p6	p6	PROPN
ejpam-628	259	20	,	,	PUNCT
ejpam-628	259	21	a	a	DET
ejpam-628	259	22	contradiction	contradiction	NOUN
ejpam-628	259	23	.	.	PUNCT
ejpam-628	260	1	thus	thus	ADV
ejpam-628	260	2	p	p	X
ejpam-628	260	3	is	be	AUX
ejpam-628	260	4	a	a	DET
ejpam-628	260	5	divisor	divisor	NOUN
ejpam-628	260	6	of	of	ADP
ejpam-628	260	7	both	both	CCONJ
ejpam-628	260	8	q1−1	q1−1	ADJ
ejpam-628	260	9	and	and	CCONJ
ejpam-628	260	10	q2−1	q2−1	NOUN
ejpam-628	260	11	.	.	PUNCT
ejpam-628	261	1	hence	hence	ADV
ejpam-628	261	2	q1	q1	PROPN
ejpam-628	261	3	−	−	NOUN
ejpam-628	261	4	1	1	NUM
ejpam-628	262	1	=	=	SYM
ejpam-628	262	2	k1p	k1p	ADJ
ejpam-628	262	3	and	and	CCONJ
ejpam-628	262	4	q2	q2	PROPN
ejpam-628	262	5	−	−	PROPN
ejpam-628	262	6	1	1	NUM
ejpam-628	262	7	=	=	SYM
ejpam-628	262	8	k2p	k2p	PROPN
ejpam-628	262	9	for	for	ADP
ejpam-628	262	10	some	some	DET
ejpam-628	262	11	positive	positive	ADJ
ejpam-628	262	12	integers	integer	NOUN
ejpam-628	262	13	k1	k1	NOUN
ejpam-628	262	14	and	and	CCONJ
ejpam-628	262	15	k2	k2	NOUN
ejpam-628	262	16	.	.	PUNCT
ejpam-628	263	1	then	then	ADV
ejpam-628	263	2	one	one	NUM
ejpam-628	263	3	obtains	obtain	VERB
ejpam-628	263	4	from	from	ADP
ejpam-628	263	5	(	(	PUNCT
ejpam-628	263	6	3	3	NUM
ejpam-628	263	7	)	)	PUNCT
ejpam-628	263	8	,	,	PUNCT
ejpam-628	263	9	pk−4	pk−4	PROPN
ejpam-628	263	10	=	=	PUNCT
ejpam-628	263	11	(	(	PUNCT
ejpam-628	263	12	k1p+	k1p+	PROPN
ejpam-628	263	13	1)(k2p+	1)(k2p+	PROPN
ejpam-628	264	1	1)−	1)−	PROPN
ejpam-628	264	2	(	(	PUNCT
ejpam-628	264	3	p−	p−	PROPN
ejpam-628	264	4	1)2k1k2	1)2k1k2	NUM
ejpam-628	264	5	,	,	PUNCT
ejpam-628	264	6	and	and	CCONJ
ejpam-628	264	7	hence	hence	ADV
ejpam-628	264	8	pk−4−	pk−4−	PROPN
ejpam-628	264	9	(	(	PUNCT
ejpam-628	264	10	k1	k1	X
ejpam-628	264	11	+	+	CCONJ
ejpam-628	264	12	k2	k2	NOUN
ejpam-628	264	13	+	+	CCONJ
ejpam-628	264	14	2k1k2)p+	2k1k2)p+	NUM
ejpam-628	264	15	k1k2	k1k2	NOUN
ejpam-628	264	16	−	−	PROPN
ejpam-628	264	17	1=	1=	X
ejpam-628	264	18	0	0	NUM
ejpam-628	264	19	.	.	PUNCT
ejpam-628	265	1	if	if	SCONJ
ejpam-628	265	2	k	k	PROPN
ejpam-628	265	3	=	=	SYM
ejpam-628	265	4	5	5	NUM
ejpam-628	265	5	,	,	PUNCT
ejpam-628	265	6	then	then	ADV
ejpam-628	265	7	p	p	NOUN
ejpam-628	265	8	=	=	PUNCT
ejpam-628	265	9	k1k2−1	k1k2−1	PROPN
ejpam-628	265	10	k1+k2	k1+k2	PROPN
ejpam-628	266	1	+	+	PROPN
ejpam-628	266	2	2k1k2−1	2k1k2−1	PROPN
ejpam-628	266	3	,	,	PUNCT
ejpam-628	266	4	a	a	DET
ejpam-628	266	5	contradiction	contradiction	NOUN
ejpam-628	266	6	.	.	PUNCT
ejpam-628	267	1	thus	thus	ADV
ejpam-628	267	2	we	we	PRON
ejpam-628	267	3	can	can	AUX
ejpam-628	267	4	assume	assume	VERB
ejpam-628	267	5	that	that	SCONJ
ejpam-628	267	6	k	k	PROPN
ejpam-628	267	7	=	=	PUNCT
ejpam-628	267	8	6	6	NUM
ejpam-628	267	9	and	and	CCONJ
ejpam-628	267	10	hence	hence	ADV
ejpam-628	267	11	p2	p2	PROPN
ejpam-628	268	1	−	−	PROPN
ejpam-628	268	2	(	(	PUNCT
ejpam-628	268	3	k1	k1	NOUN
ejpam-628	268	4	+	+	CCONJ
ejpam-628	268	5	k2	k2	NOUN
ejpam-628	268	6	+	+	CCONJ
ejpam-628	268	7	2k1k2)p+	2k1k2)p+	NUM
ejpam-628	268	8	k1k2	k1k2	NOUN
ejpam-628	268	9	−	−	PROPN
ejpam-628	268	10	1=	1=	X
ejpam-628	268	11	0	0	NUM
ejpam-628	268	12	.	.	PUNCT
ejpam-628	269	1	(	(	PUNCT
ejpam-628	269	2	4	4	NUM
ejpam-628	269	3	)	)	PUNCT
ejpam-628	269	4	thus	thus	ADV
ejpam-628	269	5	the	the	DET
ejpam-628	269	6	equation	equation	NOUN
ejpam-628	269	7	(	(	PUNCT
ejpam-628	269	8	4	4	X
ejpam-628	269	9	)	)	PUNCT
ejpam-628	269	10	shows	show	VERB
ejpam-628	269	11	that	that	SCONJ
ejpam-628	269	12	the	the	DET
ejpam-628	269	13	integer	integer	NOUN
ejpam-628	269	14	p	p	NOUN
ejpam-628	269	15	is	be	AUX
ejpam-628	269	16	a	a	DET
ejpam-628	269	17	solution	solution	NOUN
ejpam-628	269	18	of	of	ADP
ejpam-628	269	19	x	x	X
ejpam-628	269	20	2−	2−	NUM
ejpam-628	269	21	(	(	PUNCT
ejpam-628	269	22	k1	k1	NOUN
ejpam-628	269	23	+	+	CCONJ
ejpam-628	269	24	k2	k2	ADJ
ejpam-628	269	25	+	+	CCONJ
ejpam-628	269	26	2k1k2)x	2k1k2)x	NOUN
ejpam-628	270	1	+	+	CCONJ
ejpam-628	270	2	k1k2	k1k2	X
ejpam-628	270	3	−	−	NUM
ejpam-628	270	4	1=	1=	NOUN
ejpam-628	270	5	0	0	NUM
ejpam-628	270	6	.	.	PUNCT
ejpam-628	271	1	(	(	PUNCT
ejpam-628	271	2	5	5	X
ejpam-628	271	3	)	)	PUNCT
ejpam-628	271	4	now	now	ADV
ejpam-628	271	5	let	let	VERB
ejpam-628	271	6	µ	µ	X
ejpam-628	271	7	be	be	AUX
ejpam-628	271	8	another	another	DET
ejpam-628	271	9	solution	solution	NOUN
ejpam-628	271	10	of	of	ADP
ejpam-628	271	11	(	(	PUNCT
ejpam-628	271	12	5	5	NUM
ejpam-628	271	13	)	)	PUNCT
ejpam-628	271	14	.	.	PUNCT
ejpam-628	272	1	clearly	clearly	ADV
ejpam-628	272	2	µ	µ	VERB
ejpam-628	272	3	6=	6=	NUM
ejpam-628	272	4	1	1	NUM
ejpam-628	272	5	,	,	PUNCT
ejpam-628	272	6	pµ	pµ	PRON
ejpam-628	272	7	=	=	PUNCT
ejpam-628	273	1	k1k2	k1k2	PROPN
ejpam-628	273	2	−	−	PROPN
ejpam-628	273	3	1	1	NUM
ejpam-628	273	4	>	>	SYM
ejpam-628	273	5	0	0	PUNCT
ejpam-628	274	1	and	and	CCONJ
ejpam-628	274	2	p	p	PROPN
ejpam-628	274	3	+	+	PROPN
ejpam-628	274	4	µ	µ	X
ejpam-628	274	5	=	=	SYM
ejpam-628	274	6	k1	k1	X
ejpam-628	274	7	+	+	X
ejpam-628	274	8	k2	k2	NOUN
ejpam-628	274	9	+	+	CCONJ
ejpam-628	274	10	2k1k2	2k1k2	NUM
ejpam-628	274	11	.	.	PUNCT
ejpam-628	275	1	it	it	PRON
ejpam-628	275	2	follows	follow	VERB
ejpam-628	275	3	that	that	SCONJ
ejpam-628	275	4	µ	µ	NOUN
ejpam-628	275	5	is	be	AUX
ejpam-628	275	6	an	an	DET
ejpam-628	275	7	integer≥	integer≥	NOUN
ejpam-628	275	8	2	2	NUM
ejpam-628	275	9	and	and	CCONJ
ejpam-628	275	10	hence	hence	ADV
ejpam-628	275	11	pµ	pµ	VERB
ejpam-628	275	12	≥	≥	NOUN
ejpam-628	275	13	p	p	NOUN
ejpam-628	275	14	+	+	X
ejpam-628	275	15	µ	µ	NUM
ejpam-628	275	16	,	,	PUNCT
ejpam-628	275	17	i.e.	i.e.	X
ejpam-628	275	18	,	,	PUNCT
ejpam-628	275	19	k1k2	k1k2	PROPN
ejpam-628	275	20	−	−	PROPN
ejpam-628	275	21	1	1	NUM
ejpam-628	275	22	>	>	X
ejpam-628	275	23	k1	k1	PROPN
ejpam-628	275	24	+	+	X
ejpam-628	275	25	k2	k2	PROPN
ejpam-628	275	26	+	+	CCONJ
ejpam-628	275	27	2k1k2	2k1k2	NUM
ejpam-628	275	28	,	,	PUNCT
ejpam-628	275	29	a	a	DET
ejpam-628	275	30	contradiction	contradiction	NOUN
ejpam-628	275	31	(	(	PUNCT
ejpam-628	275	32	since	since	SCONJ
ejpam-628	275	33	k1	k1	PROPN
ejpam-628	275	34	,	,	PUNCT
ejpam-628	275	35	k2	k2	X
ejpam-628	275	36	≥	≥	NOUN
ejpam-628	275	37	1	1	NUM
ejpam-628	275	38	)	)	PUNCT
ejpam-628	275	39	.	.	PUNCT
ejpam-628	276	1	thus	thus	ADV
ejpam-628	276	2	t	t	X
ejpam-628	276	3	=	=	PUNCT
ejpam-628	276	4	1	1	NUM
ejpam-628	276	5	and	and	CCONJ
ejpam-628	276	6	since	since	SCONJ
ejpam-628	276	7	|z(r1)|=	|z(r1)|=	NUM
ejpam-628	276	8	|z(r2)|=	|z(r2)|=	X
ejpam-628	276	9	p	p	X
ejpam-628	276	10	,	,	PUNCT
ejpam-628	276	11	we	we	PRON
ejpam-628	276	12	have	have	VERB
ejpam-628	276	13	p4	p4	ADJ
ejpam-628	276	14	=	=	NOUN
ejpam-628	276	15	p2q1−	p2q1−	NOUN
ejpam-628	276	16	(	(	PUNCT
ejpam-628	276	17	p−	p−	NOUN
ejpam-628	276	18	1)2(q1	1)2(q1	NUM
ejpam-628	277	1	−	−	NOUN
ejpam-628	277	2	1	1	NUM
ejpam-628	277	3	)	)	PUNCT
ejpam-628	277	4	and	and	CCONJ
ejpam-628	277	5	so	so	ADV
ejpam-628	277	6	q1(2p−	q1(2p−	NOUN
ejpam-628	277	7	1	1	X
ejpam-628	277	8	)	)	PUNCT
ejpam-628	277	9	=	=	VERB
ejpam-628	277	10	p4	p4	ADJ
ejpam-628	277	11	−	−	NOUN
ejpam-628	277	12	p2	p2	NOUN
ejpam-628	277	13	+	+	X
ejpam-628	278	1	2p−	2p−	NOUN
ejpam-628	278	2	1	1	NUM
ejpam-628	278	3	.	.	PUNCT
ejpam-628	279	1	thus	thus	ADV
ejpam-628	279	2	2p−	2p−	NUM
ejpam-628	279	3	1	1	NUM
ejpam-628	279	4	is	be	AUX
ejpam-628	279	5	a	a	DET
ejpam-628	279	6	divisor	divisor	NOUN
ejpam-628	279	7	of	of	ADP
ejpam-628	279	8	p2	p2	PROPN
ejpam-628	279	9	−	−	PROPN
ejpam-628	279	10	1	1	NUM
ejpam-628	279	11	,	,	PUNCT
ejpam-628	279	12	i.e.	i.e.	X
ejpam-628	279	13	,	,	PUNCT
ejpam-628	279	14	p2	p2	PROPN
ejpam-628	279	15	−	−	PROPN
ejpam-628	279	16	1	1	NUM
ejpam-628	279	17	=	=	SYM
ejpam-628	279	18	(	(	PUNCT
ejpam-628	279	19	2p−1)a	2p−1)a	NUM
ejpam-628	279	20	for	for	ADP
ejpam-628	279	21	some	some	DET
ejpam-628	279	22	positive	positive	ADJ
ejpam-628	279	23	integer	integer	NOUN
ejpam-628	279	24	a.	a.	NOUN
ejpam-628	279	25	then	then	ADV
ejpam-628	279	26	the	the	DET
ejpam-628	279	27	equation	equation	NOUN
ejpam-628	279	28	p2−2ap+a−1=	p2−2ap+a−1=	PROPN
ejpam-628	279	29	0	0	NUM
ejpam-628	279	30	implies	imply	VERB
ejpam-628	279	31	that	that	SCONJ
ejpam-628	279	32	p	p	PROPN
ejpam-628	279	33	is	be	AUX
ejpam-628	279	34	a	a	DET
ejpam-628	279	35	divisor	divisor	NOUN
ejpam-628	279	36	of	of	ADP
ejpam-628	279	37	a−1	a−1	PROPN
ejpam-628	279	38	,	,	PUNCT
ejpam-628	279	39	i.e.	i.e.	X
ejpam-628	279	40	,	,	PUNCT
ejpam-628	279	41	a−1=	a−1=	NOUN
ejpam-628	279	42	pλ	pλ	NOUN
ejpam-628	279	43	for	for	ADP
ejpam-628	279	44	some	some	DET
ejpam-628	279	45	non	non	ADJ
ejpam-628	279	46	-	-	ADJ
ejpam-628	279	47	negative	negative	ADJ
ejpam-628	279	48	integer	integer	NOUN
ejpam-628	279	49	λ	λ	PROPN
ejpam-628	279	50	.	.	PUNCT
ejpam-628	280	1	it	it	PRON
ejpam-628	280	2	follows	follow	VERB
ejpam-628	280	3	that	that	SCONJ
ejpam-628	280	4	p	p	PROPN
ejpam-628	280	5	and	and	CCONJ
ejpam-628	280	6	λ	λ	PROPN
ejpam-628	280	7	are	be	AUX
ejpam-628	280	8	solutions	solution	NOUN
ejpam-628	280	9	of	of	ADP
ejpam-628	280	10	x2−	x2−	PROPN
ejpam-628	280	11	2ax	2ax	NOUN
ejpam-628	281	1	+	+	CCONJ
ejpam-628	281	2	a−	a−	PROPN
ejpam-628	281	3	1=	1=	X
ejpam-628	281	4	0	0	NUM
ejpam-628	282	1	and	and	CCONJ
ejpam-628	282	2	so	so	ADV
ejpam-628	282	3	p+λ	p+λ	PROPN
ejpam-628	282	4	=	=	SYM
ejpam-628	282	5	2a	2a	NUM
ejpam-628	282	6	.	.	PUNCT
ejpam-628	283	1	if	if	SCONJ
ejpam-628	283	2	λ=	λ=	NOUN
ejpam-628	283	3	1	1	NUM
ejpam-628	283	4	,	,	PUNCT
ejpam-628	283	5	then	then	ADV
ejpam-628	283	6	p	p	NOUN
ejpam-628	283	7	=	=	SYM
ejpam-628	283	8	a−	a−	PROPN
ejpam-628	283	9	1	1	NUM
ejpam-628	283	10	and	and	CCONJ
ejpam-628	283	11	p+1=	p+1=	PROPN
ejpam-628	283	12	2a	2a	NUM
ejpam-628	283	13	and	and	CCONJ
ejpam-628	283	14	hence	hence	ADV
ejpam-628	283	15	a	a	DET
ejpam-628	283	16	=	=	SYM
ejpam-628	283	17	0	0	NUM
ejpam-628	283	18	,	,	PUNCT
ejpam-628	283	19	a	a	DET
ejpam-628	283	20	contradiction	contradiction	NOUN
ejpam-628	283	21	.	.	PUNCT
ejpam-628	284	1	also	also	ADV
ejpam-628	284	2	,	,	PUNCT
ejpam-628	284	3	if	if	SCONJ
ejpam-628	284	4	λ	λ	X
ejpam-628	284	5	>	>	X
ejpam-628	284	6	1	1	NUM
ejpam-628	284	7	,	,	PUNCT
ejpam-628	284	8	then	then	ADV
ejpam-628	284	9	pλ	pλ	PROPN
ejpam-628	284	10	≥	≥	NOUN
ejpam-628	284	11	p+λ	p+λ	PROPN
ejpam-628	284	12	and	and	CCONJ
ejpam-628	284	13	so	so	ADV
ejpam-628	284	14	a	a	DET
ejpam-628	284	15	≤	≤	NUM
ejpam-628	284	16	−1	−1	NOUN
ejpam-628	284	17	,	,	PUNCT
ejpam-628	284	18	a	a	DET
ejpam-628	284	19	contradiction	contradiction	NOUN
ejpam-628	284	20	.	.	PUNCT
ejpam-628	285	1	m.	m.	NOUN
ejpam-628	285	2	behboodi	behboodi	PROPN
ejpam-628	285	3	,	,	PUNCT
ejpam-628	285	4	r.	r.	PROPN
ejpam-628	285	5	beyranvand	beyranvand	PROPN
ejpam-628	285	6	/	/	SYM
ejpam-628	285	7	eur	eur	PROPN
ejpam-628	285	8	.	.	PUNCT
ejpam-628	286	1	j.	j.	PROPN
ejpam-628	286	2	pure	pure	PROPN
ejpam-628	286	3	appl	appl	PROPN
ejpam-628	286	4	.	.	PROPN
ejpam-628	286	5	math	math	PROPN
ejpam-628	286	6	,	,	PUNCT
ejpam-628	286	7	3	3	NUM
ejpam-628	286	8	(	(	PUNCT
ejpam-628	286	9	2010	2010	NUM
ejpam-628	286	10	)	)	PUNCT
ejpam-628	286	11	,	,	PUNCT
ejpam-628	286	12	303	303	NUM
ejpam-628	286	13	-	-	SYM
ejpam-628	286	14	316	316	NUM
ejpam-628	286	15	309	309	NUM
ejpam-628	286	16	finally	finally	ADV
ejpam-628	286	17	,	,	PUNCT
ejpam-628	286	18	if	if	SCONJ
ejpam-628	286	19	λ	λ	X
ejpam-628	286	20	=	=	SYM
ejpam-628	286	21	0	0	NUM
ejpam-628	286	22	,	,	PUNCT
ejpam-628	286	23	then	then	ADV
ejpam-628	286	24	p	p	X
ejpam-628	286	25	=	=	SYM
ejpam-628	286	26	2	2	NUM
ejpam-628	286	27	,	,	PUNCT
ejpam-628	286	28	which	which	PRON
ejpam-628	286	29	yields	yield	VERB
ejpam-628	286	30	q1	q1	PROPN
ejpam-628	286	31	=	=	SYM
ejpam-628	286	32	5	5	NUM
ejpam-628	286	33	,	,	PUNCT
ejpam-628	286	34	i.e.	i.e.	X
ejpam-628	286	35	,	,	PUNCT
ejpam-628	286	36	r	r	NOUN
ejpam-628	286	37	∼=	∼=	PROPN
ejpam-628	286	38	r1	r1	NOUN
ejpam-628	286	39	×	×	NOUN
ejpam-628	286	40	r2	r2	NOUN
ejpam-628	286	41	×	×	NOUN
ejpam-628	286	42	f5	f5	NOUN
ejpam-628	286	43	where	where	SCONJ
ejpam-628	286	44	r1	r1	PROPN
ejpam-628	286	45	and	and	CCONJ
ejpam-628	286	46	r2	r2	PROPN
ejpam-628	286	47	are	be	AUX
ejpam-628	286	48	local	local	ADJ
ejpam-628	286	49	rings	ring	NOUN
ejpam-628	286	50	of	of	ADP
ejpam-628	286	51	order	order	NOUN
ejpam-628	286	52	4	4	NUM
ejpam-628	286	53	with	with	ADP
ejpam-628	286	54	2	2	NUM
ejpam-628	286	55	zero	zero	NUM
ejpam-628	286	56	-	-	PUNCT
ejpam-628	286	57	divisors	divisor	NOUN
ejpam-628	286	58	.	.	PUNCT
ejpam-628	287	1	now	now	ADV
ejpam-628	287	2	by	by	ADP
ejpam-628	287	3	[	[	X
ejpam-628	287	4	2	2	NUM
ejpam-628	287	5	,	,	PUNCT
ejpam-628	287	6	page	page	NOUN
ejpam-628	287	7	687	687	NUM
ejpam-628	287	8	]	]	PUNCT
ejpam-628	287	9	,	,	PUNCT
ejpam-628	287	10	each	each	DET
ejpam-628	287	11	ri	ri	PROPN
ejpam-628	287	12	is	be	AUX
ejpam-628	287	13	isomorphic	isomorphic	ADJ
ejpam-628	287	14	to	to	ADP
ejpam-628	287	15	z4	z4	PROPN
ejpam-628	287	16	or	or	CCONJ
ejpam-628	287	17	z2[x]/(x	z2[x]/(x	NUM
ejpam-628	287	18	2	2	NUM
ejpam-628	287	19	)	)	PUNCT
ejpam-628	287	20	.	.	PUNCT
ejpam-628	288	1	corollary	corollary	ADJ
ejpam-628	288	2	1	1	NUM
ejpam-628	288	3	.	.	PUNCT
ejpam-628	289	1	let	let	VERB
ejpam-628	289	2	r	r	PRON
ejpam-628	289	3	be	be	AUX
ejpam-628	289	4	a	a	DET
ejpam-628	289	5	commutative	commutative	ADJ
ejpam-628	289	6	ring	ring	NOUN
ejpam-628	289	7	with	with	ADP
ejpam-628	289	8	|z(r)|	|z(r)|	PROPN
ejpam-628	289	9	=	=	SYM
ejpam-628	289	10	p	p	PROPN
ejpam-628	289	11	,	,	PUNCT
ejpam-628	289	12	where	where	SCONJ
ejpam-628	289	13	p	p	NOUN
ejpam-628	289	14	is	be	AUX
ejpam-628	289	15	a	a	DET
ejpam-628	289	16	prime	prime	ADJ
ejpam-628	289	17	number	number	NOUN
ejpam-628	289	18	.	.	PUNCT
ejpam-628	290	1	then	then	ADV
ejpam-628	290	2	r	r	NOUN
ejpam-628	290	3	is	be	AUX
ejpam-628	290	4	isomorphic	isomorphic	ADJ
ejpam-628	290	5	to	to	ADP
ejpam-628	290	6	one	one	NUM
ejpam-628	290	7	of	of	ADP
ejpam-628	290	8	the	the	DET
ejpam-628	290	9	rings	ring	NOUN
ejpam-628	290	10	zp2	zp2	PROPN
ejpam-628	290	11	,	,	PUNCT
ejpam-628	290	12	zp[x]/(x	zp[x]/(x	PROPN
ejpam-628	290	13	2	2	NUM
ejpam-628	290	14	)	)	PUNCT
ejpam-628	290	15	or	or	CCONJ
ejpam-628	290	16	fq1	fq1	NUM
ejpam-628	290	17	×	×	NOUN
ejpam-628	290	18	.	.	PUNCT
ejpam-628	290	19	.	.	PUNCT
ejpam-628	291	1	.×	.×	PROPN
ejpam-628	291	2	fqt	fqt	VERB
ejpam-628	291	3	where	where	SCONJ
ejpam-628	291	4	p	p	NOUN
ejpam-628	291	5	=	=	X
ejpam-628	291	6	q1q2	q1q2	PROPN
ejpam-628	291	7	.	.	PUNCT
ejpam-628	291	8	.	.	PUNCT
ejpam-628	291	9	.	.	PUNCT
ejpam-628	292	1	qt	qt	INTJ
ejpam-628	292	2	−	−	PROPN
ejpam-628	292	3	(	(	PUNCT
ejpam-628	292	4	q1−	q1−	VERB
ejpam-628	292	5	1)(q2−	1)(q2−	NUM
ejpam-628	292	6	1	1	NUM
ejpam-628	292	7	)	)	PUNCT
ejpam-628	292	8	.	.	PUNCT
ejpam-628	292	9	.	.	PUNCT
ejpam-628	292	10	.	.	PUNCT
ejpam-628	293	1	(	(	PUNCT
ejpam-628	293	2	qt	qt	INTJ
ejpam-628	293	3	−	−	NOUN
ejpam-628	293	4	1	1	NUM
ejpam-628	293	5	)	)	PUNCT
ejpam-628	293	6	.	.	PUNCT
ejpam-628	294	1	proof	proof	NOUN
ejpam-628	294	2	.	.	PUNCT
ejpam-628	295	1	if	if	SCONJ
ejpam-628	295	2	r	r	NOUN
ejpam-628	295	3	is	be	AUX
ejpam-628	295	4	a	a	DET
ejpam-628	295	5	local	local	ADJ
ejpam-628	295	6	ring	ring	NOUN
ejpam-628	295	7	,	,	PUNCT
ejpam-628	295	8	then	then	ADV
ejpam-628	295	9	by	by	ADP
ejpam-628	295	10	lemma	lemma	PROPN
ejpam-628	295	11	1	1	NUM
ejpam-628	295	12	,	,	PUNCT
ejpam-628	295	13	|r|	|r|	NOUN
ejpam-628	295	14	=	=	NOUN
ejpam-628	295	15	p2	p2	PROPN
ejpam-628	295	16	and	and	CCONJ
ejpam-628	295	17	hence	hence	ADV
ejpam-628	295	18	by	by	ADP
ejpam-628	295	19	[	[	X
ejpam-628	295	20	2	2	NUM
ejpam-628	295	21	,	,	PUNCT
ejpam-628	295	22	page	page	NOUN
ejpam-628	295	23	687	687	NUM
ejpam-628	295	24	]	]	PUNCT
ejpam-628	295	25	,	,	PUNCT
ejpam-628	295	26	r	r	NOUN
ejpam-628	295	27	is	be	AUX
ejpam-628	295	28	isomorphic	isomorphic	ADJ
ejpam-628	295	29	to	to	ADP
ejpam-628	295	30	zp2	zp2	PROPN
ejpam-628	295	31	or	or	CCONJ
ejpam-628	295	32	zp[x]/(x	zp[x]/(x	PROPN
ejpam-628	295	33	2	2	NUM
ejpam-628	295	34	)	)	PUNCT
ejpam-628	295	35	.	.	PUNCT
ejpam-628	296	1	if	if	SCONJ
ejpam-628	296	2	r	r	NOUN
ejpam-628	296	3	is	be	AUX
ejpam-628	296	4	not	not	PART
ejpam-628	296	5	local	local	ADJ
ejpam-628	296	6	,	,	PUNCT
ejpam-628	296	7	then	then	ADV
ejpam-628	296	8	by	by	ADP
ejpam-628	296	9	theorem	theorem	NOUN
ejpam-628	296	10	4	4	NUM
ejpam-628	296	11	,	,	PUNCT
ejpam-628	296	12	r	r	NOUN
ejpam-628	296	13	∼=	∼=	NOUN
ejpam-628	296	14	fq1	fq1	NUM
ejpam-628	296	15	×	×	NOUN
ejpam-628	296	16	.	.	PUNCT
ejpam-628	296	17	.	.	PUNCT
ejpam-628	297	1	.	.	PUNCT
ejpam-628	298	1	×	×	NOUN
ejpam-628	298	2	fqt	fqt	NOUN
ejpam-628	298	3	where	where	SCONJ
ejpam-628	298	4	t	t	PROPN
ejpam-628	298	5	≥	≥	NOUN
ejpam-628	298	6	2	2	NUM
ejpam-628	298	7	and	and	CCONJ
ejpam-628	298	8	p	p	NOUN
ejpam-628	298	9	=	=	SYM
ejpam-628	298	10	q1q2	q1q2	PROPN
ejpam-628	298	11	.	.	PUNCT
ejpam-628	298	12	.	.	PUNCT
ejpam-628	298	13	.	.	PUNCT
ejpam-628	299	1	qt	qt	INTJ
ejpam-628	299	2	−	−	PROPN
ejpam-628	299	3	(	(	PUNCT
ejpam-628	299	4	q1−	q1−	VERB
ejpam-628	299	5	1)(q2−	1)(q2−	NUM
ejpam-628	299	6	1	1	NUM
ejpam-628	299	7	)	)	PUNCT
ejpam-628	299	8	.	.	PUNCT
ejpam-628	299	9	.	.	PUNCT
ejpam-628	299	10	.	.	PUNCT
ejpam-628	300	1	(	(	PUNCT
ejpam-628	300	2	qt	qt	INTJ
ejpam-628	300	3	−	−	NOUN
ejpam-628	300	4	1	1	NUM
ejpam-628	300	5	)	)	PUNCT
ejpam-628	300	6	.	.	PUNCT
ejpam-628	301	1	in	in	ADP
ejpam-628	301	2	[	[	X
ejpam-628	301	3	5	5	NUM
ejpam-628	301	4	]	]	PUNCT
ejpam-628	301	5	,	,	PUNCT
ejpam-628	301	6	it	it	PRON
ejpam-628	301	7	was	be	AUX
ejpam-628	301	8	shown	show	VERB
ejpam-628	301	9	that	that	SCONJ
ejpam-628	301	10	any	any	DET
ejpam-628	301	11	commutative	commutative	ADJ
ejpam-628	301	12	ring	ring	NOUN
ejpam-628	301	13	r	r	NOUN
ejpam-628	301	14	with	with	ADP
ejpam-628	301	15	m	m	PROPN
ejpam-628	301	16	zero	zero	NUM
ejpam-628	301	17	-	-	PUNCT
ejpam-628	301	18	divisors	divisor	NOUN
ejpam-628	301	19	has	have	VERB
ejpam-628	301	20	m2	m2	PROPN
ejpam-628	301	21	or	or	CCONJ
ejpam-628	301	22	fewer	few	ADJ
ejpam-628	301	23	elements	element	NOUN
ejpam-628	301	24	.	.	PUNCT
ejpam-628	302	1	it	it	PRON
ejpam-628	302	2	was	be	AUX
ejpam-628	302	3	proved	prove	VERB
ejpam-628	302	4	in	in	ADP
ejpam-628	302	5	[	[	X
ejpam-628	302	6	7	7	X
ejpam-628	302	7	]	]	PUNCT
ejpam-628	302	8	that	that	SCONJ
ejpam-628	302	9	if	if	SCONJ
ejpam-628	302	10	|z(r)|	|z(r)|	PROPN
ejpam-628	302	11	=	=	PUNCT
ejpam-628	302	12	m	m	PROPN
ejpam-628	302	13	and	and	CCONJ
ejpam-628	302	14	|r|	|r|	NOUN
ejpam-628	302	15	=	=	PROPN
ejpam-628	302	16	m2	m2	PROPN
ejpam-628	302	17	,	,	PUNCT
ejpam-628	302	18	then	then	ADV
ejpam-628	302	19	m	m	VERB
ejpam-628	302	20	=	=	ADJ
ejpam-628	302	21	pr	pr	NOUN
ejpam-628	302	22	for	for	ADP
ejpam-628	302	23	some	some	DET
ejpam-628	302	24	integer	integer	NOUN
ejpam-628	302	25	r	r	NOUN
ejpam-628	302	26	≥	≥	NUM
ejpam-628	302	27	1	1	NUM
ejpam-628	302	28	and	and	CCONJ
ejpam-628	302	29	some	some	DET
ejpam-628	302	30	prime	prime	ADJ
ejpam-628	302	31	p.	p.	NOUN
ejpam-628	302	32	these	these	DET
ejpam-628	302	33	rings	ring	NOUN
ejpam-628	302	34	were	be	AUX
ejpam-628	302	35	categorized	categorize	VERB
ejpam-628	302	36	in	in	ADP
ejpam-628	302	37	[	[	X
ejpam-628	302	38	4	4	X
ejpam-628	302	39	]	]	PUNCT
ejpam-628	302	40	by	by	ADP
ejpam-628	302	41	the	the	DET
ejpam-628	302	42	use	use	NOUN
ejpam-628	302	43	of	of	ADP
ejpam-628	302	44	two	two	NUM
ejpam-628	302	45	constructions	construction	NOUN
ejpam-628	302	46	.	.	PUNCT
ejpam-628	303	1	when	when	SCONJ
ejpam-628	303	2	the	the	DET
ejpam-628	303	3	ring	ring	NOUN
ejpam-628	303	4	r	r	NOUN
ejpam-628	303	5	is	be	AUX
ejpam-628	303	6	commutative	commutative	ADJ
ejpam-628	303	7	with	with	ADP
ejpam-628	303	8	1	1	NUM
ejpam-628	303	9	,	,	PUNCT
ejpam-628	303	10	then	then	ADV
ejpam-628	303	11	there	there	PRON
ejpam-628	303	12	are	be	VERB
ejpam-628	303	13	only	only	ADV
ejpam-628	303	14	two	two	NUM
ejpam-628	303	15	such	such	ADJ
ejpam-628	303	16	rings	ring	NOUN
ejpam-628	303	17	(	(	PUNCT
ejpam-628	303	18	up	up	ADP
ejpam-628	303	19	to	to	ADP
ejpam-628	303	20	isomorphism	isomorphism	NOUN
ejpam-628	303	21	)	)	PUNCT
ejpam-628	303	22	for	for	ADP
ejpam-628	303	23	m	m	PROPN
ejpam-628	303	24	=	=	VERB
ejpam-628	303	25	pr	pr	NOUN
ejpam-628	303	26	:	:	PUNCT
ejpam-628	303	27	fpr	fpr	PRON
ejpam-628	303	28	[	[	X
ejpam-628	303	29	x]/(x2	x]/(x2	X
ejpam-628	303	30	)	)	PUNCT
ejpam-628	303	31	and	and	CCONJ
ejpam-628	303	32	zp2[x]/	zp2[x]/	NOUN
ejpam-628	303	33	(	(	PUNCT
ejpam-628	303	34	f	f	PROPN
ejpam-628	303	35	(	(	PUNCT
ejpam-628	303	36	x	x	NOUN
ejpam-628	303	37	)	)	PUNCT
ejpam-628	303	38	)	)	PUNCT
ejpam-628	303	39	,	,	PUNCT
ejpam-628	303	40	where	where	SCONJ
ejpam-628	303	41	f	f	PROPN
ejpam-628	303	42	(	(	PUNCT
ejpam-628	303	43	x	x	X
ejpam-628	303	44	)	)	PUNCT
ejpam-628	303	45	is	be	AUX
ejpam-628	303	46	an	an	DET
ejpam-628	303	47	irreducible	irreducible	ADJ
ejpam-628	303	48	polynomial	polynomial	NOUN
ejpam-628	303	49	of	of	ADP
ejpam-628	303	50	degree	degree	NOUN
ejpam-628	303	51	r	r	NOUN
ejpam-628	303	52	over	over	ADP
ejpam-628	303	53	fp	fp	NOUN
ejpam-628	303	54	.	.	PUNCT
ejpam-628	304	1	the	the	DET
ejpam-628	304	2	rings	ring	NOUN
ejpam-628	304	3	from	from	ADP
ejpam-628	304	4	the	the	DET
ejpam-628	304	5	second	second	ADJ
ejpam-628	304	6	construction	construction	NOUN
ejpam-628	304	7	in	in	ADP
ejpam-628	304	8	[	[	X
ejpam-628	304	9	4	4	NUM
ejpam-628	304	10	]	]	PUNCT
ejpam-628	304	11	are	be	AUX
ejpam-628	304	12	shown	show	VERB
ejpam-628	304	13	by	by	ADP
ejpam-628	304	14	raghavendran	raghavendran	NOUN
ejpam-628	304	15	[	[	X
ejpam-628	304	16	9	9	NUM
ejpam-628	304	17	]	]	PUNCT
ejpam-628	304	18	to	to	PART
ejpam-628	304	19	all	all	PRON
ejpam-628	304	20	be	be	AUX
ejpam-628	304	21	isomorphic	isomorphic	ADJ
ejpam-628	304	22	to	to	ADP
ejpam-628	304	23	the	the	DET
ejpam-628	304	24	ring	ring	NOUN
ejpam-628	304	25	zp2[x]/	zp2[x]/	NOUN
ejpam-628	304	26	(	(	PUNCT
ejpam-628	304	27	f	f	PROPN
ejpam-628	304	28	(	(	PUNCT
ejpam-628	304	29	x	x	NOUN
ejpam-628	304	30	)	)	PUNCT
ejpam-628	304	31	)	)	PUNCT
ejpam-628	304	32	given	give	VERB
ejpam-628	304	33	above	above	ADV
ejpam-628	304	34	,	,	PUNCT
ejpam-628	304	35	which	which	PRON
ejpam-628	304	36	is	be	AUX
ejpam-628	304	37	called	call	VERB
ejpam-628	304	38	the	the	DET
ejpam-628	304	39	galois	galois	PROPN
ejpam-628	304	40	ring	ring	NOUN
ejpam-628	304	41	of	of	ADP
ejpam-628	304	42	order	order	NOUN
ejpam-628	304	43	p2r	p2r	NOUN
ejpam-628	304	44	and	and	CCONJ
ejpam-628	304	45	characteristic	characteristic	ADJ
ejpam-628	304	46	p2	p2	NOUN
ejpam-628	304	47	,	,	PUNCT
ejpam-628	304	48	denoted	denote	VERB
ejpam-628	304	49	gr(p2r	gr(p2r	NUM
ejpam-628	304	50	,	,	PUNCT
ejpam-628	304	51	p2	p2	PROPN
ejpam-628	304	52	)	)	PUNCT
ejpam-628	304	53	.	.	PUNCT
ejpam-628	305	1	let	let	VERB
ejpam-628	305	2	p	p	PRON
ejpam-628	305	3	be	be	AUX
ejpam-628	305	4	a	a	DET
ejpam-628	305	5	prime	prime	ADJ
ejpam-628	305	6	number	number	NOUN
ejpam-628	305	7	.	.	PUNCT
ejpam-628	306	1	we	we	PRON
ejpam-628	306	2	write	write	VERB
ejpam-628	306	3	σm	σm	NOUN
ejpam-628	306	4	for	for	ADP
ejpam-628	306	5	a	a	DET
ejpam-628	306	6	set	set	NOUN
ejpam-628	306	7	of	of	ADP
ejpam-628	306	8	coset	coset	NOUN
ejpam-628	306	9	representatives	representative	NOUN
ejpam-628	306	10	of	of	ADP
ejpam-628	306	11	(	(	PUNCT
ejpam-628	306	12	f∗p	f∗p	X
ejpam-628	306	13	)	)	PUNCT
ejpam-628	306	14	m	m	PROPN
ejpam-628	306	15	in	in	ADP
ejpam-628	306	16	f∗p	f∗p	NUM
ejpam-628	306	17	,	,	PUNCT
ejpam-628	306	18	and	and	CCONJ
ejpam-628	306	19	σ0	σ0	NOUN
ejpam-628	306	20	m	m	NOUN
ejpam-628	306	21	=	=	NOUN
ejpam-628	306	22	σm	σm	X
ejpam-628	306	23	∪	∪	X
ejpam-628	306	24	{	{	PUNCT
ejpam-628	306	25	0	0	NUM
ejpam-628	306	26	}	}	PUNCT
ejpam-628	306	27	.	.	PUNCT
ejpam-628	307	1	since	since	SCONJ
ejpam-628	307	2	f∗p	f∗p	NUM
ejpam-628	307	3	is	be	AUX
ejpam-628	307	4	cyclic	cyclic	ADJ
ejpam-628	307	5	,	,	PUNCT
ejpam-628	307	6	|σm|	|σm|	PROPN
ejpam-628	307	7	=	=	SYM
ejpam-628	307	8	(	(	PUNCT
ejpam-628	307	9	m	m	PROPN
ejpam-628	307	10	,	,	PUNCT
ejpam-628	307	11	p−	p−	NOUN
ejpam-628	307	12	1	1	NUM
ejpam-628	307	13	)	)	PUNCT
ejpam-628	307	14	.	.	PUNCT
ejpam-628	308	1	corollary	corollary	ADJ
ejpam-628	308	2	2	2	NUM
ejpam-628	308	3	.	.	PUNCT
ejpam-628	309	1	let	let	VERB
ejpam-628	309	2	r	r	PRON
ejpam-628	309	3	be	be	AUX
ejpam-628	309	4	a	a	DET
ejpam-628	309	5	commutative	commutative	ADJ
ejpam-628	309	6	ring	ring	NOUN
ejpam-628	309	7	with	with	ADP
ejpam-628	309	8	|z(r)|	|z(r)|	PROPN
ejpam-628	309	9	=	=	PUNCT
ejpam-628	309	10	p2	p2	NOUN
ejpam-628	309	11	,	,	PUNCT
ejpam-628	309	12	where	where	SCONJ
ejpam-628	309	13	p	p	NOUN
ejpam-628	309	14	is	be	AUX
ejpam-628	309	15	a	a	DET
ejpam-628	309	16	prime	prime	ADJ
ejpam-628	309	17	number	number	NOUN
ejpam-628	309	18	.	.	PUNCT
ejpam-628	310	1	then	then	ADV
ejpam-628	310	2	r	r	NOUN
ejpam-628	310	3	is	be	AUX
ejpam-628	310	4	isomorphic	isomorphic	ADJ
ejpam-628	310	5	to	to	ADP
ejpam-628	310	6	one	one	NUM
ejpam-628	310	7	of	of	ADP
ejpam-628	310	8	the	the	DET
ejpam-628	310	9	rings	ring	NOUN
ejpam-628	310	10	zp3	zp3	PROPN
ejpam-628	310	11	,	,	PUNCT
ejpam-628	310	12	fp[x	fp[x	PROPN
ejpam-628	310	13	,	,	PUNCT
ejpam-628	310	14	y]/(x	y]/(x	PROPN
ejpam-628	310	15	,	,	PUNCT
ejpam-628	310	16	y)2	y)2	NOUN
ejpam-628	310	17	,	,	PUNCT
ejpam-628	310	18	fp[x]/(x	fp[x]/(x	NOUN
ejpam-628	310	19	3	3	NUM
ejpam-628	310	20	)	)	PUNCT
ejpam-628	310	21	,	,	PUNCT
ejpam-628	310	22	zp2[x]/(px	zp2[x]/(px	PROPN
ejpam-628	310	23	,	,	PUNCT
ejpam-628	310	24	x2	x2	PROPN
ejpam-628	310	25	−	−	PROPN
ejpam-628	310	26	ǫp	ǫp	NOUN
ejpam-628	310	27	)	)	PUNCT
ejpam-628	311	1	where	where	SCONJ
ejpam-628	311	2	ǫ	ǫ	PROPN
ejpam-628	311	3	∈	∈	PROPN
ejpam-628	311	4	σ0	σ0	NOUN
ejpam-628	311	5	2	2	NUM
ejpam-628	311	6	,	,	PUNCT
ejpam-628	311	7	fp2[x]/(x2	fp2[x]/(x2	PROPN
ejpam-628	311	8	)	)	PUNCT
ejpam-628	311	9	,	,	PUNCT
ejpam-628	311	10	the	the	DET
ejpam-628	311	11	galois	galois	PROPN
ejpam-628	311	12	ring	ring	NOUN
ejpam-628	311	13	gr(p4	gr(p4	NOUN
ejpam-628	311	14	,	,	PUNCT
ejpam-628	311	15	p2	p2	PROPN
ejpam-628	311	16	)	)	PUNCT
ejpam-628	311	17	or	or	CCONJ
ejpam-628	311	18	fq1	fq1	NUM
ejpam-628	311	19	×	×	NOUN
ejpam-628	311	20	.	.	PUNCT
ejpam-628	311	21	.	.	PUNCT
ejpam-628	312	1	.×	.×	PROPN
ejpam-628	312	2	fqt	fqt	PROPN
ejpam-628	312	3	where	where	SCONJ
ejpam-628	312	4	t	t	PROPN
ejpam-628	312	5	≥	≥	NOUN
ejpam-628	312	6	2	2	NUM
ejpam-628	312	7	and	and	CCONJ
ejpam-628	312	8	p2	p2	PROPN
ejpam-628	312	9	=	=	SYM
ejpam-628	312	10	q1q2	q1q2	PROPN
ejpam-628	312	11	.	.	PUNCT
ejpam-628	312	12	.	.	PUNCT
ejpam-628	312	13	.	.	PUNCT
ejpam-628	313	1	qt	qt	INTJ
ejpam-628	313	2	−	−	PROPN
ejpam-628	313	3	(	(	PUNCT
ejpam-628	313	4	q1−	q1−	VERB
ejpam-628	313	5	1)(q2−	1)(q2−	NUM
ejpam-628	313	6	1	1	NUM
ejpam-628	313	7	)	)	PUNCT
ejpam-628	313	8	.	.	PUNCT
ejpam-628	313	9	.	.	PUNCT
ejpam-628	313	10	.	.	PUNCT
ejpam-628	314	1	(	(	PUNCT
ejpam-628	314	2	qt	qt	INTJ
ejpam-628	314	3	−	−	NOUN
ejpam-628	314	4	1	1	NUM
ejpam-628	314	5	)	)	PUNCT
ejpam-628	314	6	.	.	PUNCT
ejpam-628	315	1	proof	proof	NOUN
ejpam-628	315	2	.	.	PUNCT
ejpam-628	316	1	suppose	suppose	VERB
ejpam-628	316	2	that	that	SCONJ
ejpam-628	316	3	r	r	NOUN
ejpam-628	316	4	is	be	AUX
ejpam-628	316	5	a	a	DET
ejpam-628	316	6	local	local	ADJ
ejpam-628	316	7	ring	ring	NOUN
ejpam-628	316	8	with	with	ADP
ejpam-628	316	9	|z(r)|	|z(r)|	PROPN
ejpam-628	316	10	=	=	PUNCT
ejpam-628	316	11	p2	p2	PROPN
ejpam-628	316	12	.	.	PUNCT
ejpam-628	317	1	then	then	ADV
ejpam-628	317	2	by	by	ADP
ejpam-628	317	3	lemma	lemma	PROPN
ejpam-628	317	4	1	1	NUM
ejpam-628	317	5	,	,	PUNCT
ejpam-628	317	6	|r|	|r|	NOUN
ejpam-628	317	7	=	=	PROPN
ejpam-628	317	8	p3	p3	PROPN
ejpam-628	317	9	or	or	CCONJ
ejpam-628	317	10	p4	p4	ADJ
ejpam-628	317	11	.	.	PUNCT
ejpam-628	318	1	if	if	SCONJ
ejpam-628	318	2	|r|	|r|	PROPN
ejpam-628	318	3	=	=	SYM
ejpam-628	318	4	p3	p3	PROPN
ejpam-628	318	5	,	,	PUNCT
ejpam-628	318	6	then	then	ADV
ejpam-628	318	7	by	by	ADP
ejpam-628	318	8	[	[	PUNCT
ejpam-628	318	9	2	2	NUM
ejpam-628	318	10	,	,	PUNCT
ejpam-628	318	11	p.687	p.687	NOUN
ejpam-628	318	12	]	]	PUNCT
ejpam-628	318	13	,	,	PUNCT
ejpam-628	318	14	r	r	NOUN
ejpam-628	318	15	is	be	AUX
ejpam-628	318	16	isomorphic	isomorphic	ADJ
ejpam-628	318	17	to	to	ADP
ejpam-628	318	18	one	one	NUM
ejpam-628	318	19	of	of	ADP
ejpam-628	318	20	the	the	DET
ejpam-628	318	21	rings	ring	NOUN
ejpam-628	318	22	zp3	zp3	PROPN
ejpam-628	318	23	,	,	PUNCT
ejpam-628	318	24	fp[x	fp[x	PROPN
ejpam-628	318	25	,	,	PUNCT
ejpam-628	318	26	y]/(x	y]/(x	PROPN
ejpam-628	318	27	,	,	PUNCT
ejpam-628	318	28	y)2	y)2	NOUN
ejpam-628	318	29	,	,	PUNCT
ejpam-628	318	30	fp[x]/(x	fp[x]/(x	NOUN
ejpam-628	318	31	3	3	NUM
ejpam-628	318	32	)	)	PUNCT
ejpam-628	318	33	,	,	PUNCT
ejpam-628	318	34	zp2[x]/(px	zp2[x]/(px	PROPN
ejpam-628	318	35	,	,	PUNCT
ejpam-628	318	36	x2	x2	PROPN
ejpam-628	318	37	−	−	PROPN
ejpam-628	318	38	ǫp	ǫp	NOUN
ejpam-628	318	39	)	)	PUNCT
ejpam-628	319	1	where	where	SCONJ
ejpam-628	319	2	ǫ	ǫ	PROPN
ejpam-628	319	3	∈	∈	PROPN
ejpam-628	319	4	σ0	σ0	NOUN
ejpam-628	319	5	2	2	NUM
ejpam-628	319	6	.	.	PUNCT
ejpam-628	319	7	if	if	SCONJ
ejpam-628	319	8	|r|	|r|	NOUN
ejpam-628	319	9	=	=	SYM
ejpam-628	319	10	p4	p4	ADJ
ejpam-628	319	11	,	,	PUNCT
ejpam-628	319	12	then	then	ADV
ejpam-628	319	13	by	by	ADP
ejpam-628	319	14	[	[	PUNCT
ejpam-628	319	15	9	9	NUM
ejpam-628	319	16	,	,	PUNCT
ejpam-628	319	17	theorem	theorem	VERB
ejpam-628	319	18	12	12	NUM
ejpam-628	319	19	]	]	PUNCT
ejpam-628	319	20	,	,	PUNCT
ejpam-628	319	21	r	r	NOUN
ejpam-628	319	22	is	be	AUX
ejpam-628	319	23	isomorphic	isomorphic	ADJ
ejpam-628	319	24	to	to	ADP
ejpam-628	319	25	the	the	DET
ejpam-628	319	26	galois	galois	PROPN
ejpam-628	319	27	ring	ring	NOUN
ejpam-628	319	28	gr(p4	gr(p4	NOUN
ejpam-628	319	29	,	,	PUNCT
ejpam-628	319	30	p2	p2	PROPN
ejpam-628	319	31	)	)	PUNCT
ejpam-628	319	32	or	or	CCONJ
ejpam-628	319	33	fp2[x]/(x2	fp2[x]/(x2	NOUN
ejpam-628	319	34	)	)	PUNCT
ejpam-628	319	35	.	.	PUNCT
ejpam-628	320	1	now	now	ADV
ejpam-628	320	2	suppose	suppose	VERB
ejpam-628	320	3	that	that	SCONJ
ejpam-628	320	4	r	r	NOUN
ejpam-628	320	5	is	be	AUX
ejpam-628	320	6	not	not	PART
ejpam-628	320	7	a	a	DET
ejpam-628	320	8	local	local	ADJ
ejpam-628	320	9	ring	ring	NOUN
ejpam-628	320	10	.	.	PUNCT
ejpam-628	321	1	then	then	ADV
ejpam-628	321	2	by	by	ADP
ejpam-628	321	3	theorem	theorem	NOUN
ejpam-628	321	4	2	2	NUM
ejpam-628	321	5	,	,	PUNCT
ejpam-628	321	6	r∼=	r∼=	ADV
ejpam-628	321	7	fq1	fq1	ADV
ejpam-628	321	8	×	×	NOUN
ejpam-628	321	9	.	.	PUNCT
ejpam-628	321	10	.	.	PUNCT
ejpam-628	322	1	.×	.×	PROPN
ejpam-628	322	2	fqt	fqt	PROPN
ejpam-628	322	3	with	with	ADP
ejpam-628	322	4	p2	p2	PROPN
ejpam-628	322	5	=	=	SYM
ejpam-628	322	6	q1q2	q1q2	PROPN
ejpam-628	322	7	.	.	PUNCT
ejpam-628	322	8	.	.	PUNCT
ejpam-628	322	9	.	.	PUNCT
ejpam-628	323	1	qt−	qt−	PUNCT
ejpam-628	323	2	(	(	PUNCT
ejpam-628	323	3	q1−1)(q2−1	q1−1)(q2−1	NOUN
ejpam-628	323	4	)	)	PUNCT
ejpam-628	323	5	.	.	PUNCT
ejpam-628	323	6	.	.	PUNCT
ejpam-628	323	7	.	.	PUNCT
ejpam-628	324	1	(	(	PUNCT
ejpam-628	324	2	qt−1	qt−1	PROPN
ejpam-628	324	3	)	)	PUNCT
ejpam-628	324	4	.	.	PUNCT
ejpam-628	325	1	if	if	SCONJ
ejpam-628	325	2	r	r	NOUN
ejpam-628	325	3	is	be	AUX
ejpam-628	325	4	a	a	DET
ejpam-628	325	5	finite	finite	ADJ
ejpam-628	325	6	ring	ring	NOUN
ejpam-628	325	7	then	then	ADV
ejpam-628	325	8	its	its	PRON
ejpam-628	325	9	additive	additive	ADJ
ejpam-628	325	10	group	group	NOUN
ejpam-628	325	11	is	be	AUX
ejpam-628	325	12	a	a	DET
ejpam-628	325	13	finite	finite	ADJ
ejpam-628	325	14	abelian	abelian	ADJ
ejpam-628	325	15	group	group	NOUN
ejpam-628	325	16	and	and	CCONJ
ejpam-628	325	17	is	be	AUX
ejpam-628	325	18	thus	thus	ADV
ejpam-628	325	19	a	a	DET
ejpam-628	325	20	direct	direct	ADJ
ejpam-628	325	21	product	product	NOUN
ejpam-628	325	22	of	of	ADP
ejpam-628	325	23	cyclic	cyclic	ADJ
ejpam-628	325	24	groups	group	NOUN
ejpam-628	325	25	.	.	PUNCT
ejpam-628	326	1	suppose	suppose	VERB
ejpam-628	326	2	these	these	PRON
ejpam-628	326	3	have	have	VERB
ejpam-628	326	4	generators	generator	NOUN
ejpam-628	326	5	a1,	a1,	NOUN
ejpam-628	326	6	...	...	PUNCT
ejpam-628	326	7	,an	,an	PUNCT
ejpam-628	326	8	of	of	ADP
ejpam-628	326	9	orders	order	NOUN
ejpam-628	326	10	m1,	m1,	NOUN
ejpam-628	326	11	...	...	PUNCT
ejpam-628	326	12	,mn	,mn	PROPN
ejpam-628	326	13	.	.	PUNCT
ejpam-628	327	1	then	then	ADV
ejpam-628	327	2	the	the	DET
ejpam-628	327	3	ring	ring	NOUN
ejpam-628	327	4	structure	structure	NOUN
ejpam-628	327	5	is	be	AUX
ejpam-628	327	6	determined	determine	VERB
ejpam-628	327	7	by	by	ADP
ejpam-628	327	8	the	the	DET
ejpam-628	327	9	n2	n2	ADJ
ejpam-628	327	10	products	product	NOUN
ejpam-628	327	11	aia	aia	PROPN
ejpam-628	327	12	j	j	PROPN
ejpam-628	327	13	=	=	PROPN
ejpam-628	328	1	n	n	PROPN
ejpam-628	328	2	∑	∑	PUNCT
ejpam-628	328	3	k=1	k=1	PROPN
ejpam-628	328	4	wi	wi	PROPN
ejpam-628	328	5	jkak	jkak	PROPN
ejpam-628	328	6	with	with	ADP
ejpam-628	328	7	wi	wi	PROPN
ejpam-628	328	8	jk	jk	PROPN
ejpam-628	328	9	∈	∈	PROPN
ejpam-628	328	10	zmk	zmk	NOUN
ejpam-628	328	11	and	and	CCONJ
ejpam-628	328	12	thus	thus	ADV
ejpam-628	328	13	by	by	ADP
ejpam-628	328	14	the	the	DET
ejpam-628	328	15	n3	n3	NOUN
ejpam-628	328	16	structure	structure	NOUN
ejpam-628	328	17	constants	constant	VERB
ejpam-628	328	18	wi	wi	PROPN
ejpam-628	328	19	jk	jk	PROPN
ejpam-628	328	20	for	for	ADP
ejpam-628	328	21	1≤	1≤	PROPN
ejpam-628	328	22	i	i	PROPN
ejpam-628	328	23	,	,	PUNCT
ejpam-628	328	24	j	j	PROPN
ejpam-628	328	25	,	,	PUNCT
ejpam-628	328	26	k	k	PROPN
ejpam-628	328	27	≤	≤	PROPN
ejpam-628	328	28	n.	n.	NOUN
ejpam-628	328	29	thus	thus	ADV
ejpam-628	328	30	we	we	PRON
ejpam-628	328	31	introduce	introduce	VERB
ejpam-628	328	32	a	a	DET
ejpam-628	328	33	convenient	convenient	ADJ
ejpam-628	328	34	notation	notation	NOUN
ejpam-628	328	35	,	,	PUNCT
ejpam-628	328	36	for	for	ADP
ejpam-628	328	37	giving	give	VERB
ejpam-628	328	38	the	the	DET
ejpam-628	328	39	structure	structure	NOUN
ejpam-628	328	40	of	of	ADP
ejpam-628	328	41	a	a	DET
ejpam-628	328	42	finite	finite	ADJ
ejpam-628	328	43	ring	ring	NOUN
ejpam-628	328	44	.	.	PUNCT
ejpam-628	329	1	a	a	DET
ejpam-628	329	2	presentation	presentation	NOUN
ejpam-628	329	3	for	for	ADP
ejpam-628	329	4	a	a	DET
ejpam-628	329	5	finite	finite	ADJ
ejpam-628	329	6	ring	ring	NOUN
ejpam-628	329	7	r	r	NOUN
ejpam-628	329	8	consists	consist	VERB
ejpam-628	329	9	of	of	ADP
ejpam-628	329	10	a	a	DET
ejpam-628	329	11	set	set	NOUN
ejpam-628	329	12	of	of	ADP
ejpam-628	329	13	generators	generator	NOUN
ejpam-628	329	14	a1	a1	PROPN
ejpam-628	329	15	,	,	PUNCT
ejpam-628	329	16	a2	a2	PROPN
ejpam-628	329	17	,	,	PUNCT
ejpam-628	329	18	.	.	PUNCT
ejpam-628	329	19	.	.	PUNCT
ejpam-628	330	1	.	.	PUNCT
ejpam-628	331	1	,	,	PUNCT
ejpam-628	331	2	an	an	PRON
ejpam-628	331	3	of	of	ADP
ejpam-628	331	4	the	the	DET
ejpam-628	331	5	additive	additive	ADJ
ejpam-628	331	6	group	group	NOUN
ejpam-628	331	7	of	of	ADP
ejpam-628	331	8	r	r	NOUN
ejpam-628	331	9	together	together	ADV
ejpam-628	331	10	with	with	ADP
ejpam-628	331	11	relations	relation	NOUN
ejpam-628	331	12	.	.	PUNCT
ejpam-628	332	1	the	the	DET
ejpam-628	332	2	relations	relation	NOUN
ejpam-628	332	3	are	be	AUX
ejpam-628	332	4	of	of	ADP
ejpam-628	332	5	two	two	NUM
ejpam-628	332	6	types	type	NOUN
ejpam-628	332	7	:	:	PUNCT
ejpam-628	332	8	m.	m.	NOUN
ejpam-628	332	9	behboodi	behboodi	PROPN
ejpam-628	332	10	,	,	PUNCT
ejpam-628	332	11	r.	r.	PROPN
ejpam-628	332	12	beyranvand	beyranvand	PROPN
ejpam-628	332	13	/	/	SYM
ejpam-628	332	14	eur	eur	PROPN
ejpam-628	332	15	.	.	PUNCT
ejpam-628	333	1	j.	j.	PROPN
ejpam-628	333	2	pure	pure	PROPN
ejpam-628	333	3	appl	appl	PROPN
ejpam-628	333	4	.	.	PROPN
ejpam-628	333	5	math	math	PROPN
ejpam-628	333	6	,	,	PUNCT
ejpam-628	333	7	3	3	NUM
ejpam-628	333	8	(	(	PUNCT
ejpam-628	333	9	2010	2010	NUM
ejpam-628	333	10	)	)	PUNCT
ejpam-628	333	11	,	,	PUNCT
ejpam-628	333	12	303	303	NUM
ejpam-628	333	13	-	-	SYM
ejpam-628	333	14	316	316	NUM
ejpam-628	333	15	310	310	NUM
ejpam-628	333	16	(	(	PUNCT
ejpam-628	333	17	i	i	NOUN
ejpam-628	333	18	)	)	PUNCT
ejpam-628	333	19	miai	miai	NOUN
ejpam-628	333	20	=	=	SYM
ejpam-628	333	21	0	0	NUM
ejpam-628	334	1	for	for	ADP
ejpam-628	334	2	i	i	PRON
ejpam-628	334	3	=	=	NOUN
ejpam-628	334	4	1	1	NUM
ejpam-628	334	5	,	,	PUNCT
ejpam-628	334	6	...	...	PUNCT
ejpam-628	334	7	,	,	PUNCT
ejpam-628	334	8	n	n	CCONJ
ejpam-628	334	9	indicating	indicate	VERB
ejpam-628	334	10	the	the	DET
ejpam-628	334	11	additive	additive	ADJ
ejpam-628	334	12	order	order	NOUN
ejpam-628	334	13	of	of	ADP
ejpam-628	334	14	ai	ai	NOUN
ejpam-628	334	15	,	,	PUNCT
ejpam-628	334	16	and	and	CCONJ
ejpam-628	334	17	(	(	PUNCT
ejpam-628	334	18	ii	ii	NOUN
ejpam-628	334	19	)	)	PUNCT
ejpam-628	334	20	aia	aia	PROPN
ejpam-628	334	21	j	j	PROPN
ejpam-628	334	22	=	=	SYM
ejpam-628	334	23	∑n	∑n	PROPN
ejpam-628	334	24	k=1	k=1	PROPN
ejpam-628	334	25	wi	wi	PROPN
ejpam-628	334	26	jkak	jkak	PROPN
ejpam-628	334	27	with	with	ADP
ejpam-628	334	28	wi	wi	PROPN
ejpam-628	334	29	jk	jk	PROPN
ejpam-628	334	30	∈	∈	PROPN
ejpam-628	334	31	zmk	zmk	NOUN
ejpam-628	334	32	for	for	ADP
ejpam-628	334	33	1≤	1≤	NUM
ejpam-628	334	34	i	i	PROPN
ejpam-628	334	35	,	,	PUNCT
ejpam-628	334	36	j	j	PROPN
ejpam-628	334	37	≤	≤	PROPN
ejpam-628	334	38	n.	n.	NOUN
ejpam-628	334	39	if	if	SCONJ
ejpam-628	334	40	the	the	DET
ejpam-628	334	41	ring	ring	NOUN
ejpam-628	334	42	r	r	NOUN
ejpam-628	334	43	has	have	VERB
ejpam-628	334	44	the	the	DET
ejpam-628	334	45	presentation	presentation	NOUN
ejpam-628	334	46	above	above	ADP
ejpam-628	334	47	we	we	PRON
ejpam-628	334	48	write	write	VERB
ejpam-628	334	49	r=	r=	ADJ
ejpam-628	334	50	d	d	NOUN
ejpam-628	334	51	a1	a1	NOUN
ejpam-628	334	52	,	,	PUNCT
ejpam-628	334	53	.	.	PUNCT
ejpam-628	334	54	.	.	PUNCT
ejpam-628	335	1	.	.	PUNCT
ejpam-628	336	1	,	,	PUNCT
ejpam-628	337	1	an	an	DET
ejpam-628	337	2	;	;	PUNCT
ejpam-628	337	3	miai	miai	ADJ
ejpam-628	337	4	=	=	SYM
ejpam-628	337	5	0	0	NUM
ejpam-628	337	6	,	,	PUNCT
ejpam-628	337	7	aia	aia	PROPN
ejpam-628	337	8	j	j	PROPN
ejpam-628	337	9	=	=	PROPN
ejpam-628	338	1	n	n	PROPN
ejpam-628	338	2	∑	∑	PUNCT
ejpam-628	338	3	k=1	k=1	PROPN
ejpam-628	338	4	wi	wi	PROPN
ejpam-628	338	5	jkak	jkak	PROPN
ejpam-628	338	6	,	,	PUNCT
ejpam-628	338	7	for	for	ADP
ejpam-628	338	8	i	i	PRON
ejpam-628	338	9	,	,	PUNCT
ejpam-628	338	10	j	j	PROPN
ejpam-628	338	11	=	=	SYM
ejpam-628	338	12	1	1	NUM
ejpam-628	338	13	,	,	PUNCT
ejpam-628	338	14	.	.	PUNCT
ejpam-628	338	15	.	.	PUNCT
ejpam-628	339	1	.	.	PUNCT
ejpam-628	340	1	,	,	PUNCT
ejpam-628	340	2	n	n	NOUN
ejpam-628	340	3	e	e	NOUN
ejpam-628	340	4	.	.	PUNCT
ejpam-628	341	1	corollary	corollary	ADJ
ejpam-628	341	2	3	3	X
ejpam-628	341	3	.	.	PUNCT
ejpam-628	342	1	let	let	VERB
ejpam-628	342	2	r	r	PRON
ejpam-628	342	3	be	be	AUX
ejpam-628	342	4	a	a	DET
ejpam-628	342	5	commutative	commutative	ADJ
ejpam-628	342	6	ring	ring	NOUN
ejpam-628	342	7	with	with	ADP
ejpam-628	342	8	|z(r)|=	|z(r)|=	VERB
ejpam-628	342	9	p3	p3	NOUN
ejpam-628	342	10	,	,	PUNCT
ejpam-628	342	11	where	where	SCONJ
ejpam-628	342	12	p	p	NOUN
ejpam-628	342	13	is	be	AUX
ejpam-628	342	14	a	a	DET
ejpam-628	342	15	prime	prime	ADJ
ejpam-628	342	16	number	number	NOUN
ejpam-628	342	17	.	.	PUNCT
ejpam-628	343	1	then	then	ADV
ejpam-628	343	2	r	r	NOUN
ejpam-628	343	3	is	be	AUX
ejpam-628	343	4	isomorphic	isomorphic	ADJ
ejpam-628	343	5	to	to	ADP
ejpam-628	343	6	one	one	NUM
ejpam-628	343	7	of	of	ADP
ejpam-628	343	8	the	the	DET
ejpam-628	343	9	rings	ring	NOUN
ejpam-628	343	10	zp2×	zp2×	PROPN
ejpam-628	343	11	fq	fq	PROPN
ejpam-628	343	12	,	,	PUNCT
ejpam-628	343	13	zp[x]/(x	zp[x]/(x	PROPN
ejpam-628	343	14	2)×	2)×	NUM
ejpam-628	343	15	fq	fq	NOUN
ejpam-628	343	16	where	where	SCONJ
ejpam-628	343	17	p2	p2	PROPN
ejpam-628	343	18	=	=	SYM
ejpam-628	343	19	p+q−1	p+q−1	PROPN
ejpam-628	343	20	,	,	PUNCT
ejpam-628	343	21	fq1	fq1	ADV
ejpam-628	343	22	×	×	NOUN
ejpam-628	343	23	.	.	PUNCT
ejpam-628	343	24	.	.	PUNCT
ejpam-628	344	1	.×	.×	PROPN
ejpam-628	344	2	fqt	fqt	VERB
ejpam-628	345	1	where	where	SCONJ
ejpam-628	345	2	p3	p3	NOUN
ejpam-628	345	3	=	=	PUNCT
ejpam-628	345	4	q1q2	q1q2	PROPN
ejpam-628	345	5	.	.	PUNCT
ejpam-628	345	6	.	.	PUNCT
ejpam-628	345	7	.	.	PUNCT
ejpam-628	346	1	qt	qt	INTJ
ejpam-628	346	2	−	−	PROPN
ejpam-628	346	3	(	(	PUNCT
ejpam-628	346	4	q1	q1	PROPN
ejpam-628	346	5	−	−	PROPN
ejpam-628	346	6	1)(q2	1)(q2	NUM
ejpam-628	346	7	−	−	PROPN
ejpam-628	346	8	1	1	NUM
ejpam-628	346	9	)	)	PUNCT
ejpam-628	346	10	.	.	PUNCT
ejpam-628	346	11	.	.	PUNCT
ejpam-628	346	12	.	.	PUNCT
ejpam-628	347	1	(	(	PUNCT
ejpam-628	347	2	qt	qt	INTJ
ejpam-628	347	3	−	−	PROPN
ejpam-628	347	4	1	1	NUM
ejpam-628	347	5	)	)	PUNCT
ejpam-628	347	6	,	,	PUNCT
ejpam-628	347	7	the	the	DET
ejpam-628	347	8	galois	galois	PROPN
ejpam-628	347	9	ring	ring	NOUN
ejpam-628	347	10	gr(p6	gr(p6	NOUN
ejpam-628	347	11	,	,	PUNCT
ejpam-628	347	12	p2	p2	PROPN
ejpam-628	347	13	)	)	PUNCT
ejpam-628	347	14	,	,	PUNCT
ejpam-628	347	15	fp3[x]/(x2	fp3[x]/(x2	NOUN
ejpam-628	347	16	)	)	PUNCT
ejpam-628	347	17	,	,	PUNCT
ejpam-628	347	18	fp[x]/(x	fp[x]/(x	NOUN
ejpam-628	347	19	4	4	NUM
ejpam-628	347	20	)	)	PUNCT
ejpam-628	347	21	,	,	PUNCT
ejpam-628	347	22	zp2[x]/(px	zp2[x]/(px	X
ejpam-628	347	23	,	,	PUNCT
ejpam-628	347	24	x3	x3	ADJ
ejpam-628	347	25	)	)	PUNCT
ejpam-628	347	26	,	,	PUNCT
ejpam-628	347	27	zp2[x]/(x2	zp2[x]/(x2	NOUN
ejpam-628	347	28	)	)	PUNCT
ejpam-628	347	29	,	,	PUNCT
ejpam-628	347	30	zp2[x]/(px	zp2[x]/(px	PROPN
ejpam-628	347	31	,	,	PUNCT
ejpam-628	347	32	x3−ap	x3−ap	NUM
ejpam-628	347	33	)	)	PUNCT
ejpam-628	347	34	where	where	SCONJ
ejpam-628	347	35	a	a	DET
ejpam-628	347	36	∈	∈	PROPN
ejpam-628	347	37	σ3	σ3	NOUN
ejpam-628	347	38	,	,	PUNCT
ejpam-628	347	39	zp2[x]/(x2−	zp2[x]/(x2−	PROPN
ejpam-628	347	40	bp	bp	PROPN
ejpam-628	347	41	)	)	PUNCT
ejpam-628	348	1	where	where	SCONJ
ejpam-628	348	2	b	b	PROPN
ejpam-628	348	3	∈	∈	PROPN
ejpam-628	348	4	σ2	σ2	PROPN
ejpam-628	348	5	and	and	CCONJ
ejpam-628	348	6	p	p	NOUN
ejpam-628	348	7	6=	6=	NUM
ejpam-628	348	8	2	2	NUM
ejpam-628	348	9	,	,	PUNCT
ejpam-628	348	10	z4[x]/(x	z4[x]/(x	NOUN
ejpam-628	348	11	2	2	NUM
ejpam-628	348	12	−	−	NOUN
ejpam-628	348	13	2	2	NUM
ejpam-628	348	14	)	)	PUNCT
ejpam-628	348	15	,	,	PUNCT
ejpam-628	348	16	z4[x]/(x	z4[x]/(x	NOUN
ejpam-628	348	17	2	2	NUM
ejpam-628	348	18	−	−	NOUN
ejpam-628	348	19	2x	2x	NUM
ejpam-628	348	20	−	−	PROPN
ejpam-628	348	21	2	2	NUM
ejpam-628	348	22	)	)	PUNCT
ejpam-628	348	23	,	,	PUNCT
ejpam-628	348	24	zp2[x	zp2[x	PROPN
ejpam-628	348	25	,	,	PUNCT
ejpam-628	348	26	y]/(p	y]/(p	PROPN
ejpam-628	348	27	,	,	PUNCT
ejpam-628	348	28	x	x	INTJ
ejpam-628	348	29	,	,	PUNCT
ejpam-628	348	30	y)2	y)2	NOUN
ejpam-628	348	31	,	,	PUNCT
ejpam-628	348	32	fp[x	fp[x	PROPN
ejpam-628	348	33	,	,	PUNCT
ejpam-628	348	34	y	y	PROPN
ejpam-628	348	35	,	,	PUNCT
ejpam-628	348	36	z]/(x	z]/(x	X
ejpam-628	348	37	,	,	PUNCT
ejpam-628	348	38	y	y	PROPN
ejpam-628	348	39	,	,	PUNCT
ejpam-628	348	40	z)2	z)2	PROPN
ejpam-628	348	41	,	,	PUNCT
ejpam-628	348	42	zp3[x]/(px	zp3[x]/(px	NUM
ejpam-628	348	43	,	,	PUNCT
ejpam-628	348	44	x2−	x2−	PROPN
ejpam-628	348	45	cp2	cp2	PROPN
ejpam-628	348	46	)	)	PUNCT
ejpam-628	348	47	where	where	SCONJ
ejpam-628	348	48	c	c	PROPN
ejpam-628	348	49	∈	∈	PROPN
ejpam-628	348	50	σ0	σ0	NOUN
ejpam-628	348	51	2	2	NUM
ejpam-628	348	52	,	,	PUNCT
ejpam-628	348	53	z4[x]/(x	z4[x]/(x	NOUN
ejpam-628	348	54	2−	2−	NUM
ejpam-628	348	55	2x	2x	NUM
ejpam-628	348	56	)	)	PUNCT
ejpam-628	348	57	,	,	PUNCT
ejpam-628	348	58	zp4	zp4	PROPN
ejpam-628	348	59	,	,	PUNCT
ejpam-628	348	60	r1	r1	PROPN
ejpam-628	348	61	:	:	PUNCT
ejpam-628	348	62	=	=	PUNCT
ejpam-628	348	63	〈	〈	PROPN
ejpam-628	348	64	1	1	NUM
ejpam-628	348	65	,	,	PUNCT
ejpam-628	348	66	x1	x1	PROPN
ejpam-628	348	67	,	,	PUNCT
ejpam-628	348	68	x2	x2	PROPN
ejpam-628	348	69	,	,	PUNCT
ejpam-628	348	70	y	y	PROPN
ejpam-628	348	71	;	;	PUNCT
ejpam-628	348	72	p1	p1	PROPN
ejpam-628	348	73	=	=	SYM
ejpam-628	348	74	0	0	PROPN
ejpam-628	348	75	,	,	PUNCT
ejpam-628	348	76	x1	x1	PROPN
ejpam-628	348	77	2	2	NUM
ejpam-628	348	78	=	=	SYM
ejpam-628	348	79	y	y	PROPN
ejpam-628	348	80	,	,	PUNCT
ejpam-628	348	81	x2	x2	PROPN
ejpam-628	348	82	2	2	NUM
ejpam-628	348	83	=	=	SYM
ejpam-628	348	84	0	0	NUM
ejpam-628	348	85	,	,	PUNCT
ejpam-628	348	86	x	x	X
ejpam-628	348	87	i	i	NOUN
ejpam-628	348	88	x	x	X
ejpam-628	348	89	j	j	NOUN
ejpam-628	348	90	=	=	PUNCT
ejpam-628	348	91	x	x	PROPN
ejpam-628	348	92	i	i	NOUN
ejpam-628	348	93	y	y	NOUN
ejpam-628	348	94	=	=	SYM
ejpam-628	348	95	y	y	PROPN
ejpam-628	348	96	x	x	PUNCT
ejpam-628	348	97	i	i	NOUN
ejpam-628	348	98	=	=	PUNCT
ejpam-628	348	99	y2	y2	NOUN
ejpam-628	348	100	=	=	SYM
ejpam-628	348	101	0	0	NUM
ejpam-628	348	102	,	,	PUNCT
ejpam-628	348	103	i	i	PROPN
ejpam-628	348	104	6=	6=	PROPN
ejpam-628	348	105	j	j	PROPN
ejpam-628	348	106	〉	〉	PROPN
ejpam-628	348	107	,	,	PUNCT
ejpam-628	348	108	r2	r2	NOUN
ejpam-628	348	109	:	:	PUNCT
ejpam-628	348	110	=	=	PUNCT
ejpam-628	348	111	〈	〈	PROPN
ejpam-628	348	112	1	1	NUM
ejpam-628	348	113	,	,	PUNCT
ejpam-628	348	114	x1	x1	PROPN
ejpam-628	348	115	,	,	PUNCT
ejpam-628	348	116	x2	x2	PROPN
ejpam-628	348	117	,	,	PUNCT
ejpam-628	348	118	y	y	PROPN
ejpam-628	348	119	;	;	PUNCT
ejpam-628	348	120	p1	p1	PROPN
ejpam-628	348	121	=	=	SYM
ejpam-628	348	122	0	0	PROPN
ejpam-628	348	123	,	,	PUNCT
ejpam-628	348	124	x1	x1	PROPN
ejpam-628	348	125	2	2	NUM
ejpam-628	348	126	=	=	SYM
ejpam-628	348	127	y	y	PROPN
ejpam-628	348	128	,	,	PUNCT
ejpam-628	348	129	x2	x2	PROPN
ejpam-628	348	130	2	2	NUM
ejpam-628	348	131	=	=	SYM
ejpam-628	348	132	y	y	PROPN
ejpam-628	348	133	,	,	PUNCT
ejpam-628	348	134	x	x	VERB
ejpam-628	348	135	i	i	NOUN
ejpam-628	348	136	x	x	X
ejpam-628	348	137	j	j	NOUN
ejpam-628	348	138	=	=	PUNCT
ejpam-628	348	139	x	x	PROPN
ejpam-628	348	140	i	i	NOUN
ejpam-628	348	141	y	y	NOUN
ejpam-628	348	142	=	=	SYM
ejpam-628	348	143	y	y	PROPN
ejpam-628	348	144	x	x	PUNCT
ejpam-628	348	145	i	i	NOUN
ejpam-628	348	146	=	=	PUNCT
ejpam-628	348	147	y2	y2	NOUN
ejpam-628	348	148	=	=	SYM
ejpam-628	348	149	0	0	NUM
ejpam-628	348	150	,	,	PUNCT
ejpam-628	348	151	i	i	PROPN
ejpam-628	348	152	6=	6=	PROPN
ejpam-628	348	153	j	j	PROPN
ejpam-628	348	154	〉	〉	PROPN
ejpam-628	348	155	,	,	PUNCT
ejpam-628	348	156	r3	r3	PROPN
ejpam-628	348	157	:	:	PUNCT
ejpam-628	348	158	=	=	PUNCT
ejpam-628	348	159	〈	〈	PROPN
ejpam-628	348	160	1	1	NUM
ejpam-628	348	161	,	,	PUNCT
ejpam-628	348	162	x1	x1	PROPN
ejpam-628	348	163	,	,	PUNCT
ejpam-628	348	164	x2	x2	PROPN
ejpam-628	348	165	,	,	PUNCT
ejpam-628	348	166	y	y	PROPN
ejpam-628	348	167	;	;	PUNCT
ejpam-628	348	168	p1	p1	PROPN
ejpam-628	348	169	=	=	SYM
ejpam-628	348	170	0	0	PROPN
ejpam-628	348	171	,	,	PUNCT
ejpam-628	348	172	x1	x1	PROPN
ejpam-628	348	173	2	2	NUM
ejpam-628	348	174	=	=	SYM
ejpam-628	348	175	y	y	PROPN
ejpam-628	348	176	,	,	PUNCT
ejpam-628	348	177	x2	x2	PROPN
ejpam-628	348	178	2	2	NUM
ejpam-628	348	179	=	=	SYM
ejpam-628	348	180	ξy	ξy	NOUN
ejpam-628	348	181	,	,	PUNCT
ejpam-628	348	182	x	x	VERB
ejpam-628	348	183	i	i	NOUN
ejpam-628	348	184	x	x	X
ejpam-628	348	185	j	j	NOUN
ejpam-628	348	186	=	=	PUNCT
ejpam-628	348	187	x	x	PROPN
ejpam-628	348	188	i	i	NOUN
ejpam-628	348	189	y	y	NOUN
ejpam-628	348	190	=	=	SYM
ejpam-628	348	191	y	y	PROPN
ejpam-628	348	192	x	x	PUNCT
ejpam-628	348	193	i	i	NOUN
ejpam-628	348	194	=	=	PUNCT
ejpam-628	348	195	y2	y2	NOUN
ejpam-628	348	196	=	=	SYM
ejpam-628	348	197	0	0	NUM
ejpam-628	348	198	,	,	PUNCT
ejpam-628	348	199	i	i	PROPN
ejpam-628	348	200	6=	6=	PROPN
ejpam-628	348	201	j	j	PROPN
ejpam-628	348	202	〉	〉	NOUN
ejpam-628	348	203	,	,	PUNCT
ejpam-628	348	204	r4	r4	VERB
ejpam-628	348	205	:	:	PUNCT
ejpam-628	348	206	=	=	PUNCT
ejpam-628	348	207	〈	〈	PROPN
ejpam-628	348	208	1	1	NUM
ejpam-628	348	209	,	,	PUNCT
ejpam-628	348	210	x1	x1	PROPN
ejpam-628	348	211	,	,	PUNCT
ejpam-628	348	212	x2	x2	PROPN
ejpam-628	348	213	;	;	PUNCT
ejpam-628	348	214	p21=	p21=	ADP
ejpam-628	348	215	px1	px1	NOUN
ejpam-628	348	216	=	=	SYM
ejpam-628	348	217	px2	px2	NOUN
ejpam-628	348	218	=	=	SYM
ejpam-628	348	219	0	0	PROPN
ejpam-628	348	220	,	,	PUNCT
ejpam-628	348	221	x1	x1	PROPN
ejpam-628	348	222	2	2	X
ejpam-628	348	223	=	=	SYM
ejpam-628	348	224	p	p	NOUN
ejpam-628	348	225	,	,	PUNCT
ejpam-628	348	226	x2	x2	PROPN
ejpam-628	348	227	2	2	NUM
ejpam-628	348	228	=	=	SYM
ejpam-628	348	229	0	0	NUM
ejpam-628	348	230	,	,	PUNCT
ejpam-628	348	231	x1	x1	PROPN
ejpam-628	349	1	x2	x2	NOUN
ejpam-628	350	1	=	=	PUNCT
ejpam-628	351	1	x2	x2	NUM
ejpam-628	352	1	x1	x1	NOUN
ejpam-628	353	1	=	=	SYM
ejpam-628	353	2	0	0	SYM
ejpam-628	353	3	〉	〉	NOUN
ejpam-628	353	4	,	,	PUNCT
ejpam-628	353	5	r5	r5	PROPN
ejpam-628	353	6	:	:	PUNCT
ejpam-628	353	7	=	=	PUNCT
ejpam-628	353	8	〈	〈	PROPN
ejpam-628	353	9	1	1	NUM
ejpam-628	353	10	,	,	PUNCT
ejpam-628	353	11	x1	x1	PROPN
ejpam-628	353	12	,	,	PUNCT
ejpam-628	353	13	x2	x2	PROPN
ejpam-628	353	14	;	;	PUNCT
ejpam-628	353	15	p21=	p21=	ADP
ejpam-628	353	16	px1	px1	NOUN
ejpam-628	353	17	=	=	SYM
ejpam-628	353	18	px2	px2	NOUN
ejpam-628	353	19	=	=	SYM
ejpam-628	353	20	0	0	PROPN
ejpam-628	353	21	,	,	PUNCT
ejpam-628	353	22	x1	x1	PROPN
ejpam-628	353	23	2	2	X
ejpam-628	353	24	=	=	SYM
ejpam-628	353	25	ξp	ξp	NOUN
ejpam-628	353	26	,	,	PUNCT
ejpam-628	353	27	x2	x2	PROPN
ejpam-628	353	28	2	2	NUM
ejpam-628	353	29	=	=	SYM
ejpam-628	353	30	0	0	NUM
ejpam-628	353	31	,	,	PUNCT
ejpam-628	353	32	x1	x1	PROPN
ejpam-628	353	33	x2	x2	NOUN
ejpam-628	354	1	=	=	PUNCT
ejpam-628	354	2	x2	x2	NUM
ejpam-628	355	1	x1	x1	NOUN
ejpam-628	355	2	=	=	SYM
ejpam-628	355	3	0	0	SYM
ejpam-628	355	4	〉	〉	NOUN
ejpam-628	355	5	,	,	PUNCT
ejpam-628	355	6	r6	r6	NOUN
ejpam-628	355	7	:	:	PUNCT
ejpam-628	355	8	=	=	PUNCT
ejpam-628	355	9	〈	〈	PROPN
ejpam-628	355	10	1	1	NUM
ejpam-628	355	11	,	,	PUNCT
ejpam-628	355	12	x1	x1	PROPN
ejpam-628	355	13	,	,	PUNCT
ejpam-628	355	14	x2	x2	PROPN
ejpam-628	355	15	;	;	PUNCT
ejpam-628	355	16	p21=	p21=	ADP
ejpam-628	355	17	px1	px1	NOUN
ejpam-628	355	18	=	=	SYM
ejpam-628	355	19	px2	px2	NOUN
ejpam-628	355	20	=	=	SYM
ejpam-628	355	21	0	0	PROPN
ejpam-628	355	22	,	,	PUNCT
ejpam-628	355	23	x1	x1	PROPN
ejpam-628	355	24	2	2	X
ejpam-628	355	25	=	=	SYM
ejpam-628	355	26	p	p	NOUN
ejpam-628	355	27	,	,	PUNCT
ejpam-628	355	28	x2	x2	PROPN
ejpam-628	355	29	2	2	X
ejpam-628	355	30	=	=	SYM
ejpam-628	355	31	p	p	NOUN
ejpam-628	355	32	,	,	PUNCT
ejpam-628	355	33	x1	x1	PROPN
ejpam-628	355	34	x2	x2	NOUN
ejpam-628	356	1	=	=	PUNCT
ejpam-628	356	2	x2	x2	NUM
ejpam-628	357	1	x1	x1	NOUN
ejpam-628	357	2	=	=	SYM
ejpam-628	357	3	0	0	SYM
ejpam-628	357	4	〉	〉	NOUN
ejpam-628	357	5	,	,	PUNCT
ejpam-628	357	6	r7	r7	NOUN
ejpam-628	357	7	:	:	PUNCT
ejpam-628	357	8	=	=	PUNCT
ejpam-628	357	9	〈	〈	PROPN
ejpam-628	357	10	1	1	NUM
ejpam-628	357	11	,	,	PUNCT
ejpam-628	357	12	x1	x1	PROPN
ejpam-628	357	13	,	,	PUNCT
ejpam-628	357	14	x2	x2	PROPN
ejpam-628	357	15	;	;	PUNCT
ejpam-628	357	16	p21=	p21=	ADP
ejpam-628	357	17	px1	px1	NOUN
ejpam-628	357	18	=	=	SYM
ejpam-628	357	19	px2	px2	NOUN
ejpam-628	357	20	=	=	SYM
ejpam-628	357	21	0	0	PROPN
ejpam-628	357	22	,	,	PUNCT
ejpam-628	357	23	x1	x1	PROPN
ejpam-628	357	24	2	2	X
ejpam-628	357	25	=	=	SYM
ejpam-628	357	26	p	p	NOUN
ejpam-628	357	27	,	,	PUNCT
ejpam-628	357	28	x2	x2	PROPN
ejpam-628	357	29	2	2	NUM
ejpam-628	357	30	=	=	SYM
ejpam-628	357	31	ξp	ξp	NOUN
ejpam-628	357	32	,	,	PUNCT
ejpam-628	357	33	x1	x1	PROPN
ejpam-628	357	34	x2	x2	NOUN
ejpam-628	358	1	=	=	SYM
ejpam-628	358	2	0	0	NUM
ejpam-628	358	3	〉	〉	NOUN
ejpam-628	358	4	,	,	PUNCT
ejpam-628	358	5	where	where	SCONJ
ejpam-628	358	6	ξ	ξ	PROPN
ejpam-628	358	7	is	be	AUX
ejpam-628	358	8	a	a	DET
ejpam-628	358	9	non	non	ADJ
ejpam-628	358	10	-	-	ADJ
ejpam-628	358	11	square	square	ADJ
ejpam-628	358	12	in	in	ADP
ejpam-628	358	13	fp	fp	PROPN
ejpam-628	358	14	and	and	CCONJ
ejpam-628	358	15	if	if	SCONJ
ejpam-628	358	16	p	p	NOUN
ejpam-628	358	17	=	=	NOUN
ejpam-628	358	18	2	2	NUM
ejpam-628	358	19	then	then	ADV
ejpam-628	358	20	instead	instead	ADV
ejpam-628	358	21	of	of	ADP
ejpam-628	358	22	r3	r3	PROPN
ejpam-628	358	23	,	,	PUNCT
ejpam-628	358	24	r5	r5	PROPN
ejpam-628	358	25	and	and	CCONJ
ejpam-628	358	26	r7	r7	NOUN
ejpam-628	358	27	,	,	PUNCT
ejpam-628	358	28	r	r	NOUN
ejpam-628	358	29	is	be	AUX
ejpam-628	358	30	isomorphic	isomorphic	ADJ
ejpam-628	358	31	to	to	PART
ejpam-628	358	32	r′3	r′3	VERB
ejpam-628	358	33	or	or	CCONJ
ejpam-628	358	34	r′5	r′5	PRON
ejpam-628	358	35	where	where	SCONJ
ejpam-628	358	36	r′3	r′3	NOUN
ejpam-628	358	37	:	:	PUNCT
ejpam-628	358	38	=	=	PUNCT
ejpam-628	358	39	〈	〈	PROPN
ejpam-628	358	40	1	1	NUM
ejpam-628	358	41	,	,	PUNCT
ejpam-628	358	42	x1	x1	PROPN
ejpam-628	358	43	,	,	PUNCT
ejpam-628	358	44	x2	x2	PROPN
ejpam-628	358	45	;	;	PUNCT
ejpam-628	358	46	4.1=	4.1=	PROPN
ejpam-628	358	47	2x1	2x1	NUM
ejpam-628	359	1	=	=	SYM
ejpam-628	359	2	2x2	2x2	NUM
ejpam-628	359	3	=	=	SYM
ejpam-628	359	4	0	0	PROPN
ejpam-628	359	5	,	,	PUNCT
ejpam-628	359	6	x1	x1	PROPN
ejpam-628	359	7	2	2	NUM
ejpam-628	359	8	=	=	SYM
ejpam-628	359	9	0	0	NUM
ejpam-628	359	10	,	,	PUNCT
ejpam-628	359	11	x2	x2	NOUN
ejpam-628	359	12	2	2	NUM
ejpam-628	359	13	=	=	SYM
ejpam-628	359	14	0	0	NUM
ejpam-628	359	15	,	,	PUNCT
ejpam-628	359	16	x1	x1	PROPN
ejpam-628	359	17	x2	x2	NOUN
ejpam-628	360	1	=	=	SYM
ejpam-628	360	2	x2	x2	NUM
ejpam-628	361	1	x1	x1	NOUN
ejpam-628	361	2	=	=	SYM
ejpam-628	361	3	2	2	NUM
ejpam-628	361	4	〉	〉	NOUN
ejpam-628	361	5	or	or	CCONJ
ejpam-628	361	6	r′5	r′5	NOUN
ejpam-628	361	7	:	:	PUNCT
ejpam-628	361	8	=	=	PUNCT
ejpam-628	361	9	〈	〈	PROPN
ejpam-628	361	10	1	1	NUM
ejpam-628	361	11	,	,	PUNCT
ejpam-628	361	12	x1	x1	PROPN
ejpam-628	361	13	,	,	PUNCT
ejpam-628	361	14	x2	x2	PROPN
ejpam-628	361	15	,	,	PUNCT
ejpam-628	361	16	y	y	PROPN
ejpam-628	361	17	;	;	PUNCT
ejpam-628	361	18	2.1	2.1	NUM
ejpam-628	361	19	=	=	SYM
ejpam-628	361	20	0	0	NUM
ejpam-628	361	21	,	,	PUNCT
ejpam-628	361	22	x1	x1	PROPN
ejpam-628	361	23	x2	x2	NOUN
ejpam-628	362	1	=	=	PUNCT
ejpam-628	362	2	x2	x2	NUM
ejpam-628	363	1	x1	x1	PROPN
ejpam-628	363	2	=	=	SYM
ejpam-628	363	3	y	y	PROPN
ejpam-628	363	4	,	,	PUNCT
ejpam-628	363	5	x1	x1	PROPN
ejpam-628	363	6	2	2	X
ejpam-628	363	7	=	=	SYM
ejpam-628	363	8	x2	x2	NOUN
ejpam-628	363	9	2	2	NUM
ejpam-628	364	1	=	=	SYM
ejpam-628	364	2	x	x	PUNCT
ejpam-628	365	1	i	i	NOUN
ejpam-628	365	2	y	y	NOUN
ejpam-628	365	3	=	=	SYM
ejpam-628	365	4	y	y	PROPN
ejpam-628	365	5	x	x	PUNCT
ejpam-628	366	1	i	i	NOUN
ejpam-628	366	2	=	=	PUNCT
ejpam-628	366	3	y2	y2	NOUN
ejpam-628	366	4	=	=	SYM
ejpam-628	366	5	0	0	NUM
ejpam-628	366	6	〉	〉	NOUN
ejpam-628	366	7	.	.	PUNCT
ejpam-628	367	1	proof	proof	NOUN
ejpam-628	367	2	.	.	PUNCT
ejpam-628	368	1	if	if	SCONJ
ejpam-628	368	2	r	r	NOUN
ejpam-628	368	3	is	be	AUX
ejpam-628	368	4	a	a	DET
ejpam-628	368	5	local	local	ADJ
ejpam-628	368	6	ring	ring	NOUN
ejpam-628	368	7	with	with	ADP
ejpam-628	368	8	|z(r)|	|z(r)|	PROPN
ejpam-628	368	9	=	=	PUNCT
ejpam-628	368	10	p3	p3	PROPN
ejpam-628	368	11	,	,	PUNCT
ejpam-628	368	12	then	then	ADV
ejpam-628	368	13	by	by	ADP
ejpam-628	368	14	lemma	lemma	PROPN
ejpam-628	368	15	1	1	NUM
ejpam-628	368	16	,	,	PUNCT
ejpam-628	368	17	either	either	CCONJ
ejpam-628	368	18	|r|	|r|	NOUN
ejpam-628	368	19	=	=	NOUN
ejpam-628	368	20	p4	p4	ADJ
ejpam-628	368	21	or	or	CCONJ
ejpam-628	368	22	|r|	|r|	NOUN
ejpam-628	368	23	=	=	PUNCT
ejpam-628	368	24	p6	p6	PROPN
ejpam-628	368	25	.	.	PUNCT
ejpam-628	369	1	if	if	SCONJ
ejpam-628	369	2	|r|	|r|	NOUN
ejpam-628	369	3	=	=	SYM
ejpam-628	369	4	p6	p6	PROPN
ejpam-628	369	5	,	,	PUNCT
ejpam-628	369	6	then	then	ADV
ejpam-628	369	7	by	by	ADP
ejpam-628	369	8	[	[	PUNCT
ejpam-628	369	9	9	9	NUM
ejpam-628	369	10	,	,	PUNCT
ejpam-628	369	11	theorem	theorem	VERB
ejpam-628	369	12	12	12	NUM
ejpam-628	369	13	]	]	PUNCT
ejpam-628	369	14	,	,	PUNCT
ejpam-628	369	15	r	r	NOUN
ejpam-628	369	16	is	be	AUX
ejpam-628	369	17	isomorphic	isomorphic	ADJ
ejpam-628	369	18	to	to	ADP
ejpam-628	369	19	fp3[x]/(x2	fp3[x]/(x2	NOUN
ejpam-628	369	20	)	)	PUNCT
ejpam-628	369	21	or	or	CCONJ
ejpam-628	369	22	the	the	DET
ejpam-628	369	23	galois	galois	PROPN
ejpam-628	369	24	ring	ring	NOUN
ejpam-628	369	25	gr(p6	gr(p6	NOUN
ejpam-628	369	26	,	,	PUNCT
ejpam-628	369	27	p2	p2	PROPN
ejpam-628	369	28	)	)	PUNCT
ejpam-628	369	29	.	.	PUNCT
ejpam-628	370	1	if	if	SCONJ
ejpam-628	370	2	|r|	|r|	NOUN
ejpam-628	370	3	=	=	SYM
ejpam-628	370	4	p4	p4	ADJ
ejpam-628	370	5	,	,	PUNCT
ejpam-628	370	6	then	then	ADV
ejpam-628	370	7	since	since	SCONJ
ejpam-628	370	8	|z(r)|	|z(r)|	PROPN
ejpam-628	370	9	=	=	PUNCT
ejpam-628	370	10	p3	p3	PROPN
ejpam-628	370	11	,	,	PUNCT
ejpam-628	370	12	by	by	ADP
ejpam-628	370	13	using	use	VERB
ejpam-628	370	14	[	[	X
ejpam-628	370	15	2	2	NUM
ejpam-628	370	16	,	,	PUNCT
ejpam-628	370	17	p.687	p.687	NOUN
ejpam-628	370	18	-	-	PUNCT
ejpam-628	370	19	690	690	NUM
ejpam-628	370	20	]	]	PUNCT
ejpam-628	370	21	,	,	PUNCT
ejpam-628	370	22	one	one	PRON
ejpam-628	370	23	can	can	AUX
ejpam-628	370	24	easily	easily	ADV
ejpam-628	370	25	see	see	VERB
ejpam-628	370	26	that	that	SCONJ
ejpam-628	370	27	r	r	NOUN
ejpam-628	370	28	is	be	AUX
ejpam-628	370	29	isomorphic	isomorphic	ADJ
ejpam-628	370	30	to	to	ADP
ejpam-628	370	31	one	one	NUM
ejpam-628	370	32	of	of	ADP
ejpam-628	370	33	the	the	DET
ejpam-628	370	34	local	local	ADJ
ejpam-628	370	35	rings	ring	NOUN
ejpam-628	370	36	in	in	ADP
ejpam-628	370	37	above	above	ADP
ejpam-628	370	38	list	list	NOUN
ejpam-628	370	39	.	.	PUNCT
ejpam-628	371	1	now	now	ADV
ejpam-628	371	2	suppose	suppose	VERB
ejpam-628	371	3	that	that	SCONJ
ejpam-628	371	4	r	r	NOUN
ejpam-628	371	5	is	be	AUX
ejpam-628	371	6	not	not	PART
ejpam-628	371	7	local	local	ADJ
ejpam-628	371	8	.	.	PUNCT
ejpam-628	372	1	then	then	ADV
ejpam-628	372	2	by	by	ADP
ejpam-628	372	3	theorem	theorem	NOUN
ejpam-628	372	4	2	2	NUM
ejpam-628	372	5	,	,	PUNCT
ejpam-628	372	6	r	r	NOUN
ejpam-628	372	7	is	be	AUX
ejpam-628	372	8	isomorphic	isomorphic	ADJ
ejpam-628	372	9	to	to	ADP
ejpam-628	372	10	one	one	NUM
ejpam-628	372	11	of	of	ADP
ejpam-628	372	12	the	the	DET
ejpam-628	372	13	rings	ring	NOUN
ejpam-628	372	14	zp2	zp2	PROPN
ejpam-628	372	15	×	×	PROPN
ejpam-628	372	16	fq	fq	PROPN
ejpam-628	372	17	,	,	PUNCT
ejpam-628	372	18	zp[x]/(x	zp[x]/(x	PROPN
ejpam-628	373	1	2)×	2)×	NUM
ejpam-628	373	2	fq	fq	NOUN
ejpam-628	373	3	where	where	SCONJ
ejpam-628	373	4	p2	p2	PROPN
ejpam-628	373	5	=	=	PUNCT
ejpam-628	374	1	p	p	X
ejpam-628	374	2	+	+	NOUN
ejpam-628	374	3	q	q	NOUN
ejpam-628	374	4	−	−	PROPN
ejpam-628	374	5	1	1	NUM
ejpam-628	374	6	or	or	CCONJ
ejpam-628	374	7	fq1	fq1	NUM
ejpam-628	374	8	×	×	NOUN
ejpam-628	374	9	.	.	PUNCT
ejpam-628	374	10	.	.	PUNCT
ejpam-628	374	11	.	.	PUNCT
ejpam-628	375	1	×	×	NOUN
ejpam-628	375	2	fqt	fqt	NOUN
ejpam-628	375	3	where	where	SCONJ
ejpam-628	375	4	p3	p3	NOUN
ejpam-628	375	5	=	=	PUNCT
ejpam-628	376	1	q1q2	q1q2	PROPN
ejpam-628	376	2	.	.	PUNCT
ejpam-628	376	3	.	.	PUNCT
ejpam-628	376	4	.	.	PUNCT
ejpam-628	377	1	qt	qt	INTJ
ejpam-628	377	2	−	−	PROPN
ejpam-628	377	3	(	(	PUNCT
ejpam-628	377	4	q1	q1	PROPN
ejpam-628	377	5	−	−	PROPN
ejpam-628	377	6	1)(q2	1)(q2	NUM
ejpam-628	377	7	−	−	PROPN
ejpam-628	377	8	1	1	NUM
ejpam-628	377	9	)	)	PUNCT
ejpam-628	377	10	.	.	PUNCT
ejpam-628	377	11	.	.	PUNCT
ejpam-628	377	12	.	.	PUNCT
ejpam-628	378	1	(	(	PUNCT
ejpam-628	378	2	qt	qt	INTJ
ejpam-628	378	3	−	−	NOUN
ejpam-628	378	4	1	1	NUM
ejpam-628	378	5	)	)	PUNCT
ejpam-628	378	6	.	.	PUNCT
ejpam-628	379	1	this	this	PRON
ejpam-628	379	2	completes	complete	VERB
ejpam-628	379	3	the	the	DET
ejpam-628	379	4	proof	proof	NOUN
ejpam-628	379	5	.	.	PUNCT
ejpam-628	380	1	now	now	ADV
ejpam-628	380	2	we	we	PRON
ejpam-628	380	3	are	be	AUX
ejpam-628	380	4	in	in	ADP
ejpam-628	380	5	position	position	NOUN
ejpam-628	380	6	to	to	PART
ejpam-628	380	7	determine	determine	VERB
ejpam-628	380	8	the	the	DET
ejpam-628	380	9	structure	structure	NOUN
ejpam-628	380	10	of	of	ADP
ejpam-628	380	11	commutative	commutative	ADJ
ejpam-628	380	12	rings	ring	NOUN
ejpam-628	380	13	r	r	NOUN
ejpam-628	380	14	with	with	ADP
ejpam-628	380	15	|z(r)|	|z(r)|	PROPN
ejpam-628	380	16	=	=	SYM
ejpam-628	380	17	p	p	PROPN
ejpam-628	380	18	k1	k1	NOUN
ejpam-628	380	19	1	1	NUM
ejpam-628	380	20	p	p	NOUN
ejpam-628	380	21	k2	k2	PROPN
ejpam-628	380	22	2	2	NUM
ejpam-628	380	23	.	.	PUNCT
ejpam-628	380	24	.	.	PUNCT
ejpam-628	380	25	.	.	PUNCT
ejpam-628	381	1	p	p	X
ejpam-628	381	2	kn	kn	PROPN
ejpam-628	381	3	n	n	PROPN
ejpam-628	381	4	,	,	PUNCT
ejpam-628	381	5	where	where	SCONJ
ejpam-628	381	6	n≥	n≥	PROPN
ejpam-628	381	7	1	1	NUM
ejpam-628	381	8	,	,	PUNCT
ejpam-628	381	9	1≤	1≤	NUM
ejpam-628	381	10	ki	ki	PROPN
ejpam-628	381	11	≤	≤	ADV
ejpam-628	381	12	3	3	NUM
ejpam-628	381	13	and	and	CCONJ
ejpam-628	381	14	pi	pi	NOUN
ejpam-628	381	15	,	,	PUNCT
ejpam-628	381	16	s	s	VERB
ejpam-628	381	17	are	be	AUX
ejpam-628	381	18	distinct	distinct	ADJ
ejpam-628	381	19	prime	prime	ADJ
ejpam-628	381	20	numbers	number	NOUN
ejpam-628	381	21	.	.	PUNCT
ejpam-628	382	1	theorem	theorem	NOUN
ejpam-628	382	2	5	5	NUM
ejpam-628	382	3	.	.	PUNCT
ejpam-628	383	1	let	let	VERB
ejpam-628	383	2	r	r	PRON
ejpam-628	383	3	be	be	AUX
ejpam-628	383	4	a	a	DET
ejpam-628	383	5	commutative	commutative	ADJ
ejpam-628	383	6	ring	ring	NOUN
ejpam-628	383	7	with	with	ADP
ejpam-628	383	8	|z(r)|	|z(r)|	PROPN
ejpam-628	383	9	=	=	SYM
ejpam-628	383	10	p	p	PROPN
ejpam-628	383	11	k1	k1	NOUN
ejpam-628	383	12	1	1	NUM
ejpam-628	383	13	p	p	NOUN
ejpam-628	383	14	k2	k2	PROPN
ejpam-628	383	15	2	2	NUM
ejpam-628	383	16	.	.	PUNCT
ejpam-628	383	17	.	.	PUNCT
ejpam-628	383	18	.	.	PUNCT
ejpam-628	384	1	p	p	X
ejpam-628	384	2	kn	kn	PROPN
ejpam-628	384	3	n	n	PROPN
ejpam-628	384	4	,	,	PUNCT
ejpam-628	384	5	where	where	SCONJ
ejpam-628	384	6	n	n	PRON
ejpam-628	384	7	≥	≥	NOUN
ejpam-628	384	8	1	1	NUM
ejpam-628	384	9	,	,	PUNCT
ejpam-628	384	10	1	1	NUM
ejpam-628	384	11	≤	≤	NUM
ejpam-628	384	12	ki	ki	PROPN
ejpam-628	385	1	≤	≤	ADV
ejpam-628	385	2	3	3	NUM
ejpam-628	385	3	and	and	CCONJ
ejpam-628	385	4	pi	pi	NOUN
ejpam-628	385	5	,	,	PUNCT
ejpam-628	385	6	s	s	VERB
ejpam-628	385	7	are	be	AUX
ejpam-628	385	8	distinct	distinct	ADJ
ejpam-628	385	9	prime	prime	ADJ
ejpam-628	385	10	numbers	number	NOUN
ejpam-628	385	11	.	.	PUNCT
ejpam-628	386	1	then	then	ADV
ejpam-628	386	2	there	there	PRON
ejpam-628	386	3	exist	exist	VERB
ejpam-628	386	4	0≤	0≤	NUM
ejpam-628	386	5	s	s	PART
ejpam-628	386	6	≤	≤	NUM
ejpam-628	386	7	σn	σn	NOUN
ejpam-628	386	8	i=1ki	i=1ki	PROPN
ejpam-628	386	9	and	and	CCONJ
ejpam-628	386	10	t	t	PROPN
ejpam-628	386	11	≥	≥	NUM
ejpam-628	386	12	0	0	NUM
ejpam-628	387	1	such	such	ADJ
ejpam-628	387	2	that	that	SCONJ
ejpam-628	387	3	r∼=	r∼=	ADV
ejpam-628	387	4	r1	r1	PROPN
ejpam-628	387	5	×	×	NOUN
ejpam-628	387	6	.	.	PUNCT
ejpam-628	387	7	.	.	PUNCT
ejpam-628	388	1	.×	.×	NOUN
ejpam-628	388	2	rs	r	VERB
ejpam-628	388	3	×	×	NOUN
ejpam-628	389	1	fq1	fq1	INTJ
ejpam-628	389	2	×	×	NOUN
ejpam-628	389	3	.	.	PUNCT
ejpam-628	389	4	.	.	PUNCT
ejpam-628	390	1	.×	.×	PROPN
ejpam-628	390	2	fqt	fqt	PROPN
ejpam-628	390	3	m.	m.	NOUN
ejpam-628	390	4	behboodi	behboodi	PROPN
ejpam-628	390	5	,	,	PUNCT
ejpam-628	390	6	r.	r.	PROPN
ejpam-628	390	7	beyranvand	beyranvand	PROPN
ejpam-628	390	8	/	/	SYM
ejpam-628	390	9	eur	eur	PROPN
ejpam-628	390	10	.	.	PUNCT
ejpam-628	391	1	j.	j.	PROPN
ejpam-628	391	2	pure	pure	PROPN
ejpam-628	391	3	appl	appl	PROPN
ejpam-628	391	4	.	.	PROPN
ejpam-628	391	5	math	math	PROPN
ejpam-628	391	6	,	,	PUNCT
ejpam-628	391	7	3	3	NUM
ejpam-628	391	8	(	(	PUNCT
ejpam-628	391	9	2010	2010	NUM
ejpam-628	391	10	)	)	PUNCT
ejpam-628	391	11	,	,	PUNCT
ejpam-628	391	12	303	303	NUM
ejpam-628	391	13	-	-	SYM
ejpam-628	391	14	316	316	NUM
ejpam-628	391	15	311	311	NUM
ejpam-628	391	16	where	where	SCONJ
ejpam-628	391	17	fqi	fqi	NOUN
ejpam-628	391	18	,	,	PUNCT
ejpam-628	391	19	s	s	VERB
ejpam-628	391	20	are	be	AUX
ejpam-628	391	21	finite	finite	ADJ
ejpam-628	391	22	fields	field	NOUN
ejpam-628	391	23	and	and	CCONJ
ejpam-628	391	24	each	each	DET
ejpam-628	391	25	ri	ri	PROPN
ejpam-628	391	26	is	be	AUX
ejpam-628	391	27	local	local	ADJ
ejpam-628	391	28	ring	ring	NOUN
ejpam-628	391	29	with	with	ADP
ejpam-628	391	30	|z(ri)|	|z(ri)|	NOUN
ejpam-628	391	31	=	=	SYM
ejpam-628	391	32	p	p	PROPN
ejpam-628	391	33	t	t	PROPN
ejpam-628	391	34	j	j	PROPN
ejpam-628	391	35	j	j	PROPN
ejpam-628	391	36	for	for	ADP
ejpam-628	391	37	some	some	DET
ejpam-628	391	38	p	p	PRON
ejpam-628	391	39	j	j	PROPN
ejpam-628	391	40	(	(	PUNCT
ejpam-628	391	41	1	1	NUM
ejpam-628	391	42	≤	≤	NUM
ejpam-628	391	43	j	j	PROPN
ejpam-628	391	44	≤	≤	NUM
ejpam-628	391	45	n	n	CCONJ
ejpam-628	391	46	)	)	PUNCT
ejpam-628	391	47	and	and	CCONJ
ejpam-628	391	48	1	1	NUM
ejpam-628	391	49	≤	≤	NOUN
ejpam-628	392	1	t	t	PROPN
ejpam-628	392	2	j	j	PROPN
ejpam-628	392	3	≤	≤	PROPN
ejpam-628	392	4	k	k	PROPN
ejpam-628	392	5	j.	j.	PROPN
ejpam-628	392	6	consequently	consequently	ADV
ejpam-628	392	7	,	,	PUNCT
ejpam-628	392	8	each	each	DET
ejpam-628	392	9	ri	ri	PROPN
ejpam-628	392	10	is	be	AUX
ejpam-628	392	11	isomorphic	isomorphic	ADJ
ejpam-628	392	12	to	to	ADP
ejpam-628	392	13	one	one	NUM
ejpam-628	392	14	of	of	ADP
ejpam-628	392	15	the	the	DET
ejpam-628	392	16	local	local	ADJ
ejpam-628	392	17	rings	ring	NOUN
ejpam-628	392	18	described	describe	VERB
ejpam-628	392	19	in	in	ADP
ejpam-628	392	20	corollaries	corollary	NOUN
ejpam-628	392	21	1	1	NUM
ejpam-628	392	22	,	,	PUNCT
ejpam-628	392	23	2	2	NUM
ejpam-628	392	24	or	or	CCONJ
ejpam-628	392	25	3	3	NUM
ejpam-628	392	26	.	.	PUNCT
ejpam-628	393	1	proof	proof	NOUN
ejpam-628	393	2	.	.	PUNCT
ejpam-628	394	1	we	we	PRON
ejpam-628	394	2	put	put	VERB
ejpam-628	394	3	r∼=	r∼=	NUM
ejpam-628	394	4	r1×	r1×	NOUN
ejpam-628	394	5	.	.	PUNCT
ejpam-628	394	6	.	.	PUNCT
ejpam-628	395	1	.×	.×	NOUN
ejpam-628	395	2	rs	r	VERB
ejpam-628	395	3	×	×	NOUN
ejpam-628	396	1	fq1	fq1	INTJ
ejpam-628	396	2	×	×	NOUN
ejpam-628	396	3	.	.	PUNCT
ejpam-628	396	4	.	.	PUNCT
ejpam-628	397	1	.×	.×	PROPN
ejpam-628	397	2	fqt	fqt	PROPN
ejpam-628	397	3	,	,	PUNCT
ejpam-628	397	4	where	where	SCONJ
ejpam-628	397	5	fq1	fq1	ADV
ejpam-628	397	6	,	,	PUNCT
ejpam-628	397	7	.	.	PUNCT
ejpam-628	397	8	.	.	PUNCT
ejpam-628	397	9	.	.	PUNCT
ejpam-628	398	1	,	,	PUNCT
ejpam-628	398	2	fqt	fqt	PROPN
ejpam-628	398	3	are	be	AUX
ejpam-628	398	4	finite	finite	ADJ
ejpam-628	398	5	fields	field	NOUN
ejpam-628	398	6	and	and	CCONJ
ejpam-628	398	7	each	each	DET
ejpam-628	398	8	ri	ri	PROPN
ejpam-628	398	9	is	be	AUX
ejpam-628	398	10	a	a	DET
ejpam-628	398	11	commutative	commutative	ADJ
ejpam-628	398	12	finite	finite	ADJ
ejpam-628	398	13	local	local	ADJ
ejpam-628	398	14	ring	ring	NOUN
ejpam-628	398	15	that	that	PRON
ejpam-628	398	16	is	be	AUX
ejpam-628	398	17	not	not	PART
ejpam-628	398	18	a	a	DET
ejpam-628	398	19	field	field	NOUN
ejpam-628	398	20	.	.	PUNCT
ejpam-628	399	1	if	if	SCONJ
ejpam-628	399	2	s	s	NOUN
ejpam-628	399	3	=	=	NOUN
ejpam-628	399	4	0	0	NUM
ejpam-628	399	5	,	,	PUNCT
ejpam-628	399	6	then	then	ADV
ejpam-628	399	7	there	there	PRON
ejpam-628	399	8	is	be	VERB
ejpam-628	399	9	nothing	nothing	PRON
ejpam-628	399	10	to	to	PART
ejpam-628	399	11	prove	prove	VERB
ejpam-628	399	12	.	.	PUNCT
ejpam-628	400	1	thus	thus	ADV
ejpam-628	400	2	we	we	PRON
ejpam-628	400	3	can	can	AUX
ejpam-628	400	4	assume	assume	VERB
ejpam-628	400	5	that	that	SCONJ
ejpam-628	400	6	s	s	VERB
ejpam-628	400	7	≥	≥	NOUN
ejpam-628	400	8	1	1	NUM
ejpam-628	400	9	and	and	CCONJ
ejpam-628	400	10	so	so	ADV
ejpam-628	400	11	by	by	ADP
ejpam-628	400	12	theorem	theorem	NOUN
ejpam-628	400	13	1	1	NUM
ejpam-628	400	14	,	,	PUNCT
ejpam-628	400	15	for	for	ADP
ejpam-628	400	16	each	each	DET
ejpam-628	400	17	i	i	PROPN
ejpam-628	400	18	,	,	PUNCT
ejpam-628	400	19	|z(ri)|	|z(ri)|	NOUN
ejpam-628	400	20	=	=	SYM
ejpam-628	400	21	pk	pk	NOUN
ejpam-628	400	22	for	for	ADP
ejpam-628	400	23	some	some	DET
ejpam-628	400	24	prime	prime	ADJ
ejpam-628	400	25	number	number	NOUN
ejpam-628	400	26	p	p	NOUN
ejpam-628	400	27	and	and	CCONJ
ejpam-628	400	28	k	k	PROPN
ejpam-628	400	29	≥	≥	NUM
ejpam-628	400	30	1	1	NUM
ejpam-628	400	31	such	such	ADJ
ejpam-628	400	32	that	that	DET
ejpam-628	400	33	pk	pk	NOUN
ejpam-628	400	34	is	be	AUX
ejpam-628	400	35	a	a	DET
ejpam-628	400	36	divisor	divisor	NOUN
ejpam-628	400	37	of	of	ADP
ejpam-628	400	38	|z(r)|	|z(r)|	PROPN
ejpam-628	400	39	and	and	CCONJ
ejpam-628	400	40	also	also	ADV
ejpam-628	400	41	0≤	0≤	NOUN
ejpam-628	400	42	s	s	PART
ejpam-628	400	43	≤	≤	NUM
ejpam-628	400	44	σn	σn	X
ejpam-628	400	45	i=1ki	i=1ki	NOUN
ejpam-628	400	46	.	.	PUNCT
ejpam-628	401	1	thus	thus	ADV
ejpam-628	401	2	|z(ri)|=	|z(ri)|=	VERB
ejpam-628	401	3	p	p	PROPN
ejpam-628	401	4	t	t	PROPN
ejpam-628	401	5	j	j	PROPN
ejpam-628	402	1	j	j	PROPN
ejpam-628	402	2	where	where	SCONJ
ejpam-628	402	3	1≤	1≤	PROPN
ejpam-628	402	4	t	t	PROPN
ejpam-628	402	5	j	j	PROPN
ejpam-628	402	6	≤	≤	PROPN
ejpam-628	402	7	k	k	PROPN
ejpam-628	402	8	j	j	PROPN
ejpam-628	402	9	,	,	PUNCT
ejpam-628	402	10	1≤	1≤	PROPN
ejpam-628	402	11	j	j	PROPN
ejpam-628	402	12	≤	≤	PROPN
ejpam-628	402	13	n	n	PRON
ejpam-628	402	14	and	and	CCONJ
ejpam-628	402	15	1	1	NUM
ejpam-628	402	16	≤	≤	NUM
ejpam-628	402	17	i	i	PRON
ejpam-628	402	18	≤	≤	ADJ
ejpam-628	402	19	s.	s.	PROPN
ejpam-628	402	20	thus	thus	ADV
ejpam-628	402	21	for	for	ADP
ejpam-628	402	22	each	each	DET
ejpam-628	402	23	1	1	NUM
ejpam-628	402	24	≤	≤	NUM
ejpam-628	402	25	i	i	PRON
ejpam-628	403	1	≤	≤	PROPN
ejpam-628	403	2	s	s	PROPN
ejpam-628	403	3	,	,	PUNCT
ejpam-628	403	4	t	t	PROPN
ejpam-628	404	1	i	i	NOUN
ejpam-628	404	2	=	=	NOUN
ejpam-628	404	3	1,2	1,2	NUM
ejpam-628	404	4	or	or	CCONJ
ejpam-628	404	5	3	3	NUM
ejpam-628	405	1	and	and	CCONJ
ejpam-628	405	2	so	so	ADV
ejpam-628	405	3	each	each	DET
ejpam-628	405	4	ri	ri	PROPN
ejpam-628	405	5	is	be	AUX
ejpam-628	405	6	isomorphic	isomorphic	ADJ
ejpam-628	405	7	to	to	ADP
ejpam-628	405	8	one	one	NUM
ejpam-628	405	9	of	of	ADP
ejpam-628	405	10	the	the	DET
ejpam-628	405	11	local	local	ADJ
ejpam-628	405	12	rings	ring	NOUN
ejpam-628	405	13	described	describe	VERB
ejpam-628	405	14	in	in	ADP
ejpam-628	405	15	corollaries	corollary	NOUN
ejpam-628	405	16	1	1	NUM
ejpam-628	405	17	,	,	PUNCT
ejpam-628	405	18	2	2	NUM
ejpam-628	405	19	or	or	CCONJ
ejpam-628	405	20	3	3	NUM
ejpam-628	405	21	next	next	ADJ
ejpam-628	405	22	,	,	PUNCT
ejpam-628	405	23	we	we	PRON
ejpam-628	405	24	determine	determine	VERB
ejpam-628	405	25	the	the	DET
ejpam-628	405	26	structure	structure	NOUN
ejpam-628	405	27	of	of	ADP
ejpam-628	405	28	commutative	commutative	ADJ
ejpam-628	405	29	nonlocal	nonlocal	ADJ
ejpam-628	405	30	rings	ring	NOUN
ejpam-628	405	31	r	r	NOUN
ejpam-628	405	32	with	with	ADP
ejpam-628	405	33	|z(r)|=	|z(r)|=	VERB
ejpam-628	405	34	p4	p4	ADJ
ejpam-628	405	35	or	or	CCONJ
ejpam-628	405	36	p5	p5	ADJ
ejpam-628	405	37	where	where	SCONJ
ejpam-628	405	38	p	p	NOUN
ejpam-628	405	39	is	be	AUX
ejpam-628	405	40	a	a	DET
ejpam-628	405	41	prime	prime	ADJ
ejpam-628	405	42	number	number	NOUN
ejpam-628	405	43	.	.	PUNCT
ejpam-628	406	1	proposition	proposition	NOUN
ejpam-628	406	2	1	1	NUM
ejpam-628	406	3	.	.	PUNCT
ejpam-628	407	1	let	let	VERB
ejpam-628	407	2	r	r	PRON
ejpam-628	407	3	be	be	AUX
ejpam-628	407	4	a	a	DET
ejpam-628	407	5	commutative	commutative	ADJ
ejpam-628	407	6	nonlocal	nonlocal	ADJ
ejpam-628	407	7	ring	ring	NOUN
ejpam-628	407	8	with	with	ADP
ejpam-628	407	9	|z(r)|	|z(r)|	PROPN
ejpam-628	407	10	=	=	PUNCT
ejpam-628	407	11	p4	p4	NOUN
ejpam-628	407	12	where	where	SCONJ
ejpam-628	407	13	p	p	NOUN
ejpam-628	407	14	is	be	AUX
ejpam-628	407	15	a	a	DET
ejpam-628	407	16	prime	prime	ADJ
ejpam-628	407	17	number	number	NOUN
ejpam-628	407	18	.	.	PUNCT
ejpam-628	408	1	then	then	ADV
ejpam-628	408	2	r∼=	r∼=	NUM
ejpam-628	408	3	fq1	fq1	ADV
ejpam-628	408	4	×.	×.	NUM
ejpam-628	408	5	.	.	PUNCT
ejpam-628	408	6	.×fqt	.×fqt	PUNCT
ejpam-628	409	1	with	with	ADP
ejpam-628	409	2	p4	p4	ADJ
ejpam-628	409	3	=	=	SYM
ejpam-628	409	4	q1q2	q1q2	NOUN
ejpam-628	409	5	.	.	PUNCT
ejpam-628	409	6	.	.	PUNCT
ejpam-628	409	7	.	.	PUNCT
ejpam-628	410	1	qt−(q1−1)(q2−1	qt−(q1−1)(q2−1	X
ejpam-628	410	2	)	)	PUNCT
ejpam-628	410	3	.	.	PUNCT
ejpam-628	410	4	.	.	PUNCT
ejpam-628	411	1	.	.	PUNCT
ejpam-628	412	1	(	(	PUNCT
ejpam-628	412	2	qt−1	qt−1	PROPN
ejpam-628	412	3	)	)	PUNCT
ejpam-628	412	4	,	,	PUNCT
ejpam-628	412	5	r∼=	r∼=	ADV
ejpam-628	412	6	r1×fq1	r1×fq1	ADJ
ejpam-628	412	7	×	×	NOUN
ejpam-628	412	8	.	.	PUNCT
ejpam-628	412	9	.	.	PUNCT
ejpam-628	413	1	.×fqt	.×fqt	PUNCT
ejpam-628	414	1	where	where	SCONJ
ejpam-628	414	2	r1	r1	NOUN
ejpam-628	414	3	∼=	∼=	PROPN
ejpam-628	414	4	zp2	zp2	NOUN
ejpam-628	414	5	or	or	CCONJ
ejpam-628	414	6	zp[x]/(x	zp[x]/(x	NUM
ejpam-628	414	7	2	2	NUM
ejpam-628	414	8	)	)	PUNCT
ejpam-628	414	9	with	with	ADP
ejpam-628	414	10	p3	p3	PROPN
ejpam-628	414	11	=	=	SYM
ejpam-628	414	12	pq1q2	pq1q2	PROPN
ejpam-628	414	13	.	.	PUNCT
ejpam-628	414	14	.	.	PUNCT
ejpam-628	414	15	.	.	PUNCT
ejpam-628	415	1	qt−(p−1)(q1−1)(q2−1	qt−(p−1)(q1−1)(q2−1	VERB
ejpam-628	415	2	)	)	PUNCT
ejpam-628	415	3	.	.	PUNCT
ejpam-628	415	4	.	.	PUNCT
ejpam-628	415	5	.	.	PUNCT
ejpam-628	416	1	(	(	PUNCT
ejpam-628	416	2	qt−1	qt−1	PROPN
ejpam-628	416	3	)	)	PUNCT
ejpam-628	416	4	or	or	CCONJ
ejpam-628	416	5	r	r	NOUN
ejpam-628	416	6	∼=	∼=	PROPN
ejpam-628	416	7	r1	r1	NOUN
ejpam-628	416	8	×	×	PROPN
ejpam-628	416	9	fq	fq	PROPN
ejpam-628	416	10	where	where	SCONJ
ejpam-628	416	11	and	and	CCONJ
ejpam-628	416	12	p2	p2	PROPN
ejpam-628	416	13	=	=	SYM
ejpam-628	416	14	p+	p+	PROPN
ejpam-628	416	15	q−	q−	PROPN
ejpam-628	416	16	1	1	NUM
ejpam-628	416	17	and	and	CCONJ
ejpam-628	416	18	r1	r1	PROPN
ejpam-628	416	19	is	be	AUX
ejpam-628	416	20	isomorphic	isomorphic	ADJ
ejpam-628	416	21	to	to	ADP
ejpam-628	416	22	one	one	NUM
ejpam-628	416	23	of	of	ADP
ejpam-628	416	24	the	the	DET
ejpam-628	416	25	local	local	ADJ
ejpam-628	416	26	rings	ring	NOUN
ejpam-628	416	27	of	of	ADP
ejpam-628	416	28	order	order	NOUN
ejpam-628	416	29	p3	p3	PROPN
ejpam-628	416	30	described	describe	VERB
ejpam-628	416	31	in	in	ADP
ejpam-628	416	32	corollary	corollary	ADJ
ejpam-628	416	33	2	2	NUM
ejpam-628	416	34	.	.	PUNCT
ejpam-628	416	35	proof	proof	NOUN
ejpam-628	416	36	.	.	PUNCT
ejpam-628	417	1	by	by	ADP
ejpam-628	417	2	theorem	theorem	NOUN
ejpam-628	417	3	4	4	NUM
ejpam-628	417	4	,	,	PUNCT
ejpam-628	417	5	either	either	CCONJ
ejpam-628	417	6	r	r	NOUN
ejpam-628	417	7	is	be	AUX
ejpam-628	417	8	reduced	reduce	VERB
ejpam-628	417	9	or	or	CCONJ
ejpam-628	417	10	r	r	NOUN
ejpam-628	417	11	∼=	∼=	PROPN
ejpam-628	417	12	r1	r1	NOUN
ejpam-628	417	13	×	×	NOUN
ejpam-628	417	14	fq1	fq1	CCONJ
ejpam-628	417	15	×	×	NOUN
ejpam-628	417	16	.	.	PUNCT
ejpam-628	417	17	.	.	PUNCT
ejpam-628	418	1	.×	.×	PROPN
ejpam-628	418	2	fqt	fqt	PROPN
ejpam-628	418	3	,	,	PUNCT
ejpam-628	418	4	where	where	SCONJ
ejpam-628	418	5	r1	r1	PROPN
ejpam-628	418	6	is	be	AUX
ejpam-628	418	7	a	a	DET
ejpam-628	418	8	local	local	ADJ
ejpam-628	418	9	ring	ring	NOUN
ejpam-628	418	10	with	with	ADP
ejpam-628	418	11	|z(r1)|	|z(r1)|	PROPN
ejpam-628	418	12	=	=	PUNCT
ejpam-628	418	13	p	p	NOUN
ejpam-628	418	14	or	or	CCONJ
ejpam-628	418	15	p2	p2	PROPN
ejpam-628	418	16	and	and	CCONJ
ejpam-628	418	17	t	t	PROPN
ejpam-628	418	18	≥	≥	NUM
ejpam-628	418	19	1	1	NUM
ejpam-628	418	20	.	.	PUNCT
ejpam-628	419	1	now	now	ADV
ejpam-628	419	2	we	we	PRON
ejpam-628	419	3	proceed	proceed	VERB
ejpam-628	419	4	by	by	ADP
ejpam-628	419	5	cases	case	NOUN
ejpam-628	419	6	.	.	PUNCT
ejpam-628	420	1	•	•	NUM
ejpam-628	420	2	case	case	NOUN
ejpam-628	420	3	1	1	NUM
ejpam-628	420	4	:	:	PUNCT
ejpam-628	420	5	r	r	NOUN
ejpam-628	420	6	is	be	AUX
ejpam-628	420	7	reduced	reduce	VERB
ejpam-628	420	8	.	.	PUNCT
ejpam-628	421	1	then	then	ADV
ejpam-628	421	2	r∼=	r∼=	ADV
ejpam-628	421	3	fq1	fq1	ADV
ejpam-628	421	4	×	×	NOUN
ejpam-628	421	5	.	.	PUNCT
ejpam-628	421	6	.	.	PUNCT
ejpam-628	422	1	.×	.×	PROPN
ejpam-628	422	2	fqt	fqt	PROPN
ejpam-628	422	3	with	with	ADP
ejpam-628	422	4	p4	p4	ADJ
ejpam-628	422	5	=	=	SYM
ejpam-628	422	6	q1q2	q1q2	NOUN
ejpam-628	422	7	.	.	PUNCT
ejpam-628	422	8	.	.	PUNCT
ejpam-628	422	9	.	.	PUNCT
ejpam-628	423	1	qt	qt	INTJ
ejpam-628	423	2	−	−	PROPN
ejpam-628	423	3	(	(	PUNCT
ejpam-628	423	4	q1−	q1−	VERB
ejpam-628	423	5	1)(q2−	1)(q2−	NUM
ejpam-628	423	6	1	1	NUM
ejpam-628	423	7	)	)	PUNCT
ejpam-628	423	8	.	.	PUNCT
ejpam-628	423	9	.	.	PUNCT
ejpam-628	423	10	.	.	PUNCT
ejpam-628	424	1	(	(	PUNCT
ejpam-628	424	2	qt	qt	INTJ
ejpam-628	424	3	−	−	NOUN
ejpam-628	424	4	1	1	NUM
ejpam-628	424	5	)	)	PUNCT
ejpam-628	424	6	.	.	PUNCT
ejpam-628	425	1	•	•	NUM
ejpam-628	425	2	case	case	NOUN
ejpam-628	425	3	2	2	NUM
ejpam-628	425	4	:	:	PUNCT
ejpam-628	425	5	r	r	NOUN
ejpam-628	425	6	is	be	AUX
ejpam-628	425	7	not	not	PART
ejpam-628	425	8	reduced	reduce	VERB
ejpam-628	425	9	and	and	CCONJ
ejpam-628	425	10	|z(r1)|	|z(r1)|	NOUN
ejpam-628	425	11	=	=	SYM
ejpam-628	426	1	p.	p.	NOUN
ejpam-628	426	2	then	then	ADV
ejpam-628	426	3	by	by	ADP
ejpam-628	426	4	corollary	corollary	ADJ
ejpam-628	426	5	1	1	NUM
ejpam-628	426	6	,	,	PUNCT
ejpam-628	426	7	r1	r1	PROPN
ejpam-628	426	8	is	be	AUX
ejpam-628	426	9	isomorphic	isomorphic	ADJ
ejpam-628	426	10	to	to	ADP
ejpam-628	426	11	zp2	zp2	PROPN
ejpam-628	426	12	or	or	CCONJ
ejpam-628	426	13	zp[x]/(x	zp[x]/(x	NUM
ejpam-628	426	14	2	2	NUM
ejpam-628	426	15	)	)	PUNCT
ejpam-628	426	16	and	and	CCONJ
ejpam-628	426	17	p3	p3	PROPN
ejpam-628	426	18	=	=	SYM
ejpam-628	426	19	pq1q2	pq1q2	PROPN
ejpam-628	426	20	.	.	PUNCT
ejpam-628	426	21	.	.	PUNCT
ejpam-628	426	22	.	.	PUNCT
ejpam-628	427	1	qt	qt	INTJ
ejpam-628	427	2	−	−	PROPN
ejpam-628	427	3	(	(	PUNCT
ejpam-628	427	4	p−	p−	NOUN
ejpam-628	427	5	1)(q1−	1)(q1−	NUM
ejpam-628	427	6	1)(q2−	1)(q2−	NUM
ejpam-628	427	7	1	1	NUM
ejpam-628	427	8	)	)	PUNCT
ejpam-628	427	9	.	.	PUNCT
ejpam-628	427	10	.	.	PUNCT
ejpam-628	427	11	.	.	PUNCT
ejpam-628	428	1	(	(	PUNCT
ejpam-628	428	2	qt	qt	INTJ
ejpam-628	428	3	−	−	NOUN
ejpam-628	428	4	1	1	NUM
ejpam-628	428	5	)	)	PUNCT
ejpam-628	428	6	.	.	PUNCT
ejpam-628	429	1	•	•	NUM
ejpam-628	429	2	case	case	NOUN
ejpam-628	429	3	3	3	NUM
ejpam-628	429	4	:	:	PUNCT
ejpam-628	429	5	r	r	NOUN
ejpam-628	429	6	is	be	AUX
ejpam-628	429	7	not	not	PART
ejpam-628	429	8	reduced	reduce	VERB
ejpam-628	429	9	and	and	CCONJ
ejpam-628	429	10	|z(r1)|	|z(r1)|	NOUN
ejpam-628	429	11	=	=	NOUN
ejpam-628	429	12	p2	p2	PROPN
ejpam-628	429	13	.	.	PUNCT
ejpam-628	430	1	then	then	ADV
ejpam-628	430	2	by	by	ADP
ejpam-628	430	3	lemma	lemma	PROPN
ejpam-628	430	4	1	1	NUM
ejpam-628	430	5	,	,	PUNCT
ejpam-628	430	6	either	either	CCONJ
ejpam-628	430	7	|r1|	|r1|	PROPN
ejpam-628	430	8	=	=	SYM
ejpam-628	430	9	p3	p3	PROPN
ejpam-628	430	10	or	or	CCONJ
ejpam-628	430	11	|r1|	|r1|	NOUN
ejpam-628	430	12	=	=	SYM
ejpam-628	430	13	p4	p4	NOUN
ejpam-628	430	14	.	.	PUNCT
ejpam-628	431	1	since	since	SCONJ
ejpam-628	431	2	t	t	PROPN
ejpam-628	431	3	≥	≥	NUM
ejpam-628	431	4	1	1	NUM
ejpam-628	431	5	,	,	PUNCT
ejpam-628	431	6	p4	p4	ADJ
ejpam-628	431	7	=	=	SYM
ejpam-628	431	8	|z(r)|	|z(r)|	PROPN
ejpam-628	431	9	>	>	X
ejpam-628	431	10	|r1|	|r1|	PROPN
ejpam-628	431	11	and	and	CCONJ
ejpam-628	431	12	hence	hence	ADV
ejpam-628	431	13	|r1|	|r1|	PROPN
ejpam-628	431	14	=	=	SYM
ejpam-628	431	15	p3	p3	PROPN
ejpam-628	431	16	.	.	PUNCT
ejpam-628	431	17	thus	thus	ADV
ejpam-628	431	18	by	by	ADP
ejpam-628	431	19	corollary	corollary	ADJ
ejpam-628	431	20	2	2	NUM
ejpam-628	431	21	,	,	PUNCT
ejpam-628	431	22	r1	r1	PROPN
ejpam-628	431	23	is	be	AUX
ejpam-628	431	24	isomorphic	isomorphic	ADJ
ejpam-628	431	25	to	to	ADP
ejpam-628	431	26	one	one	NUM
ejpam-628	431	27	of	of	ADP
ejpam-628	431	28	the	the	DET
ejpam-628	431	29	rings	ring	NOUN
ejpam-628	431	30	fp[x	fp[x	PROPN
ejpam-628	431	31	,	,	PUNCT
ejpam-628	431	32	y]/(x	y]/(x	PROPN
ejpam-628	431	33	,	,	PUNCT
ejpam-628	431	34	y)2	y)2	NOUN
ejpam-628	431	35	,	,	PUNCT
ejpam-628	431	36	fp[x]/(x	fp[x]/(x	NOUN
ejpam-628	431	37	3	3	NUM
ejpam-628	431	38	)	)	PUNCT
ejpam-628	431	39	,	,	PUNCT
ejpam-628	431	40	zp3	zp3	PROPN
ejpam-628	431	41	or	or	CCONJ
ejpam-628	431	42	zp2[x]/(px	zp2[x]/(px	PROPN
ejpam-628	431	43	,	,	PUNCT
ejpam-628	431	44	x2−ǫp	x2−ǫp	PROPN
ejpam-628	431	45	)	)	PUNCT
ejpam-628	431	46	where	where	SCONJ
ejpam-628	431	47	ǫ	ǫ	PROPN
ejpam-628	431	48	∈	∈	PROPN
ejpam-628	431	49	σ0	σ0	NOUN
ejpam-628	431	50	2	2	NUM
ejpam-628	431	51	with	with	ADP
ejpam-628	431	52	p2	p2	PROPN
ejpam-628	431	53	=	=	SYM
ejpam-628	431	54	p+	p+	PROPN
ejpam-628	431	55	q−	q−	PROPN
ejpam-628	431	56	1	1	NUM
ejpam-628	431	57	.	.	PUNCT
ejpam-628	431	58	proposition	proposition	NOUN
ejpam-628	431	59	2	2	NUM
ejpam-628	431	60	.	.	PUNCT
ejpam-628	432	1	let	let	VERB
ejpam-628	432	2	r	r	PRON
ejpam-628	432	3	be	be	AUX
ejpam-628	432	4	a	a	DET
ejpam-628	432	5	commutative	commutative	ADJ
ejpam-628	432	6	nonlocal	nonlocal	ADJ
ejpam-628	432	7	ring	ring	NOUN
ejpam-628	432	8	with	with	ADP
ejpam-628	432	9	|z(r)|	|z(r)|	PROPN
ejpam-628	432	10	=	=	PUNCT
ejpam-628	432	11	p5	p5	PROPN
ejpam-628	432	12	where	where	SCONJ
ejpam-628	432	13	p	p	NOUN
ejpam-628	432	14	is	be	AUX
ejpam-628	432	15	a	a	DET
ejpam-628	432	16	prime	prime	ADJ
ejpam-628	432	17	number	number	NOUN
ejpam-628	432	18	.	.	PUNCT
ejpam-628	433	1	then	then	ADV
ejpam-628	433	2	(	(	PUNCT
ejpam-628	433	3	i	i	NOUN
ejpam-628	433	4	)	)	PUNCT
ejpam-628	433	5	r∼=	r∼=	PUNCT
ejpam-628	433	6	fq1	fq1	ADV
ejpam-628	433	7	×	×	NOUN
ejpam-628	433	8	.	.	PUNCT
ejpam-628	433	9	.	.	PUNCT
ejpam-628	434	1	.×	.×	PROPN
ejpam-628	434	2	fqt	fqt	VERB
ejpam-628	434	3	with	with	ADP
ejpam-628	434	4	p5	p5	NOUN
ejpam-628	434	5	=	=	PUNCT
ejpam-628	434	6	q1q2	q1q2	NOUN
ejpam-628	434	7	.	.	PUNCT
ejpam-628	434	8	.	.	PUNCT
ejpam-628	434	9	.	.	PUNCT
ejpam-628	435	1	qt	qt	INTJ
ejpam-628	435	2	−	−	PROPN
ejpam-628	435	3	(	(	PUNCT
ejpam-628	435	4	q1−	q1−	VERB
ejpam-628	435	5	1)(q2−	1)(q2−	NUM
ejpam-628	435	6	1	1	NUM
ejpam-628	435	7	)	)	PUNCT
ejpam-628	435	8	.	.	PUNCT
ejpam-628	435	9	.	.	PUNCT
ejpam-628	435	10	.	.	PUNCT
ejpam-628	436	1	(	(	PUNCT
ejpam-628	436	2	qt	qt	INTJ
ejpam-628	436	3	−	−	PROPN
ejpam-628	436	4	1	1	NUM
ejpam-628	436	5	)	)	PUNCT
ejpam-628	436	6	,	,	PUNCT
ejpam-628	436	7	(	(	PUNCT
ejpam-628	436	8	ii	ii	NOUN
ejpam-628	436	9	)	)	PUNCT
ejpam-628	436	10	r	r	NOUN
ejpam-628	436	11	∼=	∼=	PROPN
ejpam-628	436	12	r1	r1	NOUN
ejpam-628	436	13	×	×	NOUN
ejpam-628	436	14	fq1	fq1	CCONJ
ejpam-628	436	15	×	×	NOUN
ejpam-628	436	16	.	.	PUNCT
ejpam-628	436	17	.	.	PUNCT
ejpam-628	436	18	.	.	PUNCT
ejpam-628	437	1	×	×	NOUN
ejpam-628	437	2	fqt	fqt	NOUN
ejpam-628	437	3	where	where	SCONJ
ejpam-628	437	4	r1	r1	NOUN
ejpam-628	437	5	∼=	∼=	PROPN
ejpam-628	437	6	zp2	zp2	NOUN
ejpam-628	437	7	or	or	CCONJ
ejpam-628	437	8	zp[x]/(x	zp[x]/(x	NUM
ejpam-628	437	9	2	2	NUM
ejpam-628	437	10	)	)	PUNCT
ejpam-628	437	11	with	with	ADP
ejpam-628	437	12	p4	p4	ADJ
ejpam-628	437	13	=	=	SYM
ejpam-628	437	14	pq1q2	pq1q2	PROPN
ejpam-628	437	15	.	.	PUNCT
ejpam-628	437	16	.	.	PUNCT
ejpam-628	437	17	.	.	PUNCT
ejpam-628	438	1	qt	qt	INTJ
ejpam-628	438	2	−	−	PROPN
ejpam-628	439	1	(	(	PUNCT
ejpam-628	439	2	p	p	NOUN
ejpam-628	439	3	−	−	PROPN
ejpam-628	439	4	1)(q1−	1)(q1−	NUM
ejpam-628	439	5	1)(q2−	1)(q2−	NUM
ejpam-628	439	6	1	1	NUM
ejpam-628	439	7	)	)	PUNCT
ejpam-628	439	8	.	.	PUNCT
ejpam-628	439	9	.	.	PUNCT
ejpam-628	440	1	.	.	PUNCT
ejpam-628	441	1	(	(	PUNCT
ejpam-628	441	2	qt	qt	INTJ
ejpam-628	441	3	−	−	PROPN
ejpam-628	441	4	1	1	NUM
ejpam-628	441	5	)	)	PUNCT
ejpam-628	441	6	,	,	PUNCT
ejpam-628	441	7	m.	m.	NOUN
ejpam-628	441	8	behboodi	behboodi	PROPN
ejpam-628	441	9	,	,	PUNCT
ejpam-628	441	10	r.	r.	PROPN
ejpam-628	441	11	beyranvand	beyranvand	PROPN
ejpam-628	441	12	/	/	SYM
ejpam-628	441	13	eur	eur	PROPN
ejpam-628	441	14	.	.	PUNCT
ejpam-628	442	1	j.	j.	PROPN
ejpam-628	442	2	pure	pure	PROPN
ejpam-628	442	3	appl	appl	PROPN
ejpam-628	442	4	.	.	PROPN
ejpam-628	442	5	math	math	PROPN
ejpam-628	442	6	,	,	PUNCT
ejpam-628	442	7	3	3	NUM
ejpam-628	442	8	(	(	PUNCT
ejpam-628	442	9	2010	2010	NUM
ejpam-628	442	10	)	)	PUNCT
ejpam-628	442	11	,	,	PUNCT
ejpam-628	442	12	303	303	NUM
ejpam-628	442	13	-	-	SYM
ejpam-628	442	14	316	316	NUM
ejpam-628	442	15	312	312	NUM
ejpam-628	442	16	(	(	PUNCT
ejpam-628	442	17	iii	iii	NOUN
ejpam-628	442	18	)	)	PUNCT
ejpam-628	442	19	r	r	NOUN
ejpam-628	442	20	∼=	∼=	PROPN
ejpam-628	442	21	r1	r1	NOUN
ejpam-628	442	22	×	×	NOUN
ejpam-628	442	23	fq1	fq1	CCONJ
ejpam-628	442	24	×	×	NOUN
ejpam-628	442	25	.	.	PUNCT
ejpam-628	442	26	.	.	PUNCT
ejpam-628	443	1	.×	.×	PROPN
ejpam-628	443	2	fqt	fqt	PROPN
ejpam-628	443	3	where	where	SCONJ
ejpam-628	443	4	r1	r1	PROPN
ejpam-628	443	5	is	be	AUX
ejpam-628	443	6	isomorphic	isomorphic	ADJ
ejpam-628	443	7	to	to	ADP
ejpam-628	443	8	one	one	NUM
ejpam-628	443	9	of	of	ADP
ejpam-628	443	10	the	the	DET
ejpam-628	443	11	rings	ring	NOUN
ejpam-628	443	12	zp3	zp3	PROPN
ejpam-628	443	13	,	,	PUNCT
ejpam-628	443	14	fp[x	fp[x	PROPN
ejpam-628	443	15	,	,	PUNCT
ejpam-628	443	16	y]/(x	y]/(x	PROPN
ejpam-628	443	17	,	,	PUNCT
ejpam-628	443	18	y)2	y)2	NOUN
ejpam-628	443	19	,	,	PUNCT
ejpam-628	443	20	fp[x]/(x	fp[x]/(x	NOUN
ejpam-628	443	21	3	3	NUM
ejpam-628	443	22	)	)	PUNCT
ejpam-628	443	23	,	,	PUNCT
ejpam-628	443	24	or	or	CCONJ
ejpam-628	443	25	zp2[x]/(px	zp2[x]/(px	X
ejpam-628	443	26	,	,	PUNCT
ejpam-628	443	27	x2−	x2−	PROPN
ejpam-628	443	28	ǫp	ǫp	NOUN
ejpam-628	443	29	)	)	PUNCT
ejpam-628	444	1	where	where	SCONJ
ejpam-628	444	2	ǫ	ǫ	PRON
ejpam-628	444	3	∈	∈	PROPN
ejpam-628	444	4	σ0	σ0	NOUN
ejpam-628	444	5	2	2	NUM
ejpam-628	444	6	and	and	CCONJ
ejpam-628	444	7	p3	p3	PROPN
ejpam-628	444	8	=	=	SYM
ejpam-628	444	9	pq1q2	pq1q2	PROPN
ejpam-628	444	10	.	.	PUNCT
ejpam-628	444	11	.	.	PUNCT
ejpam-628	444	12	.	.	PUNCT
ejpam-628	445	1	qt	qt	INTJ
ejpam-628	445	2	−	−	PROPN
ejpam-628	445	3	(	(	PUNCT
ejpam-628	445	4	p−	p−	NOUN
ejpam-628	445	5	1)(q1−	1)(q1−	NUM
ejpam-628	445	6	1)(q2−	1)(q2−	NUM
ejpam-628	445	7	1	1	NUM
ejpam-628	445	8	)	)	PUNCT
ejpam-628	445	9	.	.	PUNCT
ejpam-628	445	10	.	.	PUNCT
ejpam-628	445	11	.	.	PUNCT
ejpam-628	446	1	(	(	PUNCT
ejpam-628	446	2	qt	qt	INTJ
ejpam-628	446	3	−	−	PROPN
ejpam-628	446	4	1	1	NUM
ejpam-628	446	5	)	)	PUNCT
ejpam-628	446	6	,	,	PUNCT
ejpam-628	446	7	(	(	PUNCT
ejpam-628	446	8	iv	iv	X
ejpam-628	446	9	)	)	PUNCT
ejpam-628	446	10	r∼=	r∼=	PUNCT
ejpam-628	446	11	r1	r1	PROPN
ejpam-628	446	12	×	×	PROPN
ejpam-628	446	13	fq	fq	PROPN
ejpam-628	446	14	where	where	SCONJ
ejpam-628	446	15	r1	r1	NOUN
ejpam-628	446	16	∼=	∼=	PROPN
ejpam-628	446	17	fp2[x]/(x2	fp2[x]/(x2	NOUN
ejpam-628	446	18	)	)	PUNCT
ejpam-628	446	19	or	or	CCONJ
ejpam-628	446	20	gr(p4	gr(p4	NOUN
ejpam-628	446	21	,	,	PUNCT
ejpam-628	446	22	p2	p2	PROPN
ejpam-628	446	23	)	)	PUNCT
ejpam-628	446	24	with	with	ADP
ejpam-628	446	25	p3	p3	PROPN
ejpam-628	446	26	=	=	PUNCT
ejpam-628	446	27	p2	p2	PROPN
ejpam-628	446	28	+	+	CCONJ
ejpam-628	446	29	q−	q−	PROPN
ejpam-628	446	30	1	1	NUM
ejpam-628	446	31	,	,	PUNCT
ejpam-628	446	32	or	or	CCONJ
ejpam-628	446	33	(	(	PUNCT
ejpam-628	446	34	v	v	NOUN
ejpam-628	446	35	)	)	PUNCT
ejpam-628	446	36	r	r	NOUN
ejpam-628	446	37	∼=	∼=	PROPN
ejpam-628	446	38	r1	r1	NOUN
ejpam-628	446	39	×	×	PROPN
ejpam-628	446	40	fq	fq	NOUN
ejpam-628	446	41	and	and	CCONJ
ejpam-628	446	42	p2	p2	PROPN
ejpam-628	446	43	=	=	SYM
ejpam-628	446	44	p+	p+	PROPN
ejpam-628	446	45	q−	q−	PROPN
ejpam-628	446	46	1	1	NUM
ejpam-628	446	47	where	where	SCONJ
ejpam-628	446	48	r1	r1	PROPN
ejpam-628	446	49	is	be	AUX
ejpam-628	446	50	isomorphic	isomorphic	ADJ
ejpam-628	446	51	to	to	ADP
ejpam-628	446	52	one	one	NUM
ejpam-628	446	53	of	of	ADP
ejpam-628	446	54	the	the	DET
ejpam-628	446	55	local	local	ADJ
ejpam-628	446	56	rings	ring	NOUN
ejpam-628	446	57	of	of	ADP
ejpam-628	446	58	order	order	NOUN
ejpam-628	446	59	p4	p4	PROPN
ejpam-628	446	60	described	describe	VERB
ejpam-628	446	61	in	in	ADP
ejpam-628	446	62	corollary	corollary	ADJ
ejpam-628	446	63	3	3	NUM
ejpam-628	446	64	.	.	PUNCT
ejpam-628	447	1	proof	proof	NOUN
ejpam-628	447	2	.	.	PUNCT
ejpam-628	448	1	by	by	ADP
ejpam-628	448	2	theorem	theorem	NOUN
ejpam-628	448	3	4	4	NUM
ejpam-628	448	4	,	,	PUNCT
ejpam-628	448	5	either	either	CCONJ
ejpam-628	448	6	r	r	NOUN
ejpam-628	448	7	is	be	AUX
ejpam-628	448	8	reduced	reduce	VERB
ejpam-628	448	9	or	or	CCONJ
ejpam-628	448	10	r	r	NOUN
ejpam-628	448	11	∼=	∼=	PROPN
ejpam-628	448	12	r1	r1	NOUN
ejpam-628	448	13	×	×	NOUN
ejpam-628	448	14	fq1	fq1	CCONJ
ejpam-628	448	15	×	×	NOUN
ejpam-628	448	16	.	.	PUNCT
ejpam-628	448	17	.	.	PUNCT
ejpam-628	449	1	.×	.×	PROPN
ejpam-628	449	2	fqt	fqt	PROPN
ejpam-628	449	3	,	,	PUNCT
ejpam-628	449	4	where	where	SCONJ
ejpam-628	449	5	r1	r1	PROPN
ejpam-628	449	6	is	be	AUX
ejpam-628	449	7	a	a	DET
ejpam-628	449	8	local	local	ADJ
ejpam-628	449	9	ring	ring	NOUN
ejpam-628	449	10	with	with	ADP
ejpam-628	449	11	|z(r1)|	|z(r1)|	PROPN
ejpam-628	449	12	=	=	SYM
ejpam-628	449	13	p	p	NOUN
ejpam-628	449	14	,	,	PUNCT
ejpam-628	449	15	p2	p2	PROPN
ejpam-628	449	16	or	or	CCONJ
ejpam-628	449	17	p3	p3	PROPN
ejpam-628	449	18	and	and	CCONJ
ejpam-628	449	19	t	t	PROPN
ejpam-628	449	20	≥	≥	NUM
ejpam-628	449	21	1	1	NUM
ejpam-628	449	22	.	.	PUNCT
ejpam-628	450	1	now	now	ADV
ejpam-628	450	2	we	we	PRON
ejpam-628	450	3	proceed	proceed	VERB
ejpam-628	450	4	by	by	ADP
ejpam-628	450	5	cases	case	NOUN
ejpam-628	450	6	.	.	PUNCT
ejpam-628	451	1	•	•	NUM
ejpam-628	451	2	case	case	NOUN
ejpam-628	451	3	1	1	NUM
ejpam-628	451	4	:	:	PUNCT
ejpam-628	451	5	r	r	NOUN
ejpam-628	451	6	is	be	AUX
ejpam-628	451	7	reduced	reduce	VERB
ejpam-628	451	8	.	.	PUNCT
ejpam-628	452	1	then	then	ADV
ejpam-628	452	2	r∼=	r∼=	ADV
ejpam-628	452	3	fq1	fq1	ADV
ejpam-628	452	4	×	×	NOUN
ejpam-628	452	5	.	.	PUNCT
ejpam-628	452	6	.	.	PUNCT
ejpam-628	453	1	.×	.×	PROPN
ejpam-628	453	2	fqt	fqt	VERB
ejpam-628	453	3	with	with	ADP
ejpam-628	453	4	p5	p5	NOUN
ejpam-628	453	5	=	=	PUNCT
ejpam-628	453	6	q1q2	q1q2	NOUN
ejpam-628	453	7	.	.	PUNCT
ejpam-628	453	8	.	.	PUNCT
ejpam-628	453	9	.	.	PUNCT
ejpam-628	454	1	qt	qt	INTJ
ejpam-628	454	2	−	−	PROPN
ejpam-628	454	3	(	(	PUNCT
ejpam-628	454	4	q1	q1	NOUN
ejpam-628	454	5	−	−	PROPN
ejpam-628	454	6	1)(q2−	1)(q2−	NUM
ejpam-628	454	7	1	1	NUM
ejpam-628	454	8	)	)	PUNCT
ejpam-628	454	9	.	.	PUNCT
ejpam-628	454	10	.	.	PUNCT
ejpam-628	454	11	.	.	PUNCT
ejpam-628	455	1	(	(	PUNCT
ejpam-628	455	2	qt	qt	INTJ
ejpam-628	455	3	−	−	PROPN
ejpam-628	455	4	1	1	NUM
ejpam-628	455	5	)	)	PUNCT
ejpam-628	455	6	•	•	NUM
ejpam-628	456	1	case	case	NOUN
ejpam-628	457	1	2	2	NUM
ejpam-628	457	2	:	:	PUNCT
ejpam-628	457	3	r	r	NOUN
ejpam-628	457	4	is	be	AUX
ejpam-628	457	5	not	not	PART
ejpam-628	457	6	reduced	reduce	VERB
ejpam-628	457	7	and	and	CCONJ
ejpam-628	457	8	|z(r1)|	|z(r1)|	NOUN
ejpam-628	457	9	=	=	SYM
ejpam-628	458	1	p.	p.	NOUN
ejpam-628	458	2	then	then	ADV
ejpam-628	458	3	by	by	ADP
ejpam-628	458	4	corollary	corollary	ADJ
ejpam-628	458	5	1	1	NUM
ejpam-628	458	6	,	,	PUNCT
ejpam-628	458	7	r1	r1	PROPN
ejpam-628	458	8	is	be	AUX
ejpam-628	458	9	isomorphic	isomorphic	ADJ
ejpam-628	458	10	to	to	ADP
ejpam-628	458	11	zp2	zp2	PROPN
ejpam-628	458	12	or	or	CCONJ
ejpam-628	458	13	zp[x]/(x	zp[x]/(x	NUM
ejpam-628	458	14	2	2	NUM
ejpam-628	458	15	)	)	PUNCT
ejpam-628	458	16	and	and	CCONJ
ejpam-628	458	17	p4	p4	ADJ
ejpam-628	458	18	=	=	SYM
ejpam-628	458	19	pq1q2	pq1q2	PROPN
ejpam-628	458	20	.	.	PUNCT
ejpam-628	458	21	.	.	PUNCT
ejpam-628	458	22	.	.	PUNCT
ejpam-628	459	1	qt	qt	INTJ
ejpam-628	459	2	−	−	PROPN
ejpam-628	459	3	(	(	PUNCT
ejpam-628	459	4	p−	p−	NOUN
ejpam-628	459	5	1)(q1−	1)(q1−	NUM
ejpam-628	459	6	1)(q2−	1)(q2−	NUM
ejpam-628	459	7	1	1	NUM
ejpam-628	459	8	)	)	PUNCT
ejpam-628	459	9	.	.	PUNCT
ejpam-628	459	10	.	.	PUNCT
ejpam-628	459	11	.	.	PUNCT
ejpam-628	460	1	(	(	PUNCT
ejpam-628	460	2	qt	qt	INTJ
ejpam-628	460	3	−	−	NOUN
ejpam-628	460	4	1	1	NUM
ejpam-628	460	5	)	)	PUNCT
ejpam-628	460	6	.	.	PUNCT
ejpam-628	461	1	•	•	NUM
ejpam-628	461	2	case	case	NOUN
ejpam-628	461	3	3	3	NUM
ejpam-628	461	4	:	:	PUNCT
ejpam-628	461	5	r	r	NOUN
ejpam-628	461	6	is	be	AUX
ejpam-628	461	7	not	not	PART
ejpam-628	461	8	reduced	reduce	VERB
ejpam-628	461	9	and	and	CCONJ
ejpam-628	461	10	|z(r1)|	|z(r1)|	NOUN
ejpam-628	461	11	=	=	NOUN
ejpam-628	461	12	p2	p2	PROPN
ejpam-628	461	13	.	.	PUNCT
ejpam-628	462	1	then	then	ADV
ejpam-628	462	2	by	by	ADP
ejpam-628	462	3	lemma	lemma	PROPN
ejpam-628	462	4	1	1	NUM
ejpam-628	462	5	,	,	PUNCT
ejpam-628	462	6	|r1|	|r1|	NOUN
ejpam-628	462	7	=	=	SYM
ejpam-628	462	8	p3	p3	PROPN
ejpam-628	462	9	or	or	CCONJ
ejpam-628	462	10	p4	p4	ADJ
ejpam-628	462	11	.	.	PUNCT
ejpam-628	463	1	if	if	SCONJ
ejpam-628	463	2	|r1|	|r1|	PROPN
ejpam-628	463	3	=	=	SYM
ejpam-628	463	4	p3	p3	PROPN
ejpam-628	463	5	,	,	PUNCT
ejpam-628	463	6	then	then	ADV
ejpam-628	463	7	by	by	ADP
ejpam-628	463	8	corollary	corollary	ADJ
ejpam-628	463	9	2	2	NUM
ejpam-628	463	10	,	,	PUNCT
ejpam-628	463	11	r1	r1	PROPN
ejpam-628	463	12	is	be	AUX
ejpam-628	463	13	isomorphic	isomorphic	ADJ
ejpam-628	463	14	to	to	ADP
ejpam-628	463	15	one	one	NUM
ejpam-628	463	16	of	of	ADP
ejpam-628	463	17	the	the	DET
ejpam-628	463	18	ring	ring	NOUN
ejpam-628	463	19	zp3	zp3	PROPN
ejpam-628	463	20	,	,	PUNCT
ejpam-628	463	21	fp[x	fp[x	PROPN
ejpam-628	463	22	,	,	PUNCT
ejpam-628	463	23	y]/(x	y]/(x	PROPN
ejpam-628	463	24	,	,	PUNCT
ejpam-628	463	25	y)2	y)2	NOUN
ejpam-628	463	26	,	,	PUNCT
ejpam-628	463	27	fp[x]/(x	fp[x]/(x	NOUN
ejpam-628	463	28	3	3	NUM
ejpam-628	463	29	)	)	PUNCT
ejpam-628	463	30	,	,	PUNCT
ejpam-628	463	31	or	or	CCONJ
ejpam-628	463	32	zp2[x]/(px	zp2[x]/(px	X
ejpam-628	463	33	,	,	PUNCT
ejpam-628	463	34	x2−	x2−	PROPN
ejpam-628	463	35	ǫp	ǫp	NOUN
ejpam-628	463	36	)	)	PUNCT
ejpam-628	464	1	where	where	SCONJ
ejpam-628	464	2	ǫ	ǫ	PRON
ejpam-628	464	3	∈	∈	PROPN
ejpam-628	464	4	σ0	σ0	NOUN
ejpam-628	464	5	2	2	NUM
ejpam-628	464	6	and	and	CCONJ
ejpam-628	464	7	p3	p3	PROPN
ejpam-628	464	8	=	=	SYM
ejpam-628	464	9	pq1q2	pq1q2	PROPN
ejpam-628	464	10	.	.	PUNCT
ejpam-628	464	11	.	.	PUNCT
ejpam-628	464	12	.	.	PUNCT
ejpam-628	465	1	qt	qt	INTJ
ejpam-628	465	2	−	−	PROPN
ejpam-628	465	3	(	(	PUNCT
ejpam-628	465	4	p−	p−	NOUN
ejpam-628	465	5	1)(q1−	1)(q1−	NUM
ejpam-628	465	6	1)(q2−	1)(q2−	NUM
ejpam-628	465	7	1	1	NUM
ejpam-628	465	8	)	)	PUNCT
ejpam-628	465	9	.	.	PUNCT
ejpam-628	465	10	.	.	PUNCT
ejpam-628	465	11	.	.	PUNCT
ejpam-628	466	1	(	(	PUNCT
ejpam-628	466	2	qt	qt	INTJ
ejpam-628	466	3	−	−	NOUN
ejpam-628	466	4	1	1	NUM
ejpam-628	466	5	)	)	PUNCT
ejpam-628	466	6	.	.	PUNCT
ejpam-628	467	1	if	if	SCONJ
ejpam-628	467	2	|r1|	|r1|	NOUN
ejpam-628	467	3	=	=	SYM
ejpam-628	467	4	p4	p4	ADJ
ejpam-628	467	5	,	,	PUNCT
ejpam-628	467	6	then	then	ADV
ejpam-628	467	7	t	t	PROPN
ejpam-628	467	8	=	=	SYM
ejpam-628	467	9	1	1	NUM
ejpam-628	467	10	,	,	PUNCT
ejpam-628	467	11	for	for	ADP
ejpam-628	467	12	if	if	SCONJ
ejpam-628	467	13	not	not	PART
ejpam-628	467	14	,	,	PUNCT
ejpam-628	467	15	then	then	ADV
ejpam-628	467	16	by	by	ADP
ejpam-628	467	17	theorem	theorem	NOUN
ejpam-628	467	18	2	2	NUM
ejpam-628	467	19	,	,	PUNCT
ejpam-628	467	20	qi	qi	X
ejpam-628	467	21	>	>	X
ejpam-628	467	22	p	p	NOUN
ejpam-628	467	23	for	for	ADP
ejpam-628	467	24	some	some	DET
ejpam-628	467	25	i	i	PRON
ejpam-628	467	26	and	and	CCONJ
ejpam-628	467	27	hence	hence	ADV
ejpam-628	467	28	|z(r)|	|z(r)|	PROPN
ejpam-628	467	29	>	>	X
ejpam-628	467	30	|r1|qi	|r1|qi	PROPN
ejpam-628	467	31	>	>	X
ejpam-628	467	32	p5	p5	PROPN
ejpam-628	467	33	,	,	PUNCT
ejpam-628	467	34	a	a	DET
ejpam-628	467	35	contradiction	contradiction	NOUN
ejpam-628	467	36	.	.	PUNCT
ejpam-628	468	1	thus	thus	ADV
ejpam-628	468	2	t	t	X
ejpam-628	468	3	=	=	SYM
ejpam-628	468	4	1	1	NUM
ejpam-628	468	5	and	and	CCONJ
ejpam-628	468	6	so	so	ADV
ejpam-628	468	7	r	r	NOUN
ejpam-628	468	8	∼=	∼=	PROPN
ejpam-628	468	9	r1	r1	NOUN
ejpam-628	468	10	×	×	PROPN
ejpam-628	468	11	fq	fq	PROPN
ejpam-628	468	12	with	with	ADP
ejpam-628	468	13	p3	p3	PROPN
ejpam-628	468	14	=	=	PUNCT
ejpam-628	468	15	p2	p2	PROPN
ejpam-628	469	1	+	+	CCONJ
ejpam-628	469	2	q−	q−	PROPN
ejpam-628	469	3	1	1	NUM
ejpam-628	469	4	.	.	PUNCT
ejpam-628	470	1	moreover	moreover	ADV
ejpam-628	470	2	,	,	PUNCT
ejpam-628	470	3	since	since	SCONJ
ejpam-628	470	4	|z(r1)|	|z(r1)|	NOUN
ejpam-628	470	5	=	=	NOUN
ejpam-628	470	6	p2	p2	PROPN
ejpam-628	470	7	and	and	CCONJ
ejpam-628	470	8	|r1|	|r1|	NOUN
ejpam-628	470	9	=	=	SYM
ejpam-628	470	10	p4	p4	ADJ
ejpam-628	470	11	,	,	PUNCT
ejpam-628	470	12	by	by	ADP
ejpam-628	470	13	[	[	PUNCT
ejpam-628	470	14	9	9	NUM
ejpam-628	470	15	,	,	PUNCT
ejpam-628	470	16	theorem	theorem	VERB
ejpam-628	470	17	12	12	NUM
ejpam-628	470	18	]	]	PUNCT
ejpam-628	470	19	,	,	PUNCT
ejpam-628	470	20	r1	r1	PROPN
ejpam-628	470	21	is	be	AUX
ejpam-628	470	22	isomorphic	isomorphic	ADJ
ejpam-628	470	23	to	to	ADP
ejpam-628	470	24	fp2[x]/(x2	fp2[x]/(x2	NOUN
ejpam-628	470	25	)	)	PUNCT
ejpam-628	470	26	or	or	CCONJ
ejpam-628	470	27	gr(p4	gr(p4	NOUN
ejpam-628	470	28	,	,	PUNCT
ejpam-628	470	29	p2	p2	PROPN
ejpam-628	470	30	)	)	PUNCT
ejpam-628	470	31	.	.	PUNCT
ejpam-628	471	1	•	•	NUM
ejpam-628	471	2	case	case	NOUN
ejpam-628	471	3	4	4	NUM
ejpam-628	471	4	:	:	PUNCT
ejpam-628	471	5	r	r	NOUN
ejpam-628	471	6	is	be	AUX
ejpam-628	471	7	not	not	PART
ejpam-628	471	8	reduced	reduce	VERB
ejpam-628	471	9	and	and	CCONJ
ejpam-628	471	10	|z(r1)|	|z(r1)|	NOUN
ejpam-628	471	11	=	=	SYM
ejpam-628	471	12	p3	p3	PROPN
ejpam-628	471	13	.	.	PUNCT
ejpam-628	472	1	then	then	ADV
ejpam-628	472	2	by	by	ADP
ejpam-628	472	3	lemma	lemma	PROPN
ejpam-628	472	4	1	1	NUM
ejpam-628	472	5	,	,	PUNCT
ejpam-628	472	6	|r1|	|r1|	NOUN
ejpam-628	472	7	=	=	SYM
ejpam-628	472	8	p4	p4	ADJ
ejpam-628	472	9	or	or	CCONJ
ejpam-628	472	10	p6	p6	ADJ
ejpam-628	472	11	.	.	PUNCT
ejpam-628	473	1	if	if	SCONJ
ejpam-628	473	2	|r1|	|r1|	PROPN
ejpam-628	473	3	=	=	SYM
ejpam-628	473	4	p6	p6	PROPN
ejpam-628	473	5	,	,	PUNCT
ejpam-628	473	6	then	then	ADV
ejpam-628	473	7	|z(r)|	|z(r)|	PROPN
ejpam-628	473	8	≥	≥	NOUN
ejpam-628	473	9	p6	p6	PROPN
ejpam-628	473	10	,	,	PUNCT
ejpam-628	473	11	a	a	DET
ejpam-628	473	12	contradiction	contradiction	NOUN
ejpam-628	473	13	(	(	PUNCT
ejpam-628	473	14	we	we	PRON
ejpam-628	473	15	note	note	VERB
ejpam-628	473	16	that	that	SCONJ
ejpam-628	473	17	t	t	PROPN
ejpam-628	473	18	≥	≥	NUM
ejpam-628	473	19	1	1	NUM
ejpam-628	473	20	)	)	PUNCT
ejpam-628	473	21	.	.	PUNCT
ejpam-628	474	1	thus	thus	ADV
ejpam-628	474	2	|r1|	|r1|	ADJ
ejpam-628	474	3	=	=	SYM
ejpam-628	474	4	p4	p4	ADJ
ejpam-628	474	5	and	and	CCONJ
ejpam-628	474	6	so	so	ADV
ejpam-628	474	7	by	by	ADP
ejpam-628	474	8	theorem	theorem	NOUN
ejpam-628	474	9	2	2	NUM
ejpam-628	474	10	,	,	PUNCT
ejpam-628	474	11	t	t	NOUN
ejpam-628	474	12	=	=	SYM
ejpam-628	474	13	1	1	NUM
ejpam-628	474	14	,	,	PUNCT
ejpam-628	474	15	i.e.	i.e.	X
ejpam-628	474	16	,	,	PUNCT
ejpam-628	474	17	r	r	NOUN
ejpam-628	474	18	∼=	∼=	PROPN
ejpam-628	474	19	r1	r1	NOUN
ejpam-628	474	20	×	×	PROPN
ejpam-628	474	21	fq	fq	PROPN
ejpam-628	474	22	with	with	ADP
ejpam-628	474	23	p2	p2	PROPN
ejpam-628	475	1	=	=	PUNCT
ejpam-628	475	2	p	p	X
ejpam-628	476	1	+	+	NOUN
ejpam-628	476	2	q	q	NOUN
ejpam-628	476	3	−	−	PROPN
ejpam-628	477	1	1	1	NUM
ejpam-628	477	2	.	.	PUNCT
ejpam-628	478	1	now	now	ADV
ejpam-628	478	2	since	since	SCONJ
ejpam-628	478	3	|r1|	|r1|	NOUN
ejpam-628	478	4	=	=	SYM
ejpam-628	478	5	p4	p4	ADJ
ejpam-628	478	6	and	and	CCONJ
ejpam-628	478	7	|z(r1)|	|z(r1)|	NOUN
ejpam-628	478	8	=	=	SYM
ejpam-628	478	9	p3	p3	PROPN
ejpam-628	478	10	,	,	PUNCT
ejpam-628	478	11	r1	r1	PROPN
ejpam-628	478	12	is	be	AUX
ejpam-628	478	13	isomorphic	isomorphic	ADJ
ejpam-628	478	14	to	to	ADP
ejpam-628	478	15	one	one	NUM
ejpam-628	478	16	of	of	ADP
ejpam-628	478	17	the	the	DET
ejpam-628	478	18	local	local	ADJ
ejpam-628	478	19	rings	ring	NOUN
ejpam-628	478	20	of	of	ADP
ejpam-628	478	21	order	order	NOUN
ejpam-628	478	22	p4	p4	PROPN
ejpam-628	478	23	described	describe	VERB
ejpam-628	478	24	in	in	ADP
ejpam-628	478	25	corollary	corollary	ADJ
ejpam-628	478	26	3	3	NUM
ejpam-628	478	27	.	.	PUNCT
ejpam-628	479	1	the	the	DET
ejpam-628	479	2	next	next	ADJ
ejpam-628	479	3	theorem	theorem	NOUN
ejpam-628	479	4	characterizes	characterize	VERB
ejpam-628	479	5	commutative	commutative	ADJ
ejpam-628	479	6	rings	ring	NOUN
ejpam-628	479	7	with	with	ADP
ejpam-628	479	8	p7	p7	ADJ
ejpam-628	479	9	zero	zero	NUM
ejpam-628	479	10	-	-	PUNCT
ejpam-628	479	11	divisors	divisor	NOUN
ejpam-628	479	12	.	.	PUNCT
ejpam-628	480	1	theorem	theorem	NOUN
ejpam-628	480	2	6	6	NUM
ejpam-628	480	3	.	.	PUNCT
ejpam-628	481	1	let	let	VERB
ejpam-628	481	2	r	r	PRON
ejpam-628	481	3	be	be	AUX
ejpam-628	481	4	a	a	DET
ejpam-628	481	5	commutative	commutative	ADJ
ejpam-628	481	6	ring	ring	NOUN
ejpam-628	481	7	with	with	ADP
ejpam-628	481	8	|z(r)|	|z(r)|	PROPN
ejpam-628	481	9	=	=	PUNCT
ejpam-628	481	10	p7	p7	PROPN
ejpam-628	481	11	where	where	SCONJ
ejpam-628	481	12	p	p	NOUN
ejpam-628	481	13	is	be	AUX
ejpam-628	481	14	a	a	DET
ejpam-628	481	15	prime	prime	ADJ
ejpam-628	481	16	number	number	NOUN
ejpam-628	481	17	.	.	PUNCT
ejpam-628	482	1	then	then	ADV
ejpam-628	482	2	either	either	CCONJ
ejpam-628	482	3	(	(	PUNCT
ejpam-628	482	4	i	i	NOUN
ejpam-628	482	5	)	)	PUNCT
ejpam-628	482	6	r	r	NOUN
ejpam-628	482	7	is	be	AUX
ejpam-628	482	8	a	a	DET
ejpam-628	482	9	local	local	ADJ
ejpam-628	482	10	ring	ring	NOUN
ejpam-628	482	11	with	with	ADP
ejpam-628	482	12	|r|=	|r|=	NOUN
ejpam-628	482	13	p8	p8	ADJ
ejpam-628	482	14	or	or	CCONJ
ejpam-628	482	15	p14	p14	VERB
ejpam-628	482	16	;	;	PUNCT
ejpam-628	482	17	(	(	PUNCT
ejpam-628	482	18	ii	ii	NOUN
ejpam-628	482	19	)	)	PUNCT
ejpam-628	482	20	r	r	NOUN
ejpam-628	482	21	is	be	AUX
ejpam-628	482	22	a	a	DET
ejpam-628	482	23	reduced	reduce	VERB
ejpam-628	482	24	ring	ring	NOUN
ejpam-628	482	25	and	and	CCONJ
ejpam-628	482	26	so	so	ADV
ejpam-628	482	27	r	r	NOUN
ejpam-628	482	28	∼=	∼=	PROPN
ejpam-628	482	29	fq1	fq1	NUM
ejpam-628	482	30	×	×	NOUN
ejpam-628	482	31	.	.	PUNCT
ejpam-628	482	32	.	.	PUNCT
ejpam-628	483	1	.×	.×	PROPN
ejpam-628	483	2	fqt	fqt	PROPN
ejpam-628	483	3	,	,	PUNCT
ejpam-628	483	4	where	where	SCONJ
ejpam-628	483	5	each	each	DET
ejpam-628	483	6	fqi	fqi	NOUN
ejpam-628	483	7	(	(	PUNCT
ejpam-628	483	8	1	1	NUM
ejpam-628	483	9	≤	≤	NUM
ejpam-628	483	10	i	i	NOUN
ejpam-628	483	11	≤	≤	PROPN
ejpam-628	483	12	t	t	PROPN
ejpam-628	483	13	)	)	PUNCT
ejpam-628	483	14	is	be	AUX
ejpam-628	483	15	a	a	DET
ejpam-628	483	16	finite	finite	ADJ
ejpam-628	483	17	field	field	NOUN
ejpam-628	483	18	and	and	CCONJ
ejpam-628	484	1	p7	p7	ADJ
ejpam-628	484	2	=	=	SYM
ejpam-628	484	3	q1q2	q1q2	PROPN
ejpam-628	484	4	.	.	PUNCT
ejpam-628	484	5	.	.	PUNCT
ejpam-628	484	6	.	.	PUNCT
ejpam-628	485	1	qt	qt	INTJ
ejpam-628	485	2	−	−	PROPN
ejpam-628	485	3	(	(	PUNCT
ejpam-628	485	4	q1−	q1−	VERB
ejpam-628	485	5	1)(q2−	1)(q2−	NUM
ejpam-628	485	6	1	1	NUM
ejpam-628	485	7	)	)	PUNCT
ejpam-628	485	8	.	.	PUNCT
ejpam-628	485	9	.	.	PUNCT
ejpam-628	485	10	.	.	PUNCT
ejpam-628	486	1	(	(	PUNCT
ejpam-628	486	2	qt	qt	INTJ
ejpam-628	486	3	−	−	PROPN
ejpam-628	486	4	1	1	NUM
ejpam-628	486	5	)	)	PUNCT
ejpam-628	486	6	;	;	PUNCT
ejpam-628	486	7	(	(	PUNCT
ejpam-628	486	8	iii	iii	X
ejpam-628	486	9	)	)	PUNCT
ejpam-628	486	10	r	r	NOUN
ejpam-628	486	11	∼=	∼=	PROPN
ejpam-628	486	12	r1	r1	NOUN
ejpam-628	486	13	×	×	NOUN
ejpam-628	486	14	fq1	fq1	CCONJ
ejpam-628	486	15	×	×	NOUN
ejpam-628	486	16	.	.	PUNCT
ejpam-628	486	17	.	.	PUNCT
ejpam-628	487	1	.×	.×	PROPN
ejpam-628	487	2	fqt	fqt	PROPN
ejpam-628	487	3	,	,	PUNCT
ejpam-628	487	4	where	where	SCONJ
ejpam-628	487	5	each	each	DET
ejpam-628	487	6	fqi	fqi	NOUN
ejpam-628	487	7	(	(	PUNCT
ejpam-628	487	8	1	1	NUM
ejpam-628	487	9	≤	≤	NUM
ejpam-628	487	10	i	i	NOUN
ejpam-628	487	11	≤	≤	PROPN
ejpam-628	487	12	t	t	PROPN
ejpam-628	487	13	)	)	PUNCT
ejpam-628	487	14	is	be	AUX
ejpam-628	487	15	a	a	DET
ejpam-628	487	16	finite	finite	ADJ
ejpam-628	487	17	field	field	NOUN
ejpam-628	487	18	and	and	CCONJ
ejpam-628	487	19	r1	r1	PROPN
ejpam-628	487	20	is	be	AUX
ejpam-628	487	21	a	a	DET
ejpam-628	487	22	local	local	ADJ
ejpam-628	487	23	ring	ring	NOUN
ejpam-628	487	24	with	with	ADP
ejpam-628	487	25	|z(r1)|	|z(r1)|	PROPN
ejpam-628	487	26	=	=	SYM
ejpam-628	487	27	pm	pm	NOUN
ejpam-628	487	28	,	,	PUNCT
ejpam-628	487	29	|r1|	|r1|	NOUN
ejpam-628	487	30	=	=	PUNCT
ejpam-628	487	31	pn	pn	PROPN
ejpam-628	488	1	such	such	ADJ
ejpam-628	488	2	that	that	SCONJ
ejpam-628	488	3	0	0	NUM
ejpam-628	488	4	<	<	X
ejpam-628	488	5	m	m	X
ejpam-628	488	6	<	<	X
ejpam-628	488	7	n≤	n≤	PRON
ejpam-628	488	8	6	6	NUM
ejpam-628	488	9	and	and	CCONJ
ejpam-628	488	10	p7	p7	ADJ
ejpam-628	488	11	=	=	SYM
ejpam-628	488	12	pnq1q2	pnq1q2	NOUN
ejpam-628	488	13	.	.	PUNCT
ejpam-628	488	14	.	.	PUNCT
ejpam-628	488	15	.	.	PUNCT
ejpam-628	489	1	qt	qt	INTJ
ejpam-628	489	2	−	−	PROPN
ejpam-628	490	1	(	(	PUNCT
ejpam-628	490	2	p	p	NOUN
ejpam-628	490	3	n−	n−	NOUN
ejpam-628	490	4	pm)(q1−	pm)(q1−	VERB
ejpam-628	490	5	1)(q2−	1)(q2−	NUM
ejpam-628	490	6	1	1	NUM
ejpam-628	490	7	)	)	PUNCT
ejpam-628	490	8	.	.	PUNCT
ejpam-628	490	9	.	.	PUNCT
ejpam-628	491	1	.	.	PUNCT
ejpam-628	492	1	(	(	PUNCT
ejpam-628	492	2	qt	qt	INTJ
ejpam-628	492	3	−	−	PROPN
ejpam-628	492	4	1	1	NUM
ejpam-628	492	5	)	)	PUNCT
ejpam-628	492	6	;	;	PUNCT
ejpam-628	492	7	m.	m.	NOUN
ejpam-628	492	8	behboodi	behboodi	PROPN
ejpam-628	492	9	,	,	PUNCT
ejpam-628	492	10	r.	r.	PROPN
ejpam-628	492	11	beyranvand	beyranvand	PROPN
ejpam-628	492	12	/	/	SYM
ejpam-628	492	13	eur	eur	PROPN
ejpam-628	492	14	.	.	PUNCT
ejpam-628	493	1	j.	j.	PROPN
ejpam-628	493	2	pure	pure	PROPN
ejpam-628	493	3	appl	appl	PROPN
ejpam-628	493	4	.	.	PROPN
ejpam-628	493	5	math	math	PROPN
ejpam-628	493	6	,	,	PUNCT
ejpam-628	493	7	3	3	NUM
ejpam-628	493	8	(	(	PUNCT
ejpam-628	493	9	2010	2010	NUM
ejpam-628	493	10	)	)	PUNCT
ejpam-628	493	11	,	,	PUNCT
ejpam-628	493	12	303	303	NUM
ejpam-628	493	13	-	-	SYM
ejpam-628	493	14	316	316	NUM
ejpam-628	493	15	313	313	NUM
ejpam-628	493	16	(	(	PUNCT
ejpam-628	493	17	iv	iv	X
ejpam-628	493	18	)	)	PUNCT
ejpam-628	493	19	r	r	NOUN
ejpam-628	493	20	∼=	∼=	PROPN
ejpam-628	493	21	r1	r1	NOUN
ejpam-628	493	22	×	×	NOUN
ejpam-628	493	23	r2	r2	NOUN
ejpam-628	493	24	×	×	NOUN
ejpam-628	493	25	fq1	fq1	CCONJ
ejpam-628	493	26	×	×	NOUN
ejpam-628	493	27	.	.	PUNCT
ejpam-628	493	28	.	.	PUNCT
ejpam-628	493	29	.	.	PUNCT
ejpam-628	494	1	×	×	NOUN
ejpam-628	494	2	fqt	fqt	NOUN
ejpam-628	494	3	,	,	PUNCT
ejpam-628	494	4	where	where	SCONJ
ejpam-628	494	5	each	each	DET
ejpam-628	494	6	fqi	fqi	NOUN
ejpam-628	494	7	(	(	PUNCT
ejpam-628	494	8	1	1	NUM
ejpam-628	494	9	≤	≤	NUM
ejpam-628	494	10	i	i	NOUN
ejpam-628	494	11	≤	≤	PROPN
ejpam-628	494	12	t	t	PROPN
ejpam-628	494	13	)	)	PUNCT
ejpam-628	494	14	is	be	AUX
ejpam-628	494	15	a	a	DET
ejpam-628	494	16	finite	finite	ADJ
ejpam-628	494	17	field	field	NOUN
ejpam-628	494	18	,	,	PUNCT
ejpam-628	494	19	each	each	DET
ejpam-628	494	20	ri	ri	PROPN
ejpam-628	494	21	is	be	AUX
ejpam-628	494	22	isomorphic	isomorphic	ADJ
ejpam-628	494	23	to	to	ADP
ejpam-628	494	24	zp2	zp2	PROPN
ejpam-628	494	25	or	or	CCONJ
ejpam-628	494	26	zp[x]/(x	zp[x]/(x	NUM
ejpam-628	494	27	2	2	NUM
ejpam-628	494	28	)	)	PUNCT
ejpam-628	494	29	and	and	CCONJ
ejpam-628	494	30	p5	p5	ADJ
ejpam-628	494	31	=	=	SYM
ejpam-628	494	32	p2q1q2	p2q1q2	PROPN
ejpam-628	494	33	.	.	PUNCT
ejpam-628	494	34	.	.	PUNCT
ejpam-628	494	35	.	.	PUNCT
ejpam-628	495	1	qt	qt	INTJ
ejpam-628	495	2	−	−	PROPN
ejpam-628	495	3	(	(	PUNCT
ejpam-628	495	4	p−	p−	NOUN
ejpam-628	495	5	1)2(q1−	1)2(q1−	NUM
ejpam-628	495	6	1)(q2−	1)(q2−	NUM
ejpam-628	495	7	1	1	NUM
ejpam-628	495	8	)	)	PUNCT
ejpam-628	495	9	.	.	PUNCT
ejpam-628	495	10	.	.	PUNCT
ejpam-628	495	11	.	.	PUNCT
ejpam-628	496	1	(	(	PUNCT
ejpam-628	496	2	qt	qt	INTJ
ejpam-628	496	3	−	−	PROPN
ejpam-628	496	4	1	1	NUM
ejpam-628	496	5	)	)	PUNCT
ejpam-628	496	6	;	;	PUNCT
ejpam-628	496	7	or	or	CCONJ
ejpam-628	496	8	(	(	PUNCT
ejpam-628	496	9	v	v	NOUN
ejpam-628	496	10	)	)	PUNCT
ejpam-628	496	11	r	r	NOUN
ejpam-628	496	12	∼=	∼=	NOUN
ejpam-628	496	13	r1	r1	NOUN
ejpam-628	496	14	×	×	NOUN
ejpam-628	496	15	r2	r2	NOUN
ejpam-628	496	16	×	×	NOUN
ejpam-628	496	17	f5	f5	NOUN
ejpam-628	496	18	where	where	SCONJ
ejpam-628	496	19	r1	r1	PROPN
ejpam-628	496	20	is	be	AUX
ejpam-628	496	21	isomorphic	isomorphic	ADJ
ejpam-628	496	22	to	to	ADP
ejpam-628	496	23	one	one	NUM
ejpam-628	496	24	of	of	ADP
ejpam-628	496	25	the	the	DET
ejpam-628	496	26	rings	ring	NOUN
ejpam-628	496	27	z4	z4	PROPN
ejpam-628	496	28	or	or	CCONJ
ejpam-628	496	29	z2[x]/(x	z2[x]/(x	NUM
ejpam-628	496	30	2	2	NUM
ejpam-628	496	31	)	)	PUNCT
ejpam-628	496	32	and	and	CCONJ
ejpam-628	496	33	r2	r2	PROPN
ejpam-628	496	34	is	be	AUX
ejpam-628	496	35	isomorphic	isomorphic	ADJ
ejpam-628	496	36	to	to	ADP
ejpam-628	496	37	one	one	NUM
ejpam-628	496	38	of	of	ADP
ejpam-628	496	39	the	the	DET
ejpam-628	496	40	rings	ring	NOUN
ejpam-628	496	41	z8	z8	PROPN
ejpam-628	496	42	,	,	PUNCT
ejpam-628	496	43	z2[x	z2[x	PROPN
ejpam-628	496	44	,	,	PUNCT
ejpam-628	496	45	y]/(x	y]/(x	PROPN
ejpam-628	496	46	,	,	PUNCT
ejpam-628	496	47	y)2	y)2	NOUN
ejpam-628	496	48	,	,	PUNCT
ejpam-628	496	49	z2[x]/(x	z2[x]/(x	NUM
ejpam-628	496	50	3	3	NUM
ejpam-628	496	51	)	)	PUNCT
ejpam-628	496	52	,	,	PUNCT
ejpam-628	496	53	or	or	CCONJ
ejpam-628	496	54	z4[x]/(2x	z4[x]/(2x	VERB
ejpam-628	496	55	,	,	PUNCT
ejpam-628	496	56	x2−	x2−	PROPN
ejpam-628	496	57	2ǫ	2ǫ	NOUN
ejpam-628	496	58	)	)	PUNCT
ejpam-628	496	59	where	where	SCONJ
ejpam-628	496	60	ǫ	ǫ	PROPN
ejpam-628	496	61	∈	∈	PROPN
ejpam-628	496	62	σ0	σ0	NOUN
ejpam-628	496	63	2	2	NUM
ejpam-628	496	64	.	.	PUNCT
ejpam-628	496	65	proof	proof	NOUN
ejpam-628	496	66	.	.	PUNCT
ejpam-628	497	1	suppose	suppose	VERB
ejpam-628	497	2	|z(r)|	|z(r)|	PROPN
ejpam-628	497	3	=	=	PUNCT
ejpam-628	497	4	p7	p7	PROPN
ejpam-628	497	5	and	and	CCONJ
ejpam-628	497	6	r	r	NOUN
ejpam-628	497	7	is	be	AUX
ejpam-628	497	8	not	not	PART
ejpam-628	497	9	a	a	DET
ejpam-628	497	10	local	local	ADJ
ejpam-628	497	11	ring	ring	NOUN
ejpam-628	497	12	.	.	PUNCT
ejpam-628	498	1	if	if	SCONJ
ejpam-628	498	2	r	r	NOUN
ejpam-628	498	3	is	be	AUX
ejpam-628	498	4	reduced	reduce	VERB
ejpam-628	498	5	,	,	PUNCT
ejpam-628	498	6	then	then	ADV
ejpam-628	498	7	we	we	PRON
ejpam-628	498	8	are	be	AUX
ejpam-628	498	9	done	do	VERB
ejpam-628	498	10	.	.	PUNCT
ejpam-628	499	1	now	now	ADV
ejpam-628	499	2	suppose	suppose	VERB
ejpam-628	499	3	r	r	NOUN
ejpam-628	499	4	is	be	AUX
ejpam-628	499	5	not	not	PART
ejpam-628	499	6	a	a	DET
ejpam-628	499	7	reduced	reduce	VERB
ejpam-628	499	8	ring	ring	NOUN
ejpam-628	499	9	.	.	PUNCT
ejpam-628	500	1	then	then	ADV
ejpam-628	500	2	by	by	ADP
ejpam-628	500	3	theorem	theorem	NOUN
ejpam-628	500	4	2	2	NUM
ejpam-628	500	5	,	,	PUNCT
ejpam-628	500	6	we	we	PRON
ejpam-628	500	7	can	can	AUX
ejpam-628	500	8	assume	assume	VERB
ejpam-628	500	9	that	that	SCONJ
ejpam-628	500	10	r∼=	r∼=	NOUN
ejpam-628	500	11	r1×	r1×	NOUN
ejpam-628	500	12	.	.	PUNCT
ejpam-628	500	13	.	.	PUNCT
ejpam-628	501	1	.×	.×	NOUN
ejpam-628	501	2	rs	r	VERB
ejpam-628	501	3	×	×	NOUN
ejpam-628	502	1	fq1	fq1	INTJ
ejpam-628	502	2	×	×	NOUN
ejpam-628	502	3	.	.	PUNCT
ejpam-628	502	4	.	.	PUNCT
ejpam-628	503	1	.×	.×	PROPN
ejpam-628	503	2	fqt	fqt	PROPN
ejpam-628	503	3	,	,	PUNCT
ejpam-628	503	4	where	where	SCONJ
ejpam-628	503	5	s	s	X
ejpam-628	503	6	,	,	PUNCT
ejpam-628	503	7	t	t	PROPN
ejpam-628	503	8	≥	≥	NUM
ejpam-628	503	9	1	1	NUM
ejpam-628	503	10	and	and	CCONJ
ejpam-628	503	11	each	each	DET
ejpam-628	503	12	ri	ri	PROPN
ejpam-628	503	13	is	be	AUX
ejpam-628	503	14	a	a	DET
ejpam-628	503	15	local	local	ADJ
ejpam-628	503	16	ring	ring	NOUN
ejpam-628	503	17	with	with	ADP
ejpam-628	503	18	|z(ri)|=	|z(ri)|=	ADJ
ejpam-628	503	19	pti	pti	PROPN
ejpam-628	503	20	,	,	PUNCT
ejpam-628	503	21	|ri|	|ri|	NOUN
ejpam-628	503	22	=	=	SYM
ejpam-628	503	23	pki	pki	NOUN
ejpam-628	503	24	for	for	ADP
ejpam-628	503	25	some	some	DET
ejpam-628	503	26	t	t	NOUN
ejpam-628	504	1	i	i	PRON
ejpam-628	504	2	,	,	PUNCT
ejpam-628	504	3	ki	ki	PROPN
ejpam-628	504	4	≥	≥	PROPN
ejpam-628	504	5	1	1	NUM
ejpam-628	504	6	such	such	ADJ
ejpam-628	504	7	that	that	SCONJ
ejpam-628	504	8	1≤	1≤	NUM
ejpam-628	504	9	s	s	VERB
ejpam-628	504	10	∑	∑	PROPN
ejpam-628	504	11	i=1	i=1	PROPN
ejpam-628	505	1	t	t	PROPN
ejpam-628	506	1	i	i	PRON
ejpam-628	506	2	≤	≤	PROPN
ejpam-628	506	3	s	s	VERB
ejpam-628	506	4	∑	∑	PROPN
ejpam-628	506	5	i=1	i=1	PROPN
ejpam-628	506	6	ki	ki	PROPN
ejpam-628	507	1	−	−	PROPN
ejpam-628	507	2	s	s	PART
ejpam-628	507	3	≤	≤	NOUN
ejpam-628	507	4	7−	7−	NUM
ejpam-628	507	5	s−	s−	PROPN
ejpam-628	507	6	1≤	1≤	PROPN
ejpam-628	507	7	7−	7−	NUM
ejpam-628	507	8	1−	1−	NUM
ejpam-628	507	9	1=	1=	NUM
ejpam-628	507	10	5	5	NUM
ejpam-628	507	11	.	.	PUNCT
ejpam-628	508	1	it	it	PRON
ejpam-628	508	2	follows	follow	VERB
ejpam-628	508	3	that	that	PRON
ejpam-628	508	4	s	s	VERB
ejpam-628	508	5	≤	≤	ADJ
ejpam-628	508	6	5	5	NUM
ejpam-628	508	7	.	.	PUNCT
ejpam-628	509	1	we	we	PRON
ejpam-628	509	2	claim	claim	VERB
ejpam-628	509	3	that	that	SCONJ
ejpam-628	509	4	s	s	VERB
ejpam-628	509	5	=	=	SYM
ejpam-628	509	6	1	1	NUM
ejpam-628	509	7	or	or	CCONJ
ejpam-628	509	8	2	2	NUM
ejpam-628	509	9	,	,	PUNCT
ejpam-628	509	10	for	for	ADP
ejpam-628	509	11	if	if	SCONJ
ejpam-628	509	12	not	not	PART
ejpam-628	509	13	either	either	CCONJ
ejpam-628	509	14	s	s	VERB
ejpam-628	509	15	≥	≥	NOUN
ejpam-628	509	16	4	4	NUM
ejpam-628	509	17	or	or	CCONJ
ejpam-628	509	18	s	s	NOUN
ejpam-628	509	19	=	=	SYM
ejpam-628	509	20	3	3	X
ejpam-628	509	21	.	.	PUNCT
ejpam-628	510	1	if	if	SCONJ
ejpam-628	510	2	s	s	X
ejpam-628	510	3	≥	≥	NOUN
ejpam-628	510	4	4	4	NUM
ejpam-628	510	5	,	,	PUNCT
ejpam-628	510	6	then	then	ADV
ejpam-628	510	7	|z(r)|	|z(r)|	PROPN
ejpam-628	510	8	>	>	X
ejpam-628	510	9	p8	p8	PROPN
ejpam-628	510	10	(	(	PUNCT
ejpam-628	510	11	because	because	SCONJ
ejpam-628	510	12	|ri|	|ri|	NOUN
ejpam-628	510	13	≥	≥	NOUN
ejpam-628	510	14	p2	p2	VERB
ejpam-628	510	15	for	for	ADP
ejpam-628	510	16	all	all	DET
ejpam-628	510	17	i	i	PROPN
ejpam-628	510	18	)	)	PUNCT
ejpam-628	510	19	,	,	PUNCT
ejpam-628	510	20	a	a	DET
ejpam-628	510	21	contradiction	contradiction	NOUN
ejpam-628	510	22	.	.	PUNCT
ejpam-628	511	1	now	now	ADV
ejpam-628	511	2	let	let	VERB
ejpam-628	511	3	s	s	NOUN
ejpam-628	511	4	=	=	NOUN
ejpam-628	511	5	3	3	X
ejpam-628	511	6	.	.	PUNCT
ejpam-628	512	1	if	if	SCONJ
ejpam-628	512	2	|z(ri)|	|z(ri)|	NOUN
ejpam-628	512	3	=	=	SYM
ejpam-628	512	4	pti	pti	PROPN
ejpam-628	512	5	with	with	ADP
ejpam-628	512	6	t	t	PROPN
ejpam-628	512	7	i	i	PRON
ejpam-628	512	8	≥	≥	NUM
ejpam-628	512	9	2	2	NUM
ejpam-628	512	10	for	for	ADP
ejpam-628	512	11	some	some	DET
ejpam-628	512	12	i	i	PROPN
ejpam-628	512	13	,	,	PUNCT
ejpam-628	512	14	then	then	ADV
ejpam-628	512	15	|z(r)|	|z(r)|	PROPN
ejpam-628	512	16	>	>	X
ejpam-628	512	17	p7	p7	PROPN
ejpam-628	512	18	,	,	PUNCT
ejpam-628	512	19	a	a	DET
ejpam-628	512	20	contradiction	contradiction	NOUN
ejpam-628	512	21	.	.	PUNCT
ejpam-628	513	1	thus	thus	ADV
ejpam-628	513	2	|z(ri)|	|z(ri)|	ADP
ejpam-628	513	3	=	=	SYM
ejpam-628	513	4	p	p	PROPN
ejpam-628	513	5	for	for	ADP
ejpam-628	513	6	i	i	PRON
ejpam-628	513	7	=	=	NOUN
ejpam-628	513	8	1,2,3	1,2,3	X
ejpam-628	513	9	.	.	PUNCT
ejpam-628	513	10	now	now	ADV
ejpam-628	513	11	by	by	ADP
ejpam-628	513	12	the	the	DET
ejpam-628	513	13	relation	relation	NOUN
ejpam-628	513	14	(	(	PUNCT
ejpam-628	513	15	1	1	NUM
ejpam-628	513	16	)	)	PUNCT
ejpam-628	513	17	of	of	ADP
ejpam-628	513	18	theorem	theorem	NOUN
ejpam-628	513	19	1	1	NUM
ejpam-628	513	20	,	,	PUNCT
ejpam-628	513	21	we	we	PRON
ejpam-628	513	22	have	have	VERB
ejpam-628	513	23	p7	p7	VERB
ejpam-628	513	24	=	=	SYM
ejpam-628	513	25	p6q1q2	p6q1q2	PROPN
ejpam-628	513	26	.	.	PUNCT
ejpam-628	513	27	.	.	PUNCT
ejpam-628	513	28	.	.	PUNCT
ejpam-628	514	1	qt	qt	INTJ
ejpam-628	514	2	−	−	PROPN
ejpam-628	515	1	(	(	PUNCT
ejpam-628	515	2	p	p	NOUN
ejpam-628	515	3	2	2	NUM
ejpam-628	515	4	−	−	NOUN
ejpam-628	515	5	p)3(q1−	p)3(q1−	NOUN
ejpam-628	515	6	1)(q2−	1)(q2−	NUM
ejpam-628	515	7	1	1	NUM
ejpam-628	515	8	)	)	PUNCT
ejpam-628	515	9	.	.	PUNCT
ejpam-628	515	10	.	.	PUNCT
ejpam-628	516	1	.	.	PUNCT
ejpam-628	517	1	(	(	PUNCT
ejpam-628	517	2	qt	qt	INTJ
ejpam-628	517	3	−	−	NOUN
ejpam-628	517	4	1	1	NUM
ejpam-628	517	5	)	)	PUNCT
ejpam-628	517	6	.	.	PUNCT
ejpam-628	518	1	thus	thus	ADV
ejpam-628	518	2	p4	p4	ADJ
ejpam-628	518	3	=	=	SYM
ejpam-628	518	4	p3q1q2	p3q1q2	NOUN
ejpam-628	518	5	.	.	PUNCT
ejpam-628	518	6	.	.	PUNCT
ejpam-628	518	7	.	.	PUNCT
ejpam-628	519	1	qt	qt	INTJ
ejpam-628	519	2	−	−	PROPN
ejpam-628	519	3	(	(	PUNCT
ejpam-628	519	4	p−	p−	NOUN
ejpam-628	519	5	1)3(q1	1)3(q1	NUM
ejpam-628	519	6	−	−	NUM
ejpam-628	519	7	1)(q2−	1)(q2−	NUM
ejpam-628	519	8	1	1	NUM
ejpam-628	519	9	)	)	PUNCT
ejpam-628	519	10	.	.	PUNCT
ejpam-628	519	11	.	.	PUNCT
ejpam-628	519	12	.	.	PUNCT
ejpam-628	520	1	(	(	PUNCT
ejpam-628	520	2	qt	qt	INTJ
ejpam-628	520	3	−	−	NOUN
ejpam-628	520	4	1	1	NUM
ejpam-628	520	5	)	)	PUNCT
ejpam-628	520	6	and	and	CCONJ
ejpam-628	520	7	so	so	ADV
ejpam-628	520	8	p	p	PRON
ejpam-628	520	9	is	be	AUX
ejpam-628	520	10	a	a	DET
ejpam-628	520	11	divisor	divisor	NOUN
ejpam-628	520	12	of	of	ADP
ejpam-628	520	13	(	(	PUNCT
ejpam-628	520	14	qi	qi	NOUN
ejpam-628	520	15	−	−	PROPN
ejpam-628	520	16	1	1	NUM
ejpam-628	520	17	)	)	PUNCT
ejpam-628	520	18	for	for	ADP
ejpam-628	520	19	some	some	DET
ejpam-628	520	20	i	i	PRON
ejpam-628	520	21	(	(	PUNCT
ejpam-628	520	22	so	so	ADV
ejpam-628	520	23	qi	qi	X
ejpam-628	520	24	>	>	X
ejpam-628	520	25	p	p	X
ejpam-628	520	26	)	)	PUNCT
ejpam-628	520	27	.	.	PUNCT
ejpam-628	521	1	now	now	ADV
ejpam-628	521	2	if	if	SCONJ
ejpam-628	521	3	t	t	PROPN
ejpam-628	521	4	≥	≥	NOUN
ejpam-628	521	5	2	2	NUM
ejpam-628	521	6	,	,	PUNCT
ejpam-628	521	7	then	then	ADV
ejpam-628	521	8	|z(r)|	|z(r)|	PROPN
ejpam-628	521	9	≥	≥	NOUN
ejpam-628	521	10	|r1||r2||r3|qi	|r1||r2||r3|qi	VERB
ejpam-628	521	11	>	>	X
ejpam-628	521	12	p7	p7	PROPN
ejpam-628	521	13	,	,	PUNCT
ejpam-628	521	14	this	this	PRON
ejpam-628	521	15	is	be	AUX
ejpam-628	521	16	a	a	DET
ejpam-628	521	17	contradiction	contradiction	NOUN
ejpam-628	521	18	.	.	PUNCT
ejpam-628	522	1	thus	thus	ADV
ejpam-628	522	2	t	t	X
ejpam-628	522	3	=	=	SYM
ejpam-628	522	4	1	1	NUM
ejpam-628	523	1	and	and	CCONJ
ejpam-628	523	2	so	so	ADV
ejpam-628	523	3	the	the	DET
ejpam-628	523	4	relation	relation	NOUN
ejpam-628	523	5	p4	p4	NOUN
ejpam-628	523	6	=	=	NOUN
ejpam-628	523	7	p3q1	p3q1	NOUN
ejpam-628	523	8	−	−	NOUN
ejpam-628	523	9	(	(	PUNCT
ejpam-628	523	10	p−	p−	NOUN
ejpam-628	523	11	1)3(q1	1)3(q1	NUM
ejpam-628	523	12	−	−	NOUN
ejpam-628	523	13	1	1	NUM
ejpam-628	523	14	)	)	PUNCT
ejpam-628	523	15	implies	imply	VERB
ejpam-628	523	16	that	that	SCONJ
ejpam-628	523	17	p3	p3	PROPN
ejpam-628	523	18	is	be	AUX
ejpam-628	523	19	a	a	DET
ejpam-628	523	20	divisor	divisor	NOUN
ejpam-628	523	21	of	of	ADP
ejpam-628	523	22	(	(	PUNCT
ejpam-628	523	23	q1−1	q1−1	NOUN
ejpam-628	523	24	)	)	PUNCT
ejpam-628	523	25	.	.	PUNCT
ejpam-628	524	1	hence	hence	ADV
ejpam-628	524	2	q1	q1	VERB
ejpam-628	524	3	>	>	X
ejpam-628	524	4	p3	p3	PROPN
ejpam-628	524	5	and	and	CCONJ
ejpam-628	524	6	so	so	ADV
ejpam-628	524	7	|z(r)|	|z(r)|	PROPN
ejpam-628	524	8	≥	≥	PRON
ejpam-628	524	9	|z(r1)||r2||r3|q1	|z(r1)||r2||r3|q1	X
ejpam-628	524	10	>	>	X
ejpam-628	524	11	p7	p7	PROPN
ejpam-628	524	12	,	,	PUNCT
ejpam-628	524	13	a	a	DET
ejpam-628	524	14	contradiction	contradiction	NOUN
ejpam-628	524	15	.	.	PUNCT
ejpam-628	525	1	thus	thus	ADV
ejpam-628	525	2	s	s	VERB
ejpam-628	525	3	≤	≤	NUM
ejpam-628	525	4	2	2	NUM
ejpam-628	525	5	.	.	PUNCT
ejpam-628	525	6	suppose	suppose	VERB
ejpam-628	525	7	s	s	X
ejpam-628	525	8	=	=	SYM
ejpam-628	525	9	1	1	NUM
ejpam-628	525	10	,	,	PUNCT
ejpam-628	525	11	i.e.	i.e.	X
ejpam-628	525	12	,	,	PUNCT
ejpam-628	525	13	r	r	NOUN
ejpam-628	525	14	∼=	∼=	PROPN
ejpam-628	525	15	r1	r1	NOUN
ejpam-628	525	16	×	×	NOUN
ejpam-628	525	17	fq1	fq1	CCONJ
ejpam-628	525	18	×	×	NOUN
ejpam-628	525	19	.	.	PUNCT
ejpam-628	525	20	.	.	PUNCT
ejpam-628	526	1	.×	.×	PROPN
ejpam-628	526	2	fqt	fqt	PROPN
ejpam-628	526	3	.	.	PUNCT
ejpam-628	527	1	then	then	ADV
ejpam-628	527	2	by	by	ADP
ejpam-628	527	3	theorem	theorem	NOUN
ejpam-628	527	4	2	2	NUM
ejpam-628	527	5	,	,	PUNCT
ejpam-628	527	6	either	either	CCONJ
ejpam-628	527	7	r1	r1	PROPN
ejpam-628	527	8	is	be	AUX
ejpam-628	527	9	a	a	DET
ejpam-628	527	10	local	local	ADJ
ejpam-628	527	11	ring	ring	NOUN
ejpam-628	527	12	with	with	ADP
ejpam-628	527	13	|z(r1)|=	|z(r1)|=	ADJ
ejpam-628	528	1	pt1	pt1	X
ejpam-628	528	2	where	where	SCONJ
ejpam-628	528	3	t1	t1	NOUN
ejpam-628	528	4	=	=	NOUN
ejpam-628	528	5	1,2,3	1,2,3	NUM
ejpam-628	528	6	or	or	CCONJ
ejpam-628	528	7	4	4	NUM
ejpam-628	528	8	such	such	ADJ
ejpam-628	528	9	that	that	DET
ejpam-628	528	10	p7	p7	NOUN
ejpam-628	528	11	=	=	PUNCT
ejpam-628	528	12	|r1|q1×	|r1|q1×	NOUN
ejpam-628	528	13	.	.	PUNCT
ejpam-628	528	14	.	.	PUNCT
ejpam-628	529	1	.×	.×	NOUN
ejpam-628	529	2	qt	qt	PROPN
ejpam-628	529	3	−	−	PROPN
ejpam-628	530	1	(	(	PUNCT
ejpam-628	530	2	|r1|	|r1|	NOUN
ejpam-628	530	3	−	−	PROPN
ejpam-628	530	4	pk)(q1−	pk)(q1−	NOUN
ejpam-628	530	5	1)×	1)×	NUM
ejpam-628	530	6	.	.	PUNCT
ejpam-628	530	7	.	.	PUNCT
ejpam-628	530	8	.	.	PUNCT
ejpam-628	531	1	(	(	PUNCT
ejpam-628	531	2	qt	qt	INTJ
ejpam-628	531	3	−	−	PROPN
ejpam-628	531	4	1	1	NUM
ejpam-628	531	5	)	)	PUNCT
ejpam-628	531	6	or	or	CCONJ
ejpam-628	531	7	r∼=	r∼=	NUM
ejpam-628	531	8	r1×	r1×	NOUN
ejpam-628	531	9	fq1	fq1	ADV
ejpam-628	531	10	where	where	SCONJ
ejpam-628	531	11	|r1|	|r1|	PROPN
ejpam-628	531	12	=	=	SYM
ejpam-628	531	13	p6	p6	PROPN
ejpam-628	531	14	,	,	PUNCT
ejpam-628	531	15	|z(r1)|	|z(r1)|	X
ejpam-628	531	16	=	=	PUNCT
ejpam-628	531	17	p5	p5	ADJ
ejpam-628	531	18	and	and	CCONJ
ejpam-628	531	19	p2	p2	PROPN
ejpam-628	531	20	−	−	PROPN
ejpam-628	532	1	p−	p−	NOUN
ejpam-628	532	2	q1	q1	NOUN
ejpam-628	532	3	+	+	CCONJ
ejpam-628	532	4	1=	1=	X
ejpam-628	532	5	0	0	NUM
ejpam-628	532	6	.	.	PUNCT
ejpam-628	533	1	now	now	ADV
ejpam-628	533	2	suppose	suppose	VERB
ejpam-628	533	3	s	s	X
ejpam-628	533	4	=	=	ADJ
ejpam-628	533	5	2	2	NUM
ejpam-628	533	6	.	.	PUNCT
ejpam-628	534	1	the	the	DET
ejpam-628	534	2	proof	proof	NOUN
ejpam-628	534	3	now	now	ADV
ejpam-628	534	4	proceeds	proceed	VERB
ejpam-628	534	5	by	by	ADP
ejpam-628	534	6	cases	case	NOUN
ejpam-628	534	7	.	.	PUNCT
ejpam-628	535	1	•	•	NUM
ejpam-628	535	2	case	case	NOUN
ejpam-628	535	3	1	1	NUM
ejpam-628	535	4	:	:	PUNCT
ejpam-628	535	5	t1	t1	NOUN
ejpam-628	535	6	≥	≥	NUM
ejpam-628	535	7	3	3	NUM
ejpam-628	535	8	or	or	CCONJ
ejpam-628	535	9	t2	t2	PROPN
ejpam-628	535	10	≥	≥	NOUN
ejpam-628	535	11	3	3	NUM
ejpam-628	535	12	.	.	PUNCT
ejpam-628	535	13	without	without	ADP
ejpam-628	535	14	loss	loss	NOUN
ejpam-628	535	15	of	of	ADP
ejpam-628	535	16	generality	generality	NOUN
ejpam-628	535	17	we	we	PRON
ejpam-628	535	18	can	can	AUX
ejpam-628	535	19	assume	assume	VERB
ejpam-628	535	20	that	that	SCONJ
ejpam-628	535	21	t1	t1	PROPN
ejpam-628	535	22	≥	≥	NUM
ejpam-628	535	23	3	3	NUM
ejpam-628	535	24	.	.	PUNCT
ejpam-628	536	1	then	then	ADV
ejpam-628	536	2	|r1|	|r1|	PROPN
ejpam-628	536	3	=	=	SYM
ejpam-628	536	4	pk1	pk1	PROPN
ejpam-628	536	5	,	,	PUNCT
ejpam-628	536	6	and	and	CCONJ
ejpam-628	536	7	|r2|	|r2|	PROPN
ejpam-628	536	8	=	=	PROPN
ejpam-628	536	9	pk2	pk2	PROPN
ejpam-628	536	10	where	where	SCONJ
ejpam-628	536	11	k1	k1	PROPN
ejpam-628	536	12	≥	≥	NUM
ejpam-628	536	13	4	4	NUM
ejpam-628	536	14	and	and	CCONJ
ejpam-628	536	15	k2	k2	ADJ
ejpam-628	536	16	≥	≥	NOUN
ejpam-628	536	17	2	2	NUM
ejpam-628	536	18	.	.	PUNCT
ejpam-628	537	1	if	if	SCONJ
ejpam-628	537	2	k1	k1	PROPN
ejpam-628	537	3	≥	≥	NUM
ejpam-628	537	4	5	5	NUM
ejpam-628	537	5	or	or	CCONJ
ejpam-628	537	6	k2	k2	ADJ
ejpam-628	537	7	≥	≥	NOUN
ejpam-628	537	8	3	3	NUM
ejpam-628	537	9	,	,	PUNCT
ejpam-628	537	10	then	then	ADV
ejpam-628	537	11	|z(r)|	|z(r)|	PROPN
ejpam-628	537	12	>	>	X
ejpam-628	537	13	|r1||r2|	|r1||r2|	PROPN
ejpam-628	537	14	≥	≥	NOUN
ejpam-628	537	15	p7	p7	PROPN
ejpam-628	537	16	,	,	PUNCT
ejpam-628	537	17	a	a	DET
ejpam-628	537	18	contradiction	contradiction	NOUN
ejpam-628	537	19	.	.	PUNCT
ejpam-628	538	1	now	now	ADV
ejpam-628	538	2	let	let	VERB
ejpam-628	538	3	k1	k1	NOUN
ejpam-628	538	4	=	=	SYM
ejpam-628	538	5	4	4	NUM
ejpam-628	538	6	and	and	CCONJ
ejpam-628	538	7	k2	k2	NOUN
ejpam-628	538	8	=	=	SYM
ejpam-628	538	9	2	2	X
ejpam-628	538	10	.	.	PUNCT
ejpam-628	538	11	by	by	ADP
ejpam-628	538	12	theorem	theorem	NOUN
ejpam-628	538	13	2	2	NUM
ejpam-628	538	14	,	,	PUNCT
ejpam-628	538	15	qi	qi	X
ejpam-628	538	16	>	>	X
ejpam-628	538	17	p	p	NOUN
ejpam-628	538	18	for	for	ADP
ejpam-628	538	19	some	some	DET
ejpam-628	538	20	1	1	NUM
ejpam-628	538	21	≤	≤	NUM
ejpam-628	538	22	i	i	PRON
ejpam-628	538	23	≤	≤	ADJ
ejpam-628	538	24	t.	t.	NOUN
ejpam-628	538	25	if	if	SCONJ
ejpam-628	538	26	t	t	PROPN
ejpam-628	538	27	≥	≥	NOUN
ejpam-628	538	28	2	2	NUM
ejpam-628	538	29	,	,	PUNCT
ejpam-628	538	30	then	then	ADV
ejpam-628	538	31	|z(r)|	|z(r)|	PROPN
ejpam-628	538	32	>	>	X
ejpam-628	538	33	|r1||r2|qi	|r1||r2|qi	VERB
ejpam-628	538	34	≥	≥	NOUN
ejpam-628	538	35	p7	p7	PROPN
ejpam-628	538	36	,	,	PUNCT
ejpam-628	538	37	a	a	DET
ejpam-628	538	38	contradiction	contradiction	NOUN
ejpam-628	538	39	.	.	PUNCT
ejpam-628	539	1	thus	thus	ADV
ejpam-628	539	2	t	t	X
ejpam-628	539	3	=	=	SYM
ejpam-628	539	4	1	1	NUM
ejpam-628	539	5	and	and	CCONJ
ejpam-628	539	6	so	so	ADV
ejpam-628	539	7	p7	p7	ADJ
ejpam-628	539	8	=	=	PUNCT
ejpam-628	539	9	p6q1	p6q1	ADP
ejpam-628	539	10	−	−	PROPN
ejpam-628	539	11	(	(	PUNCT
ejpam-628	539	12	p	p	NOUN
ejpam-628	539	13	4	4	NUM
ejpam-628	539	14	−	−	NOUN
ejpam-628	539	15	p3)(p2	p3)(p2	NOUN
ejpam-628	539	16	−	−	PROPN
ejpam-628	539	17	p)(q1	p)(q1	NOUN
ejpam-628	539	18	−	−	NOUN
ejpam-628	539	19	1	1	NUM
ejpam-628	539	20	)	)	PUNCT
ejpam-628	539	21	.	.	PUNCT
ejpam-628	540	1	it	it	PRON
ejpam-628	540	2	follows	follow	VERB
ejpam-628	540	3	that	that	SCONJ
ejpam-628	540	4	p2	p2	PROPN
ejpam-628	540	5	is	be	AUX
ejpam-628	540	6	a	a	DET
ejpam-628	540	7	divisor	divisor	NOUN
ejpam-628	540	8	of	of	ADP
ejpam-628	540	9	q1	q1	PROPN
ejpam-628	540	10	−	−	PROPN
ejpam-628	540	11	1	1	NUM
ejpam-628	540	12	and	and	CCONJ
ejpam-628	540	13	so	so	ADV
ejpam-628	540	14	q1	q1	PROPN
ejpam-628	540	15	>	>	X
ejpam-628	540	16	p2	p2	PROPN
ejpam-628	540	17	.	.	PUNCT
ejpam-628	541	1	hence	hence	ADV
ejpam-628	541	2	|z(r)|	|z(r)|	NOUN
ejpam-628	541	3	>	>	X
ejpam-628	541	4	|z(r1)||r2||fq1	|z(r1)||r2||fq1	PROPN
ejpam-628	541	5	|	|	ADV
ejpam-628	541	6	>	>	X
ejpam-628	541	7	p7	p7	PROPN
ejpam-628	541	8	,	,	PUNCT
ejpam-628	541	9	a	a	DET
ejpam-628	541	10	contradiction	contradiction	NOUN
ejpam-628	541	11	.	.	PUNCT
ejpam-628	542	1	m.	m.	NOUN
ejpam-628	542	2	behboodi	behboodi	PROPN
ejpam-628	542	3	,	,	PUNCT
ejpam-628	542	4	r.	r.	PROPN
ejpam-628	542	5	beyranvand	beyranvand	PROPN
ejpam-628	542	6	/	/	SYM
ejpam-628	542	7	eur	eur	PROPN
ejpam-628	542	8	.	.	PUNCT
ejpam-628	543	1	j.	j.	PROPN
ejpam-628	543	2	pure	pure	PROPN
ejpam-628	543	3	appl	appl	PROPN
ejpam-628	543	4	.	.	PROPN
ejpam-628	543	5	math	math	PROPN
ejpam-628	543	6	,	,	PUNCT
ejpam-628	543	7	3	3	NUM
ejpam-628	543	8	(	(	PUNCT
ejpam-628	543	9	2010	2010	NUM
ejpam-628	543	10	)	)	PUNCT
ejpam-628	543	11	,	,	PUNCT
ejpam-628	543	12	303	303	NUM
ejpam-628	543	13	-	-	SYM
ejpam-628	543	14	316	316	NUM
ejpam-628	543	15	314	314	NUM
ejpam-628	543	16	•	•	NOUN
ejpam-628	543	17	case	case	NOUN
ejpam-628	543	18	2	2	NUM
ejpam-628	543	19	:	:	PUNCT
ejpam-628	543	20	t1	t1	NOUN
ejpam-628	543	21	=	=	SYM
ejpam-628	543	22	t2	t2	NOUN
ejpam-628	543	23	=	=	SYM
ejpam-628	543	24	2	2	NUM
ejpam-628	543	25	i.e.	i.e.	X
ejpam-628	543	26	,	,	PUNCT
ejpam-628	543	27	|z(r1)|	|z(r1)|	NOUN
ejpam-628	543	28	=	=	SYM
ejpam-628	543	29	|z(r2)|	|z(r2)|	PROPN
ejpam-628	543	30	=	=	NOUN
ejpam-628	543	31	p2	p2	NOUN
ejpam-628	543	32	.	.	PUNCT
ejpam-628	544	1	then	then	ADV
ejpam-628	544	2	by	by	ADP
ejpam-628	544	3	lemma	lemma	PROPN
ejpam-628	544	4	1	1	NUM
ejpam-628	544	5	,	,	PUNCT
ejpam-628	544	6	|r1|	|r1|	NOUN
ejpam-628	544	7	=	=	SYM
ejpam-628	544	8	pk1	pk1	NOUN
ejpam-628	544	9	and	and	CCONJ
ejpam-628	544	10	|r2|	|r2|	NOUN
ejpam-628	544	11	=	=	PROPN
ejpam-628	544	12	pk2	pk2	PROPN
ejpam-628	544	13	where	where	SCONJ
ejpam-628	544	14	3	3	NUM
ejpam-628	544	15	≤	≤	NUM
ejpam-628	544	16	k1	k1	NOUN
ejpam-628	544	17	,	,	PUNCT
ejpam-628	544	18	k2	k2	ADJ
ejpam-628	544	19	≤	≤	ADJ
ejpam-628	544	20	4	4	NUM
ejpam-628	544	21	.	.	PUNCT
ejpam-628	545	1	if	if	SCONJ
ejpam-628	545	2	k1	k1	NOUN
ejpam-628	545	3	=	=	SYM
ejpam-628	545	4	4	4	NUM
ejpam-628	545	5	or	or	CCONJ
ejpam-628	545	6	k2	k2	NOUN
ejpam-628	545	7	=	=	SYM
ejpam-628	545	8	4	4	NUM
ejpam-628	545	9	,	,	PUNCT
ejpam-628	545	10	then	then	ADV
ejpam-628	545	11	|z(r)|	|z(r)|	PROPN
ejpam-628	545	12	>	>	X
ejpam-628	545	13	|r1||r2|	|r1||r2|	PROPN
ejpam-628	545	14	≥	≥	NOUN
ejpam-628	545	15	p7	p7	PROPN
ejpam-628	545	16	,	,	PUNCT
ejpam-628	545	17	a	a	DET
ejpam-628	545	18	contradiction	contradiction	NOUN
ejpam-628	545	19	.	.	PUNCT
ejpam-628	546	1	now	now	ADV
ejpam-628	546	2	let	let	VERB
ejpam-628	546	3	k1	k1	NOUN
ejpam-628	546	4	=	=	SYM
ejpam-628	546	5	k2	k2	PROPN
ejpam-628	546	6	=	=	NOUN
ejpam-628	546	7	3	3	X
ejpam-628	546	8	.	.	PUNCT
ejpam-628	546	9	by	by	ADP
ejpam-628	546	10	theorem	theorem	NOUN
ejpam-628	546	11	2	2	NUM
ejpam-628	546	12	,	,	PUNCT
ejpam-628	546	13	qi	qi	X
ejpam-628	546	14	>	>	X
ejpam-628	546	15	p	p	NOUN
ejpam-628	546	16	for	for	ADP
ejpam-628	546	17	some	some	DET
ejpam-628	546	18	1	1	NUM
ejpam-628	546	19	≤	≤	NUM
ejpam-628	546	20	i	i	PRON
ejpam-628	546	21	≤	≤	ADJ
ejpam-628	546	22	t.	t.	NOUN
ejpam-628	546	23	if	if	SCONJ
ejpam-628	546	24	t	t	PROPN
ejpam-628	546	25	≥	≥	NOUN
ejpam-628	546	26	2	2	NUM
ejpam-628	546	27	,	,	PUNCT
ejpam-628	546	28	then	then	ADV
ejpam-628	546	29	|z(r)|	|z(r)|	PROPN
ejpam-628	546	30	>	>	X
ejpam-628	546	31	|r1||r2|qi	|r1||r2|qi	VERB
ejpam-628	546	32	≥	≥	NOUN
ejpam-628	546	33	p7	p7	PROPN
ejpam-628	546	34	,	,	PUNCT
ejpam-628	546	35	a	a	DET
ejpam-628	546	36	contradiction	contradiction	NOUN
ejpam-628	546	37	.	.	PUNCT
ejpam-628	547	1	thus	thus	ADV
ejpam-628	547	2	t	t	X
ejpam-628	547	3	=	=	SYM
ejpam-628	547	4	1	1	NUM
ejpam-628	547	5	and	and	CCONJ
ejpam-628	547	6	so	so	ADV
ejpam-628	547	7	p7	p7	ADJ
ejpam-628	547	8	=	=	PUNCT
ejpam-628	547	9	p6q1−	p6q1−	NOUN
ejpam-628	547	10	(	(	PUNCT
ejpam-628	547	11	p	p	NOUN
ejpam-628	547	12	3−	3−	NUM
ejpam-628	547	13	p2)2(q1−	p2)2(q1−	NUM
ejpam-628	547	14	1	1	NUM
ejpam-628	547	15	)	)	PUNCT
ejpam-628	547	16	.	.	PUNCT
ejpam-628	548	1	it	it	PRON
ejpam-628	548	2	follows	follow	VERB
ejpam-628	548	3	that	that	SCONJ
ejpam-628	548	4	p2	p2	PROPN
ejpam-628	548	5	is	be	AUX
ejpam-628	548	6	a	a	DET
ejpam-628	548	7	divisor	divisor	NOUN
ejpam-628	548	8	of	of	ADP
ejpam-628	548	9	q1−	q1−	NOUN
ejpam-628	548	10	1	1	NUM
ejpam-628	548	11	and	and	CCONJ
ejpam-628	548	12	so	so	ADV
ejpam-628	548	13	q1	q1	PROPN
ejpam-628	548	14	>	>	X
ejpam-628	548	15	p2	p2	PROPN
ejpam-628	548	16	.	.	PUNCT
ejpam-628	549	1	hence	hence	ADV
ejpam-628	549	2	|z(r)|	|z(r)|	NOUN
ejpam-628	549	3	>	>	X
ejpam-628	549	4	|z(r1)||r2||fq1	|z(r1)||r2||fq1	PROPN
ejpam-628	549	5	|	|	ADV
ejpam-628	549	6	>	>	X
ejpam-628	549	7	p7	p7	PROPN
ejpam-628	549	8	,	,	PUNCT
ejpam-628	549	9	a	a	DET
ejpam-628	549	10	contradiction	contradiction	NOUN
ejpam-628	549	11	.	.	PUNCT
ejpam-628	550	1	•	•	NUM
ejpam-628	550	2	case	case	NOUN
ejpam-628	550	3	3	3	NUM
ejpam-628	550	4	:	:	PUNCT
ejpam-628	550	5	t1	t1	NOUN
ejpam-628	550	6	=	=	SYM
ejpam-628	550	7	2	2	NUM
ejpam-628	550	8	and	and	CCONJ
ejpam-628	550	9	t2	t2	NOUN
ejpam-628	550	10	=	=	SYM
ejpam-628	550	11	1	1	NUM
ejpam-628	550	12	or	or	CCONJ
ejpam-628	550	13	t1	t1	NUM
ejpam-628	550	14	=	=	SYM
ejpam-628	550	15	1	1	NUM
ejpam-628	550	16	and	and	CCONJ
ejpam-628	550	17	t2	t2	NOUN
ejpam-628	550	18	=	=	SYM
ejpam-628	550	19	2	2	X
ejpam-628	550	20	.	.	NOUN
ejpam-628	550	21	without	without	ADP
ejpam-628	550	22	loss	loss	NOUN
ejpam-628	550	23	of	of	ADP
ejpam-628	550	24	generality	generality	NOUN
ejpam-628	550	25	we	we	PRON
ejpam-628	550	26	can	can	AUX
ejpam-628	550	27	assume	assume	VERB
ejpam-628	550	28	that	that	SCONJ
ejpam-628	550	29	t1	t1	NOUN
ejpam-628	550	30	=	=	SYM
ejpam-628	550	31	2	2	NUM
ejpam-628	550	32	and	and	CCONJ
ejpam-628	550	33	t2	t2	NOUN
ejpam-628	550	34	=	=	SYM
ejpam-628	550	35	1	1	X
ejpam-628	550	36	.	.	PUNCT
ejpam-628	550	37	by	by	ADP
ejpam-628	550	38	lemma	lemma	PROPN
ejpam-628	550	39	1	1	NUM
ejpam-628	550	40	,	,	PUNCT
ejpam-628	550	41	either	either	CCONJ
ejpam-628	550	42	|r1|	|r1|	PROPN
ejpam-628	550	43	=	=	SYM
ejpam-628	550	44	p3	p3	PROPN
ejpam-628	550	45	or	or	CCONJ
ejpam-628	550	46	|r1|	|r1|	NOUN
ejpam-628	550	47	=	=	PUNCT
ejpam-628	550	48	p4	p4	ADJ
ejpam-628	550	49	and	and	CCONJ
ejpam-628	550	50	|r2|	|r2|	ADJ
ejpam-628	550	51	=	=	ADJ
ejpam-628	550	52	p2	p2	NOUN
ejpam-628	550	53	.	.	PUNCT
ejpam-628	551	1	by	by	ADP
ejpam-628	551	2	the	the	DET
ejpam-628	551	3	proof	proof	NOUN
ejpam-628	551	4	in	in	ADP
ejpam-628	551	5	case	case	NOUN
ejpam-628	551	6	1	1	NUM
ejpam-628	551	7	,	,	PUNCT
ejpam-628	551	8	|r1|	|r1|	PROPN
ejpam-628	551	9	6=	6=	SYM
ejpam-628	551	10	p4	p4	ADJ
ejpam-628	551	11	.	.	PUNCT
ejpam-628	552	1	thus	thus	ADV
ejpam-628	552	2	|r1|	|r1|	PROPN
ejpam-628	552	3	=	=	SYM
ejpam-628	552	4	p3	p3	PROPN
ejpam-628	552	5	and	and	CCONJ
ejpam-628	552	6	|r2|	|r2|	ADJ
ejpam-628	552	7	=	=	NOUN
ejpam-628	552	8	p2	p2	PROPN
ejpam-628	552	9	and	and	CCONJ
ejpam-628	552	10	hence	hence	ADV
ejpam-628	552	11	by	by	ADP
ejpam-628	552	12	the	the	DET
ejpam-628	552	13	relation	relation	NOUN
ejpam-628	552	14	(	(	PUNCT
ejpam-628	552	15	2	2	NUM
ejpam-628	552	16	)	)	PUNCT
ejpam-628	552	17	of	of	ADP
ejpam-628	552	18	theorem	theorem	NOUN
ejpam-628	552	19	2	2	NUM
ejpam-628	552	20	we	we	PRON
ejpam-628	552	21	have	have	VERB
ejpam-628	552	22	p4	p4	ADJ
ejpam-628	552	23	=	=	SYM
ejpam-628	552	24	p2q1q2	p2q1q2	PROPN
ejpam-628	552	25	.	.	PUNCT
ejpam-628	552	26	.	.	PUNCT
ejpam-628	552	27	.	.	PUNCT
ejpam-628	553	1	qt	qt	INTJ
ejpam-628	553	2	−	−	PROPN
ejpam-628	553	3	(	(	PUNCT
ejpam-628	553	4	p−	p−	NOUN
ejpam-628	553	5	1)2(q1−	1)2(q1−	NUM
ejpam-628	553	6	1)(q2−	1)(q2−	NUM
ejpam-628	553	7	1	1	NUM
ejpam-628	553	8	)	)	PUNCT
ejpam-628	553	9	.	.	PUNCT
ejpam-628	553	10	.	.	PUNCT
ejpam-628	553	11	.	.	PUNCT
ejpam-628	554	1	(	(	PUNCT
ejpam-628	554	2	qt	qt	INTJ
ejpam-628	554	3	−	−	NOUN
ejpam-628	554	4	1	1	NUM
ejpam-628	554	5	)	)	PUNCT
ejpam-628	554	6	.	.	PUNCT
ejpam-628	555	1	(	(	PUNCT
ejpam-628	555	2	6	6	X
ejpam-628	555	3	)	)	PUNCT
ejpam-628	555	4	it	it	PRON
ejpam-628	555	5	follows	follow	VERB
ejpam-628	555	6	that	that	SCONJ
ejpam-628	555	7	p2	p2	PROPN
ejpam-628	555	8	is	be	AUX
ejpam-628	555	9	a	a	DET
ejpam-628	555	10	divisor	divisor	NOUN
ejpam-628	555	11	of	of	ADP
ejpam-628	555	12	(	(	PUNCT
ejpam-628	555	13	q1	q1	PROPN
ejpam-628	555	14	−	−	PROPN
ejpam-628	555	15	1)(q2	1)(q2	NUM
ejpam-628	555	16	−	−	PROPN
ejpam-628	555	17	1	1	NUM
ejpam-628	555	18	)	)	PUNCT
ejpam-628	555	19	.	.	PUNCT
ejpam-628	555	20	.	.	PUNCT
ejpam-628	556	1	.	.	PUNCT
ejpam-628	557	1	(	(	PUNCT
ejpam-628	557	2	qt	qt	INTJ
ejpam-628	557	3	−	−	NOUN
ejpam-628	557	4	1	1	NUM
ejpam-628	557	5	)	)	PUNCT
ejpam-628	557	6	.	.	PUNCT
ejpam-628	558	1	without	without	ADP
ejpam-628	558	2	loss	loss	NOUN
ejpam-628	558	3	of	of	ADP
ejpam-628	558	4	generality	generality	NOUN
ejpam-628	558	5	we	we	PRON
ejpam-628	558	6	may	may	AUX
ejpam-628	558	7	assume	assume	VERB
ejpam-628	558	8	that	that	SCONJ
ejpam-628	558	9	p2	p2	PROPN
ejpam-628	558	10	is	be	AUX
ejpam-628	558	11	a	a	DET
ejpam-628	558	12	divisor	divisor	NOUN
ejpam-628	558	13	of	of	ADP
ejpam-628	558	14	(	(	PUNCT
ejpam-628	558	15	q1	q1	NOUN
ejpam-628	558	16	−	−	NOUN
ejpam-628	558	17	1	1	NUM
ejpam-628	558	18	)	)	PUNCT
ejpam-628	558	19	or	or	CCONJ
ejpam-628	558	20	p	p	NOUN
ejpam-628	558	21	is	be	AUX
ejpam-628	558	22	a	a	DET
ejpam-628	558	23	divisor	divisor	NOUN
ejpam-628	558	24	of	of	ADP
ejpam-628	558	25	(	(	PUNCT
ejpam-628	558	26	q1−	q1−	NOUN
ejpam-628	558	27	1	1	NUM
ejpam-628	558	28	)	)	PUNCT
ejpam-628	558	29	and	and	CCONJ
ejpam-628	558	30	(	(	PUNCT
ejpam-628	558	31	q2	q2	NOUN
ejpam-628	558	32	−	−	PROPN
ejpam-628	558	33	1	1	NUM
ejpam-628	558	34	)	)	PUNCT
ejpam-628	558	35	.	.	PUNCT
ejpam-628	559	1	if	if	SCONJ
ejpam-628	559	2	t	t	PROPN
ejpam-628	559	3	>	>	X
ejpam-628	559	4	2	2	NUM
ejpam-628	559	5	,	,	PUNCT
ejpam-628	559	6	then	then	ADV
ejpam-628	559	7	|z(r)|	|z(r)|	PROPN
ejpam-628	559	8	>	>	X
ejpam-628	559	9	|r1||r2||fq1	|r1||r2||fq1	NUM
ejpam-628	559	10	||fq2	||fq2	PROPN
ejpam-628	559	11	|	|	ADV
ejpam-628	559	12	≥	≥	NOUN
ejpam-628	559	13	p7	p7	PROPN
ejpam-628	559	14	,	,	PUNCT
ejpam-628	559	15	a	a	DET
ejpam-628	559	16	contradiction	contradiction	NOUN
ejpam-628	559	17	.	.	PUNCT
ejpam-628	560	1	thus	thus	ADV
ejpam-628	560	2	t	t	X
ejpam-628	560	3	≤	≤	NUM
ejpam-628	560	4	2	2	NUM
ejpam-628	560	5	and	and	CCONJ
ejpam-628	560	6	so	so	ADV
ejpam-628	560	7	the	the	DET
ejpam-628	560	8	proof	proof	NOUN
ejpam-628	560	9	now	now	ADV
ejpam-628	560	10	proceeds	proceed	VERB
ejpam-628	560	11	by	by	ADP
ejpam-628	560	12	subcases	subcase	NOUN
ejpam-628	560	13	.	.	PUNCT
ejpam-628	561	1	–	–	PUNCT
ejpam-628	561	2	subcase	subcase	NOUN
ejpam-628	561	3	1	1	NUM
ejpam-628	561	4	:	:	PUNCT
ejpam-628	561	5	t	t	NOUN
ejpam-628	561	6	=	=	SYM
ejpam-628	561	7	1	1	NUM
ejpam-628	561	8	i.e.	i.e.	X
ejpam-628	561	9	,	,	PUNCT
ejpam-628	561	10	r∼=	r∼=	NUM
ejpam-628	561	11	r1×	r1×	VERB
ejpam-628	561	12	r2	r2	PROPN
ejpam-628	561	13	×	×	NOUN
ejpam-628	562	1	fq1	fq1	INTJ
ejpam-628	562	2	.	.	PUNCT
ejpam-628	563	1	then	then	ADV
ejpam-628	563	2	by	by	ADP
ejpam-628	563	3	(	(	PUNCT
ejpam-628	563	4	6	6	NUM
ejpam-628	563	5	)	)	PUNCT
ejpam-628	563	6	,	,	PUNCT
ejpam-628	563	7	we	we	PRON
ejpam-628	563	8	have	have	VERB
ejpam-628	563	9	p4	p4	ADJ
ejpam-628	563	10	=	=	SYM
ejpam-628	563	11	p2q1	p2q1	NOUN
ejpam-628	563	12	−	−	PROPN
ejpam-628	563	13	(	(	PUNCT
ejpam-628	563	14	p−	p−	NOUN
ejpam-628	563	15	1)2(q1−	1)2(q1−	NUM
ejpam-628	563	16	1	1	NUM
ejpam-628	563	17	)	)	PUNCT
ejpam-628	563	18	.	.	PUNCT
ejpam-628	564	1	(	(	PUNCT
ejpam-628	564	2	7	7	X
ejpam-628	564	3	)	)	PUNCT
ejpam-628	564	4	this	this	PRON
ejpam-628	564	5	implies	imply	VERB
ejpam-628	564	6	that	that	SCONJ
ejpam-628	564	7	p2	p2	PROPN
ejpam-628	564	8	is	be	AUX
ejpam-628	564	9	a	a	DET
ejpam-628	564	10	divisor	divisor	NOUN
ejpam-628	564	11	of	of	ADP
ejpam-628	564	12	(	(	PUNCT
ejpam-628	564	13	q1	q1	NOUN
ejpam-628	564	14	−	−	NOUN
ejpam-628	564	15	1	1	NUM
ejpam-628	564	16	)	)	PUNCT
ejpam-628	564	17	.	.	PUNCT
ejpam-628	565	1	thus	thus	ADV
ejpam-628	565	2	q1	q1	VERB
ejpam-628	565	3	−	−	PROPN
ejpam-628	565	4	1	1	NUM
ejpam-628	565	5	=	=	SYM
ejpam-628	565	6	p2k	p2k	PROPN
ejpam-628	565	7	for	for	ADP
ejpam-628	565	8	some	some	DET
ejpam-628	565	9	positive	positive	ADJ
ejpam-628	565	10	integer	integer	NOUN
ejpam-628	565	11	k	k	PROPN
ejpam-628	565	12	and	and	CCONJ
ejpam-628	565	13	so	so	ADV
ejpam-628	565	14	by	by	ADP
ejpam-628	565	15	using	use	VERB
ejpam-628	565	16	(	(	PUNCT
ejpam-628	565	17	7	7	NUM
ejpam-628	565	18	)	)	PUNCT
ejpam-628	565	19	we	we	PRON
ejpam-628	565	20	obtain	obtain	VERB
ejpam-628	565	21	p2−2pk+k−1=	p2−2pk+k−1=	PROPN
ejpam-628	565	22	0	0	NUM
ejpam-628	565	23	.	.	PUNCT
ejpam-628	566	1	this	this	DET
ejpam-628	566	2	equation	equation	NOUN
ejpam-628	566	3	implies	imply	VERB
ejpam-628	566	4	that	that	SCONJ
ejpam-628	566	5	p	p	NOUN
ejpam-628	566	6	is	be	AUX
ejpam-628	566	7	a	a	DET
ejpam-628	566	8	divisor	divisor	NOUN
ejpam-628	566	9	of	of	ADP
ejpam-628	566	10	k	k	PROPN
ejpam-628	566	11	−	−	PROPN
ejpam-628	566	12	1	1	NUM
ejpam-628	566	13	,	,	PUNCT
ejpam-628	566	14	i.e.	i.e.	X
ejpam-628	566	15	,	,	PUNCT
ejpam-628	566	16	k	k	PROPN
ejpam-628	566	17	−	−	PROPN
ejpam-628	566	18	1	1	NUM
ejpam-628	566	19	=	=	SYM
ejpam-628	566	20	pλ	pλ	NOUN
ejpam-628	566	21	for	for	ADP
ejpam-628	566	22	some	some	DET
ejpam-628	566	23	non	non	ADJ
ejpam-628	566	24	-	-	ADJ
ejpam-628	566	25	negative	negative	ADJ
ejpam-628	566	26	integer	integer	NOUN
ejpam-628	566	27	λ	λ	PROPN
ejpam-628	566	28	.	.	PUNCT
ejpam-628	567	1	it	it	PRON
ejpam-628	567	2	follows	follow	VERB
ejpam-628	567	3	that	that	SCONJ
ejpam-628	567	4	p	p	PROPN
ejpam-628	567	5	and	and	CCONJ
ejpam-628	567	6	λ	λ	PROPN
ejpam-628	567	7	are	be	AUX
ejpam-628	567	8	solutions	solution	NOUN
ejpam-628	567	9	of	of	ADP
ejpam-628	567	10	x2	x2	NOUN
ejpam-628	567	11	−	−	PROPN
ejpam-628	567	12	2kx	2kx	NOUN
ejpam-628	568	1	+	+	CCONJ
ejpam-628	568	2	k−	k−	NOUN
ejpam-628	568	3	1	1	NUM
ejpam-628	568	4	=	=	SYM
ejpam-628	568	5	0	0	NUM
ejpam-628	569	1	and	and	CCONJ
ejpam-628	569	2	so	so	ADV
ejpam-628	569	3	p+	p+	VERB
ejpam-628	569	4	λ	λ	PROPN
ejpam-628	569	5	=	=	SYM
ejpam-628	569	6	2k	2k	NUM
ejpam-628	569	7	.	.	PUNCT
ejpam-628	570	1	if	if	SCONJ
ejpam-628	570	2	λ	λ	X
ejpam-628	570	3	=	=	SYM
ejpam-628	570	4	1	1	NUM
ejpam-628	570	5	,	,	PUNCT
ejpam-628	570	6	then	then	ADV
ejpam-628	570	7	p	p	NOUN
ejpam-628	570	8	=	=	PUNCT
ejpam-628	571	1	k	k	PROPN
ejpam-628	571	2	−	−	PROPN
ejpam-628	571	3	1	1	NUM
ejpam-628	571	4	and	and	CCONJ
ejpam-628	571	5	p	p	X
ejpam-628	572	1	+	+	NOUN
ejpam-628	572	2	1	1	NUM
ejpam-628	572	3	=	=	SYM
ejpam-628	572	4	2k	2k	NOUN
ejpam-628	572	5	and	and	CCONJ
ejpam-628	572	6	hence	hence	ADV
ejpam-628	572	7	k	k	PROPN
ejpam-628	572	8	=	=	SYM
ejpam-628	572	9	0	0	PROPN
ejpam-628	572	10	,	,	PUNCT
ejpam-628	572	11	a	a	DET
ejpam-628	572	12	contradiction	contradiction	NOUN
ejpam-628	572	13	.	.	PUNCT
ejpam-628	573	1	also	also	ADV
ejpam-628	573	2	,	,	PUNCT
ejpam-628	573	3	if	if	SCONJ
ejpam-628	573	4	λ	λ	X
ejpam-628	573	5	>	>	X
ejpam-628	573	6	1	1	NUM
ejpam-628	573	7	,	,	PUNCT
ejpam-628	573	8	then	then	ADV
ejpam-628	573	9	pλ	pλ	VERB
ejpam-628	573	10	≥	≥	NOUN
ejpam-628	573	11	p	p	NOUN
ejpam-628	573	12	+	+	X
ejpam-628	573	13	λ	λ	PROPN
ejpam-628	573	14	and	and	CCONJ
ejpam-628	573	15	so	so	ADV
ejpam-628	573	16	k	k	PROPN
ejpam-628	573	17	≤	≤	PROPN
ejpam-628	573	18	−1	−1	NOUN
ejpam-628	573	19	,	,	PUNCT
ejpam-628	573	20	a	a	DET
ejpam-628	573	21	contradiction	contradiction	NOUN
ejpam-628	573	22	.	.	PUNCT
ejpam-628	574	1	finally	finally	ADV
ejpam-628	574	2	,	,	PUNCT
ejpam-628	574	3	if	if	SCONJ
ejpam-628	574	4	λ	λ	PROPN
ejpam-628	574	5	=	=	SYM
ejpam-628	574	6	0	0	NUM
ejpam-628	574	7	,	,	PUNCT
ejpam-628	574	8	then	then	ADV
ejpam-628	574	9	p	p	NOUN
ejpam-628	574	10	=	=	PROPN
ejpam-628	574	11	2	2	NUM
ejpam-628	574	12	which	which	PRON
ejpam-628	574	13	yields	yield	VERB
ejpam-628	574	14	q1	q1	PROPN
ejpam-628	574	15	=	=	SYM
ejpam-628	574	16	5	5	NUM
ejpam-628	574	17	,	,	PUNCT
ejpam-628	574	18	i.e.	i.e.	X
ejpam-628	574	19	,	,	PUNCT
ejpam-628	574	20	r	r	NOUN
ejpam-628	574	21	∼=	∼=	PROPN
ejpam-628	574	22	r1	r1	NOUN
ejpam-628	574	23	×	×	NOUN
ejpam-628	574	24	r2	r2	NOUN
ejpam-628	574	25	×	×	NOUN
ejpam-628	574	26	f5	f5	NOUN
ejpam-628	574	27	where	where	SCONJ
ejpam-628	574	28	r1	r1	PROPN
ejpam-628	574	29	and	and	CCONJ
ejpam-628	574	30	r2	r2	PROPN
ejpam-628	574	31	are	be	AUX
ejpam-628	574	32	local	local	ADJ
ejpam-628	574	33	rings	ring	NOUN
ejpam-628	574	34	with	with	ADP
ejpam-628	574	35	|r1|	|r1|	PROPN
ejpam-628	574	36	=	=	SYM
ejpam-628	574	37	8	8	NUM
ejpam-628	574	38	and	and	CCONJ
ejpam-628	574	39	|r2|	|r2|	VERB
ejpam-628	574	40	=	=	ADJ
ejpam-628	574	41	4	4	X
ejpam-628	574	42	.	.	PUNCT
ejpam-628	575	1	since	since	SCONJ
ejpam-628	575	2	|z(r1)|	|z(r1)|	PROPN
ejpam-628	575	3	=	=	NOUN
ejpam-628	575	4	4	4	NUM
ejpam-628	575	5	,	,	PUNCT
ejpam-628	575	6	by	by	ADP
ejpam-628	575	7	[	[	X
ejpam-628	575	8	2	2	NUM
ejpam-628	575	9	,	,	PUNCT
ejpam-628	575	10	p.687	p.687	NOUN
ejpam-628	575	11	]	]	PUNCT
ejpam-628	575	12	,	,	PUNCT
ejpam-628	575	13	r1	r1	PROPN
ejpam-628	575	14	is	be	AUX
ejpam-628	575	15	isomorphic	isomorphic	ADJ
ejpam-628	575	16	to	to	ADP
ejpam-628	575	17	one	one	NUM
ejpam-628	575	18	of	of	ADP
ejpam-628	575	19	the	the	DET
ejpam-628	575	20	rings	ring	NOUN
ejpam-628	575	21	z8	z8	PROPN
ejpam-628	575	22	,	,	PUNCT
ejpam-628	575	23	z2[x	z2[x	PROPN
ejpam-628	575	24	,	,	PUNCT
ejpam-628	575	25	y]/(x	y]/(x	PROPN
ejpam-628	575	26	,	,	PUNCT
ejpam-628	575	27	y)2	y)2	NOUN
ejpam-628	575	28	,	,	PUNCT
ejpam-628	575	29	z2[x]/(x	z2[x]/(x	NUM
ejpam-628	575	30	3	3	NUM
ejpam-628	575	31	)	)	PUNCT
ejpam-628	575	32	,	,	PUNCT
ejpam-628	575	33	or	or	CCONJ
ejpam-628	575	34	z4[x]/(2x	z4[x]/(2x	VERB
ejpam-628	575	35	,	,	PUNCT
ejpam-628	575	36	x2	x2	PROPN
ejpam-628	575	37	−	−	PROPN
ejpam-628	575	38	2ǫ	2ǫ	NOUN
ejpam-628	575	39	)	)	PUNCT
ejpam-628	575	40	where	where	SCONJ
ejpam-628	575	41	ǫ	ǫ	PROPN
ejpam-628	575	42	∈	∈	PROPN
ejpam-628	575	43	σ0	σ0	NOUN
ejpam-628	575	44	2	2	NUM
ejpam-628	575	45	.	.	PUNCT
ejpam-628	576	1	also	also	ADV
ejpam-628	576	2	,	,	PUNCT
ejpam-628	576	3	r2	r2	PROPN
ejpam-628	576	4	is	be	AUX
ejpam-628	576	5	isomorphic	isomorphic	ADJ
ejpam-628	576	6	to	to	ADP
ejpam-628	576	7	z4	z4	PROPN
ejpam-628	576	8	or	or	CCONJ
ejpam-628	576	9	z2[x]/(x	z2[x]/(x	NUM
ejpam-628	576	10	2	2	NUM
ejpam-628	576	11	)	)	PUNCT
ejpam-628	576	12	.	.	PUNCT
ejpam-628	577	1	–	–	PUNCT
ejpam-628	577	2	subcase	subcase	NOUN
ejpam-628	577	3	2	2	NUM
ejpam-628	577	4	:	:	PUNCT
ejpam-628	577	5	t	t	NOUN
ejpam-628	577	6	=	=	SYM
ejpam-628	577	7	2	2	NUM
ejpam-628	577	8	i.e.	i.e.	X
ejpam-628	577	9	,	,	PUNCT
ejpam-628	577	10	r∼=	r∼=	NUM
ejpam-628	577	11	r1×	r1×	VERB
ejpam-628	577	12	r2	r2	PROPN
ejpam-628	577	13	×	×	NOUN
ejpam-628	578	1	fq1	fq1	ADV
ejpam-628	578	2	×	×	NOUN
ejpam-628	578	3	fq2	fq2	NOUN
ejpam-628	578	4	.	.	PUNCT
ejpam-628	579	1	then	then	ADV
ejpam-628	579	2	by	by	ADP
ejpam-628	579	3	(	(	PUNCT
ejpam-628	579	4	6	6	NUM
ejpam-628	579	5	)	)	PUNCT
ejpam-628	579	6	,	,	PUNCT
ejpam-628	579	7	we	we	PRON
ejpam-628	579	8	have	have	VERB
ejpam-628	579	9	p4	p4	ADJ
ejpam-628	579	10	=	=	NOUN
ejpam-628	579	11	p2q1q2−	p2q1q2−	NOUN
ejpam-628	579	12	(	(	PUNCT
ejpam-628	579	13	p−	p−	NOUN
ejpam-628	579	14	1)2(q1−	1)2(q1−	NUM
ejpam-628	579	15	1)(q2−	1)(q2−	NUM
ejpam-628	579	16	1	1	NUM
ejpam-628	579	17	)	)	PUNCT
ejpam-628	579	18	.	.	PUNCT
ejpam-628	580	1	thus	thus	ADV
ejpam-628	580	2	p2	p2	PROPN
ejpam-628	580	3	is	be	AUX
ejpam-628	580	4	a	a	DET
ejpam-628	580	5	divisor	divisor	NOUN
ejpam-628	580	6	of	of	ADP
ejpam-628	580	7	(	(	PUNCT
ejpam-628	580	8	q1−	q1−	NOUN
ejpam-628	580	9	1)(q2−	1)(q2−	NUM
ejpam-628	580	10	1	1	NUM
ejpam-628	580	11	)	)	PUNCT
ejpam-628	580	12	.	.	PUNCT
ejpam-628	581	1	if	if	SCONJ
ejpam-628	581	2	p2	p2	PROPN
ejpam-628	581	3	is	be	AUX
ejpam-628	581	4	a	a	DET
ejpam-628	581	5	divisor	divisor	NOUN
ejpam-628	581	6	of	of	ADP
ejpam-628	581	7	q1−	q1−	NOUN
ejpam-628	581	8	1	1	NUM
ejpam-628	581	9	(	(	PUNCT
ejpam-628	581	10	or	or	CCONJ
ejpam-628	581	11	q2−	q2−	PRON
ejpam-628	581	12	1	1	NUM
ejpam-628	581	13	)	)	PUNCT
ejpam-628	581	14	,	,	PUNCT
ejpam-628	581	15	then	then	ADV
ejpam-628	581	16	q1	q1	VERB
ejpam-628	581	17	>	>	X
ejpam-628	581	18	p2	p2	PROPN
ejpam-628	581	19	(	(	PUNCT
ejpam-628	581	20	or	or	CCONJ
ejpam-628	581	21	q2	q2	NOUN
ejpam-628	581	22	>	>	X
ejpam-628	581	23	p2	p2	PROPN
ejpam-628	581	24	)	)	PUNCT
ejpam-628	581	25	and	and	CCONJ
ejpam-628	581	26	so	so	ADV
ejpam-628	581	27	|z(r)|	|z(r)|	PROPN
ejpam-628	581	28	>	>	X
ejpam-628	581	29	|r1||r2||fqi	|r1||r2||fqi	PROPN
ejpam-628	581	30	|	|	ADV
ejpam-628	581	31	>	>	X
ejpam-628	581	32	p7	p7	PROPN
ejpam-628	581	33	where	where	SCONJ
ejpam-628	581	34	i	i	PRON
ejpam-628	581	35	=	=	VERB
ejpam-628	581	36	1	1	NUM
ejpam-628	581	37	or	or	CCONJ
ejpam-628	581	38	2	2	NUM
ejpam-628	581	39	,	,	PUNCT
ejpam-628	581	40	this	this	PRON
ejpam-628	581	41	is	be	AUX
ejpam-628	581	42	a	a	DET
ejpam-628	581	43	contradiction	contradiction	NOUN
ejpam-628	581	44	.	.	PUNCT
ejpam-628	582	1	thus	thus	ADV
ejpam-628	582	2	p	p	X
ejpam-628	582	3	is	be	AUX
ejpam-628	582	4	a	a	DET
ejpam-628	582	5	divisor	divisor	NOUN
ejpam-628	582	6	of	of	ADP
ejpam-628	582	7	both	both	DET
ejpam-628	582	8	q1	q1	PROPN
ejpam-628	582	9	−	−	PROPN
ejpam-628	582	10	1	1	NUM
ejpam-628	582	11	and	and	CCONJ
ejpam-628	582	12	q2	q2	NOUN
ejpam-628	582	13	−	−	PROPN
ejpam-628	583	1	1	1	X
ejpam-628	583	2	.	.	X
ejpam-628	583	3	hence	hence	ADV
ejpam-628	583	4	q1	q1	VERB
ejpam-628	583	5	−	−	NOUN
ejpam-628	583	6	1	1	NUM
ejpam-628	583	7	=	=	SYM
ejpam-628	583	8	k1p	k1p	ADJ
ejpam-628	583	9	and	and	CCONJ
ejpam-628	583	10	q2−	q2−	VERB
ejpam-628	583	11	1=	1=	X
ejpam-628	583	12	k2p	k2p	X
ejpam-628	583	13	for	for	ADP
ejpam-628	583	14	some	some	DET
ejpam-628	583	15	positive	positive	ADJ
ejpam-628	583	16	integers	integer	NOUN
ejpam-628	583	17	k1	k1	NOUN
ejpam-628	583	18	and	and	CCONJ
ejpam-628	583	19	k2	k2	NOUN
ejpam-628	583	20	.	.	PUNCT
ejpam-628	584	1	then	then	ADV
ejpam-628	584	2	one	one	NUM
ejpam-628	584	3	obtains	obtain	VERB
ejpam-628	584	4	from	from	ADP
ejpam-628	584	5	(	(	PUNCT
ejpam-628	584	6	6	6	NUM
ejpam-628	584	7	)	)	PUNCT
ejpam-628	584	8	,	,	PUNCT
ejpam-628	584	9	p2	p2	PROPN
ejpam-628	584	10	=	=	PUNCT
ejpam-628	585	1	(	(	PUNCT
ejpam-628	585	2	k1p+	k1p+	PROPN
ejpam-628	585	3	1)(k2p+	1)(k2p+	PROPN
ejpam-628	585	4	1)−	1)−	PROPN
ejpam-628	585	5	(	(	PUNCT
ejpam-628	585	6	p−	p−	PROPN
ejpam-628	585	7	1)2k1k2	1)2k1k2	NUM
ejpam-628	585	8	,	,	PUNCT
ejpam-628	585	9	and	and	CCONJ
ejpam-628	585	10	hence	hence	ADV
ejpam-628	585	11	p2	p2	PROPN
ejpam-628	585	12	−	−	PROPN
ejpam-628	586	1	(	(	PUNCT
ejpam-628	586	2	k1	k1	NOUN
ejpam-628	586	3	+	+	CCONJ
ejpam-628	586	4	k2	k2	NOUN
ejpam-628	586	5	+	+	CCONJ
ejpam-628	586	6	2k1k2)p+	2k1k2)p+	NUM
ejpam-628	586	7	k1k2	k1k2	NOUN
ejpam-628	586	8	−	−	PROPN
ejpam-628	586	9	1=	1=	X
ejpam-628	586	10	0	0	NUM
ejpam-628	586	11	.	.	PUNCT
ejpam-628	587	1	(	(	PUNCT
ejpam-628	587	2	8)	8)	NUM
ejpam-628	587	3	references	reference	NOUN
ejpam-628	587	4	315	315	NUM
ejpam-628	587	5	now	now	ADV
ejpam-628	587	6	the	the	DET
ejpam-628	587	7	equation	equation	NOUN
ejpam-628	587	8	(	(	PUNCT
ejpam-628	587	9	8)	8)	NUM
ejpam-628	587	10	shows	show	VERB
ejpam-628	587	11	that	that	SCONJ
ejpam-628	587	12	the	the	DET
ejpam-628	587	13	integer	integer	NOUN
ejpam-628	587	14	p	p	NOUN
ejpam-628	587	15	is	be	AUX
ejpam-628	587	16	a	a	DET
ejpam-628	587	17	solution	solution	NOUN
ejpam-628	587	18	of	of	ADP
ejpam-628	587	19	x	x	X
ejpam-628	587	20	2−	2−	NUM
ejpam-628	587	21	(	(	PUNCT
ejpam-628	587	22	k1	k1	NOUN
ejpam-628	587	23	+	+	CCONJ
ejpam-628	587	24	k2	k2	ADJ
ejpam-628	587	25	+	+	CCONJ
ejpam-628	587	26	2k1k2)x	2k1k2)x	NOUN
ejpam-628	588	1	+	+	CCONJ
ejpam-628	588	2	k1k2	k1k2	X
ejpam-628	588	3	−	−	NUM
ejpam-628	588	4	1=	1=	NOUN
ejpam-628	588	5	0	0	NUM
ejpam-628	588	6	.	.	PUNCT
ejpam-628	589	1	(	(	PUNCT
ejpam-628	589	2	9	9	X
ejpam-628	589	3	)	)	PUNCT
ejpam-628	589	4	now	now	ADV
ejpam-628	589	5	let	let	VERB
ejpam-628	589	6	µ	µ	X
ejpam-628	589	7	be	be	AUX
ejpam-628	589	8	another	another	DET
ejpam-628	589	9	solution	solution	NOUN
ejpam-628	589	10	of	of	ADP
ejpam-628	589	11	(	(	PUNCT
ejpam-628	589	12	9	9	NUM
ejpam-628	589	13	)	)	PUNCT
ejpam-628	589	14	.	.	PUNCT
ejpam-628	590	1	clearly	clearly	ADV
ejpam-628	590	2	µ	µ	VERB
ejpam-628	590	3	6=	6=	NUM
ejpam-628	590	4	1	1	NUM
ejpam-628	590	5	,	,	PUNCT
ejpam-628	590	6	pµ	pµ	PRON
ejpam-628	590	7	=	=	PUNCT
ejpam-628	591	1	k1k2	k1k2	PROPN
ejpam-628	591	2	−	−	PROPN
ejpam-628	591	3	1	1	NUM
ejpam-628	591	4	>	>	SYM
ejpam-628	591	5	0	0	PUNCT
ejpam-628	591	6	and	and	CCONJ
ejpam-628	591	7	p+	p+	PROPN
ejpam-628	591	8	µ	µ	X
ejpam-628	591	9	=	=	SYM
ejpam-628	591	10	k1	k1	PROPN
ejpam-628	591	11	+	+	X
ejpam-628	591	12	k2	k2	NOUN
ejpam-628	591	13	+	+	CCONJ
ejpam-628	591	14	2k1k2	2k1k2	NUM
ejpam-628	591	15	.	.	PUNCT
ejpam-628	592	1	it	it	PRON
ejpam-628	592	2	follows	follow	VERB
ejpam-628	592	3	that	that	SCONJ
ejpam-628	592	4	µ	µ	NOUN
ejpam-628	592	5	is	be	AUX
ejpam-628	592	6	an	an	DET
ejpam-628	592	7	integer≥	integer≥	NOUN
ejpam-628	592	8	2	2	NUM
ejpam-628	592	9	and	and	CCONJ
ejpam-628	592	10	hence	hence	ADV
ejpam-628	592	11	pµ	pµ	ADV
ejpam-628	592	12	≥	≥	NUM
ejpam-628	592	13	p+µ	p+µ	PROPN
ejpam-628	592	14	,	,	PUNCT
ejpam-628	592	15	i.e.	i.e.	X
ejpam-628	592	16	,	,	PUNCT
ejpam-628	592	17	k1k2	k1k2	PROPN
ejpam-628	592	18	−	−	PROPN
ejpam-628	592	19	1	1	NUM
ejpam-628	592	20	>	>	X
ejpam-628	592	21	k1	k1	NOUN
ejpam-628	592	22	+	+	X
ejpam-628	592	23	k2	k2	PROPN
ejpam-628	592	24	+	+	CCONJ
ejpam-628	592	25	2k1k2	2k1k2	NUM
ejpam-628	592	26	,	,	PUNCT
ejpam-628	592	27	a	a	DET
ejpam-628	592	28	contradiction	contradiction	NOUN
ejpam-628	592	29	(	(	PUNCT
ejpam-628	592	30	since	since	SCONJ
ejpam-628	592	31	k1	k1	PROPN
ejpam-628	592	32	,	,	PUNCT
ejpam-628	592	33	k2	k2	X
ejpam-628	592	34	≥	≥	NOUN
ejpam-628	592	35	1	1	NUM
ejpam-628	592	36	)	)	PUNCT
ejpam-628	592	37	.	.	PUNCT
ejpam-628	593	1	•	•	NUM
ejpam-628	593	2	case	case	NOUN
ejpam-628	593	3	4	4	NUM
ejpam-628	593	4	:	:	PUNCT
ejpam-628	593	5	t1	t1	NOUN
ejpam-628	593	6	=	=	SYM
ejpam-628	593	7	t2	t2	NOUN
ejpam-628	593	8	=	=	SYM
ejpam-628	593	9	1	1	NUM
ejpam-628	593	10	i.e.	i.e.	X
ejpam-628	593	11	,	,	PUNCT
ejpam-628	593	12	|z(r1)|	|z(r1)|	NOUN
ejpam-628	593	13	=	=	SYM
ejpam-628	593	14	|z(r2)|	|z(r2)|	PROPN
ejpam-628	593	15	=	=	PUNCT
ejpam-628	593	16	p	p	NOUN
ejpam-628	593	17	and	and	CCONJ
ejpam-628	593	18	r	r	NOUN
ejpam-628	593	19	∼=	∼=	PROPN
ejpam-628	593	20	r1	r1	NOUN
ejpam-628	593	21	×	×	NOUN
ejpam-628	593	22	r2	r2	NOUN
ejpam-628	593	23	×	×	NOUN
ejpam-628	593	24	fq1	fq1	CCONJ
ejpam-628	593	25	×	×	NOUN
ejpam-628	593	26	.	.	PUNCT
ejpam-628	593	27	.	.	PUNCT
ejpam-628	594	1	.×	.×	PROPN
ejpam-628	594	2	fqt	fqt	VERB
ejpam-628	594	3	with	with	ADP
ejpam-628	594	4	p5	p5	PROPN
ejpam-628	594	5	=	=	SYM
ejpam-628	594	6	p2q1q2	p2q1q2	PROPN
ejpam-628	594	7	.	.	PUNCT
ejpam-628	594	8	.	.	PUNCT
ejpam-628	594	9	.	.	PUNCT
ejpam-628	595	1	qt	qt	INTJ
ejpam-628	595	2	−	−	PROPN
ejpam-628	596	1	(	(	PUNCT
ejpam-628	596	2	p	p	X
ejpam-628	596	3	−	−	PROPN
ejpam-628	596	4	1)(q1	1)(q1	NUM
ejpam-628	597	1	−	−	PROPN
ejpam-628	597	2	1)(q2	1)(q2	NUM
ejpam-628	597	3	−	−	NOUN
ejpam-628	597	4	1	1	NUM
ejpam-628	597	5	)	)	PUNCT
ejpam-628	597	6	.	.	PUNCT
ejpam-628	597	7	.	.	PUNCT
ejpam-628	597	8	.	.	PUNCT
ejpam-628	598	1	(	(	PUNCT
ejpam-628	598	2	qt	qt	INTJ
ejpam-628	598	3	−	−	NOUN
ejpam-628	598	4	1	1	NUM
ejpam-628	598	5	)	)	PUNCT
ejpam-628	598	6	.	.	PUNCT
ejpam-628	599	1	on	on	ADP
ejpam-628	599	2	the	the	DET
ejpam-628	599	3	other	other	ADJ
ejpam-628	599	4	hand	hand	NOUN
ejpam-628	599	5	by	by	ADP
ejpam-628	599	6	by	by	ADP
ejpam-628	599	7	[	[	X
ejpam-628	599	8	2	2	NUM
ejpam-628	599	9	,	,	PUNCT
ejpam-628	599	10	p.687	p.687	NOUN
ejpam-628	599	11	]	]	PUNCT
ejpam-628	599	12	,	,	PUNCT
ejpam-628	599	13	r1	r1	PROPN
ejpam-628	599	14	is	be	AUX
ejpam-628	599	15	isomorphic	isomorphic	ADJ
ejpam-628	599	16	to	to	ADP
ejpam-628	599	17	zp2	zp2	PROPN
ejpam-628	599	18	or	or	CCONJ
ejpam-628	599	19	zp[x]/(x	zp[x]/(x	PROPN
ejpam-628	599	20	2	2	NUM
ejpam-628	599	21	)	)	PUNCT
ejpam-628	599	22	.	.	PUNCT
ejpam-628	600	1	finally	finally	ADV
ejpam-628	600	2	,	,	PUNCT
ejpam-628	600	3	we	we	PRON
ejpam-628	600	4	conclude	conclude	VERB
ejpam-628	600	5	the	the	DET
ejpam-628	600	6	article	article	NOUN
ejpam-628	600	7	with	with	ADP
ejpam-628	600	8	the	the	DET
ejpam-628	600	9	next	next	ADJ
ejpam-628	600	10	remark	remark	NOUN
ejpam-628	600	11	which	which	PRON
ejpam-628	600	12	is	be	AUX
ejpam-628	600	13	a	a	DET
ejpam-628	600	14	good	good	ADJ
ejpam-628	600	15	justification	justification	NOUN
ejpam-628	600	16	for	for	ADP
ejpam-628	600	17	the	the	DET
ejpam-628	600	18	classification	classification	NOUN
ejpam-628	600	19	up	up	ADP
ejpam-628	600	20	to	to	AUX
ejpam-628	600	21	isomorphism	isomorphism	VERB
ejpam-628	600	22	commutative	commutative	ADJ
ejpam-628	600	23	rings	ring	NOUN
ejpam-628	600	24	with	with	ADP
ejpam-628	600	25	p1	p1	PROPN
ejpam-628	600	26	k1	k1	NOUN
ejpam-628	600	27	.	.	PUNCT
ejpam-628	600	28	.	.	PUNCT
ejpam-628	600	29	.	.	PUNCT
ejpam-628	601	1	pn	pn	PROPN
ejpam-628	601	2	kn	kn	PROPN
ejpam-628	601	3	zero	zero	NUM
ejpam-628	601	4	-	-	PUNCT
ejpam-628	601	5	divisors	divisor	NOUN
ejpam-628	601	6	,	,	PUNCT
ejpam-628	601	7	where	where	SCONJ
ejpam-628	601	8	1≤	1≤	X
ejpam-628	601	9	ki	ki	PROPN
ejpam-628	601	10	≤	≤	PROPN
ejpam-628	601	11	5	5	NUM
ejpam-628	601	12	.	.	PUNCT
ejpam-628	601	13	remark	remark	NOUN
ejpam-628	601	14	1	1	NUM
ejpam-628	601	15	.	.	PUNCT
ejpam-628	602	1	we	we	PRON
ejpam-628	602	2	remark	remark	VERB
ejpam-628	602	3	that	that	SCONJ
ejpam-628	602	4	by	by	ADP
ejpam-628	602	5	propositions	proposition	NOUN
ejpam-628	602	6	1	1	NUM
ejpam-628	602	7	and	and	CCONJ
ejpam-628	602	8	2	2	NUM
ejpam-628	602	9	,	,	PUNCT
ejpam-628	602	10	for	for	ADP
ejpam-628	602	11	classifying	classify	VERB
ejpam-628	602	12	up	up	ADP
ejpam-628	602	13	to	to	ADP
ejpam-628	602	14	isomorphism	isomorphism	NOUN
ejpam-628	602	15	commutative	commutative	ADJ
ejpam-628	602	16	rings	ring	NOUN
ejpam-628	602	17	with	with	ADP
ejpam-628	602	18	p1	p1	PROPN
ejpam-628	602	19	k1	k1	NOUN
ejpam-628	602	20	.	.	PUNCT
ejpam-628	602	21	.	.	PUNCT
ejpam-628	602	22	.	.	PUNCT
ejpam-628	603	1	pn	pn	PROPN
ejpam-628	603	2	kn	kn	PROPN
ejpam-628	603	3	zero	zero	NUM
ejpam-628	603	4	-	-	PUNCT
ejpam-628	603	5	divisors	divisor	NOUN
ejpam-628	603	6	,	,	PUNCT
ejpam-628	603	7	where	where	SCONJ
ejpam-628	603	8	1	1	NUM
ejpam-628	603	9	≤	≤	NUM
ejpam-628	603	10	ki	ki	X
ejpam-628	603	11	≤	≤	ADV
ejpam-628	603	12	5	5	NUM
ejpam-628	603	13	,	,	PUNCT
ejpam-628	603	14	it	it	PRON
ejpam-628	603	15	suffices	suffice	VERB
ejpam-628	603	16	to	to	PART
ejpam-628	603	17	find	find	VERB
ejpam-628	603	18	local	local	ADJ
ejpam-628	603	19	rings	ring	NOUN
ejpam-628	603	20	with	with	ADP
ejpam-628	603	21	|r|=	|r|=	NOUN
ejpam-628	603	22	p6	p6	PROPN
ejpam-628	603	23	,	,	PUNCT
ejpam-628	603	24	|z(r)|=	|z(r)|=	VERB
ejpam-628	603	25	p4	p4	ADJ
ejpam-628	603	26	and	and	CCONJ
ejpam-628	603	27	|r|=	|r|=	NOUN
ejpam-628	603	28	p6	p6	PROPN
ejpam-628	603	29	,	,	PUNCT
ejpam-628	603	30	|z(r)|=	|z(r)|=	VERB
ejpam-628	603	31	p5	p5	NOUN
ejpam-628	603	32	.	.	PUNCT
ejpam-628	604	1	in	in	ADP
ejpam-628	604	2	fact	fact	NOUN
ejpam-628	604	3	,	,	PUNCT
ejpam-628	604	4	if	if	SCONJ
ejpam-628	604	5	r	r	NOUN
ejpam-628	604	6	is	be	AUX
ejpam-628	604	7	a	a	DET
ejpam-628	604	8	local	local	ADJ
ejpam-628	604	9	ring	ring	NOUN
ejpam-628	604	10	with	with	ADP
ejpam-628	604	11	|z(r)|=	|z(r)|=	VERB
ejpam-628	604	12	p4	p4	ADJ
ejpam-628	604	13	,	,	PUNCT
ejpam-628	604	14	then	then	ADV
ejpam-628	604	15	by	by	ADP
ejpam-628	604	16	lemma	lemma	PROPN
ejpam-628	604	17	1	1	NUM
ejpam-628	604	18	,	,	PUNCT
ejpam-628	604	19	|r|=	|r|=	NOUN
ejpam-628	604	20	p5	p5	ADJ
ejpam-628	604	21	,	,	PUNCT
ejpam-628	604	22	p6	p6	ADJ
ejpam-628	604	23	or	or	CCONJ
ejpam-628	604	24	p8	p8	ADJ
ejpam-628	604	25	.	.	PUNCT
ejpam-628	605	1	the	the	DET
ejpam-628	605	2	local	local	ADJ
ejpam-628	605	3	rings	ring	NOUN
ejpam-628	605	4	of	of	ADP
ejpam-628	605	5	order	order	NOUN
ejpam-628	605	6	p5	p5	ADJ
ejpam-628	605	7	is	be	AUX
ejpam-628	605	8	determined	determine	VERB
ejpam-628	605	9	in	in	ADP
ejpam-628	605	10	[	[	X
ejpam-628	605	11	3	3	NUM
ejpam-628	605	12	]	]	PUNCT
ejpam-628	605	13	.	.	PUNCT
ejpam-628	606	1	also	also	ADV
ejpam-628	606	2	,	,	PUNCT
ejpam-628	606	3	if	if	SCONJ
ejpam-628	606	4	|r|=	|r|=	PRON
ejpam-628	606	5	p8	p8	VERB
ejpam-628	606	6	,	,	PUNCT
ejpam-628	606	7	then	then	ADV
ejpam-628	606	8	by	by	ADP
ejpam-628	606	9	[	[	PUNCT
ejpam-628	606	10	9	9	NUM
ejpam-628	606	11	,	,	PUNCT
ejpam-628	606	12	theorem	theorem	VERB
ejpam-628	606	13	12	12	NUM
ejpam-628	606	14	]	]	PUNCT
ejpam-628	606	15	,	,	PUNCT
ejpam-628	606	16	r	r	NOUN
ejpam-628	606	17	is	be	AUX
ejpam-628	606	18	isomorphic	isomorphic	ADJ
ejpam-628	606	19	to	to	ADP
ejpam-628	606	20	the	the	DET
ejpam-628	606	21	galois	galois	PROPN
ejpam-628	606	22	ring	ring	NOUN
ejpam-628	606	23	gr(p8	gr(p8	NOUN
ejpam-628	606	24	,	,	PUNCT
ejpam-628	606	25	p2	p2	PROPN
ejpam-628	606	26	)	)	PUNCT
ejpam-628	606	27	or	or	CCONJ
ejpam-628	606	28	fp4[x]/(x2	fp4[x]/(x2	NOUN
ejpam-628	606	29	)	)	PUNCT
ejpam-628	606	30	.	.	PUNCT
ejpam-628	607	1	on	on	ADP
ejpam-628	607	2	the	the	DET
ejpam-628	607	3	other	other	ADJ
ejpam-628	607	4	hand	hand	NOUN
ejpam-628	607	5	,	,	PUNCT
ejpam-628	607	6	if	if	SCONJ
ejpam-628	607	7	r	r	NOUN
ejpam-628	607	8	is	be	AUX
ejpam-628	607	9	a	a	DET
ejpam-628	607	10	local	local	ADJ
ejpam-628	607	11	ring	ring	NOUN
ejpam-628	607	12	with	with	ADP
ejpam-628	607	13	|z(r)|=	|z(r)|=	VERB
ejpam-628	607	14	p5	p5	NOUN
ejpam-628	607	15	,	,	PUNCT
ejpam-628	607	16	then	then	ADV
ejpam-628	607	17	by	by	ADP
ejpam-628	607	18	lemma	lemma	PROPN
ejpam-628	607	19	1	1	NUM
ejpam-628	607	20	,	,	PUNCT
ejpam-628	607	21	|r|=	|r|=	NOUN
ejpam-628	607	22	p6	p6	NOUN
ejpam-628	607	23	or	or	CCONJ
ejpam-628	607	24	p10	p10	NOUN
ejpam-628	607	25	.	.	PUNCT
ejpam-628	608	1	if	if	SCONJ
ejpam-628	608	2	|r|=	|r|=	PRON
ejpam-628	608	3	p10	p10	NOUN
ejpam-628	608	4	,	,	PUNCT
ejpam-628	608	5	then	then	ADV
ejpam-628	608	6	by	by	ADP
ejpam-628	608	7	[	[	PUNCT
ejpam-628	608	8	9	9	NUM
ejpam-628	608	9	,	,	PUNCT
ejpam-628	608	10	theorem	theorem	VERB
ejpam-628	608	11	12	12	NUM
ejpam-628	608	12	]	]	PUNCT
ejpam-628	608	13	,	,	PUNCT
ejpam-628	608	14	r	r	NOUN
ejpam-628	608	15	is	be	AUX
ejpam-628	608	16	isomorphic	isomorphic	ADJ
ejpam-628	608	17	to	to	ADP
ejpam-628	608	18	the	the	DET
ejpam-628	608	19	galois	galois	PROPN
ejpam-628	608	20	ring	ring	NOUN
ejpam-628	608	21	gr(p10	gr(p10	PROPN
ejpam-628	608	22	,	,	PUNCT
ejpam-628	608	23	p2	p2	PROPN
ejpam-628	608	24	)	)	PUNCT
ejpam-628	608	25	or	or	CCONJ
ejpam-628	608	26	fp5[x]/(x2	fp5[x]/(x2	NOUN
ejpam-628	608	27	)	)	PUNCT
ejpam-628	608	28	.	.	PUNCT
ejpam-628	609	1	acknowledgements	acknowledgement	NOUN
ejpam-628	609	2	this	this	DET
ejpam-628	609	3	work	work	NOUN
ejpam-628	609	4	was	be	AUX
ejpam-628	609	5	partially	partially	ADV
ejpam-628	609	6	supported	support	VERB
ejpam-628	609	7	by	by	ADP
ejpam-628	609	8	iut	iut	PROPN
ejpam-628	609	9	(	(	PUNCT
ejpam-628	609	10	ceama	ceama	PROPN
ejpam-628	609	11	)	)	PUNCT
ejpam-628	609	12	.	.	PUNCT
ejpam-628	610	1	the	the	DET
ejpam-628	610	2	research	research	NOUN
ejpam-628	610	3	of	of	ADP
ejpam-628	610	4	the	the	DET
ejpam-628	610	5	first	first	ADJ
ejpam-628	610	6	author	author	NOUN
ejpam-628	610	7	was	be	AUX
ejpam-628	610	8	in	in	ADP
ejpam-628	610	9	part	part	NOUN
ejpam-628	610	10	supported	support	VERB
ejpam-628	610	11	by	by	ADP
ejpam-628	610	12	a	a	DET
ejpam-628	610	13	grant	grant	NOUN
ejpam-628	610	14	from	from	ADP
ejpam-628	610	15	ipm	ipm	NOUN
ejpam-628	610	16	(	(	PUNCT
ejpam-628	610	17	no	no	NOUN
ejpam-628	610	18	.	.	NOUN
ejpam-628	610	19	87160026	87160026	NUM
ejpam-628	610	20	)	)	PUNCT
ejpam-628	610	21	.	.	PUNCT
ejpam-628	611	1	references	reference	NOUN
ejpam-628	611	2	[	[	X
ejpam-628	611	3	1	1	NUM
ejpam-628	611	4	]	]	X
ejpam-628	611	5	d.f	d.f	PROPN
ejpam-628	611	6	.	.	PROPN
ejpam-628	611	7	anderson	anderson	PROPN
ejpam-628	611	8	,	,	PUNCT
ejpam-628	611	9	a.	a.	NOUN
ejpam-628	611	10	farzier	farzier	PROPN
ejpam-628	611	11	,	,	PUNCT
ejpam-628	611	12	a.	a.	NOUN
ejpam-628	611	13	lauve	lauve	PROPN
ejpam-628	611	14	,	,	PUNCT
ejpam-628	611	15	and	and	CCONJ
ejpam-628	611	16	p.s	p.s	PROPN
ejpam-628	611	17	.	.	PROPN
ejpam-628	611	18	livingston	livingston	PROPN
ejpam-628	611	19	,	,	PUNCT
ejpam-628	611	20	the	the	DET
ejpam-628	611	21	zero	zero	NUM
ejpam-628	611	22	-	-	PUNCT
ejpam-628	611	23	divisor	divisor	NOUN
ejpam-628	611	24	graph	graph	NOUN
ejpam-628	611	25	of	of	ADP
ejpam-628	611	26	a	a	DET
ejpam-628	611	27	commutative	commutative	ADJ
ejpam-628	611	28	ring	ring	NOUN
ejpam-628	611	29	,	,	PUNCT
ejpam-628	611	30	ii	ii	PROPN
ejpam-628	611	31	,	,	PUNCT
ejpam-628	611	32	lecture	lecture	NOUN
ejpam-628	611	33	notes	note	NOUN
ejpam-628	611	34	in	in	ADP
ejpam-628	611	35	pure	pure	ADJ
ejpam-628	611	36	and	and	CCONJ
ejpam-628	611	37	appl	appl	NOUN
ejpam-628	611	38	.	.	PUNCT
ejpam-628	611	39	math	math	NOUN
ejpam-628	611	40	.	.	PUNCT
ejpam-628	612	1	202	202	NUM
ejpam-628	612	2	(	(	PUNCT
ejpam-628	612	3	2001	2001	NUM
ejpam-628	612	4	)	)	PUNCT
ejpam-628	612	5	,	,	PUNCT
ejpam-628	612	6	61	61	NUM
ejpam-628	612	7	-	-	SYM
ejpam-628	612	8	72	72	NUM
ejpam-628	612	9	,	,	PUNCT
ejpam-628	612	10	marcel	marcel	PROPN
ejpam-628	612	11	dekker	dekker	PROPN
ejpam-628	612	12	,	,	PUNCT
ejpam-628	612	13	new	new	PROPN
ejpam-628	612	14	york	york	PROPN
ejpam-628	612	15	.	.	PUNCT
ejpam-628	613	1	[	[	X
ejpam-628	613	2	2	2	X
ejpam-628	613	3	]	]	PUNCT
ejpam-628	613	4	b.	b.	PROPN
ejpam-628	613	5	corbas	corbas	PROPN
ejpam-628	613	6	and	and	CCONJ
ejpam-628	613	7	g.d	g.d	PROPN
ejpam-628	613	8	.	.	PROPN
ejpam-628	613	9	williams	williams	PROPN
ejpam-628	613	10	,	,	PUNCT
ejpam-628	613	11	rings	ring	NOUN
ejpam-628	613	12	of	of	ADP
ejpam-628	613	13	order	order	NOUN
ejpam-628	613	14	p5	p5	ADJ
ejpam-628	613	15	part	part	NOUN
ejpam-628	613	16	i.	i.	PROPN
ejpam-628	613	17	nonlocal	nonlocal	ADJ
ejpam-628	613	18	rings	ring	NOUN
ejpam-628	613	19	,	,	PUNCT
ejpam-628	613	20	j.	j.	PROPN
ejpam-628	613	21	algebra	algebra	PROPN
ejpam-628	613	22	,	,	PUNCT
ejpam-628	613	23	231	231	NUM
ejpam-628	613	24	(	(	PUNCT
ejpam-628	613	25	2000	2000	NUM
ejpam-628	613	26	)	)	PUNCT
ejpam-628	613	27	,	,	PUNCT
ejpam-628	613	28	677	677	NUM
ejpam-628	613	29	-	-	SYM
ejpam-628	613	30	690	690	NUM
ejpam-628	613	31	.	.	PUNCT
ejpam-628	614	1	[	[	X
ejpam-628	614	2	3	3	X
ejpam-628	614	3	]	]	X
ejpam-628	614	4	b.	b.	PROPN
ejpam-628	614	5	corbas	corbas	PROPN
ejpam-628	614	6	and	and	CCONJ
ejpam-628	614	7	g.d	g.d	PROPN
ejpam-628	614	8	.	.	PROPN
ejpam-628	614	9	williams	williams	PROPN
ejpam-628	614	10	,	,	PUNCT
ejpam-628	614	11	rings	ring	NOUN
ejpam-628	614	12	of	of	ADP
ejpam-628	614	13	order	order	NOUN
ejpam-628	614	14	p5	p5	PROPN
ejpam-628	614	15	part	part	PROPN
ejpam-628	614	16	ii	ii	PROPN
ejpam-628	614	17	.	.	PUNCT
ejpam-628	615	1	local	local	ADJ
ejpam-628	615	2	rings	ring	NOUN
ejpam-628	615	3	,	,	PUNCT
ejpam-628	615	4	j.	j.	PROPN
ejpam-628	615	5	algebra	algebra	PROPN
ejpam-628	615	6	,	,	PUNCT
ejpam-628	615	7	231	231	NUM
ejpam-628	615	8	(	(	PUNCT
ejpam-628	615	9	2000	2000	NUM
ejpam-628	615	10	)	)	PUNCT
ejpam-628	615	11	,	,	PUNCT
ejpam-628	615	12	691	691	NUM
ejpam-628	615	13	-	-	SYM
ejpam-628	615	14	704	704	NUM
ejpam-628	615	15	.	.	PUNCT
ejpam-628	616	1	[	[	X
ejpam-628	616	2	4	4	X
ejpam-628	616	3	]	]	X
ejpam-628	616	4	b.	b.	PROPN
ejpam-628	616	5	corbas	corbas	PROPN
ejpam-628	616	6	,	,	PUNCT
ejpam-628	616	7	rings	ring	NOUN
ejpam-628	616	8	with	with	ADP
ejpam-628	616	9	few	few	ADJ
ejpam-628	616	10	zero	zero	NUM
ejpam-628	616	11	-	-	PUNCT
ejpam-628	616	12	divisors	divisor	NOUN
ejpam-628	616	13	,	,	PUNCT
ejpam-628	616	14	math	math	NOUN
ejpam-628	616	15	.	.	PUNCT
ejpam-628	617	1	ann	ann	PROPN
ejpam-628	617	2	.	.	PROPN
ejpam-628	618	1	181	181	NUM
ejpam-628	618	2	(	(	PUNCT
ejpam-628	618	3	1	1	NUM
ejpam-628	618	4	)	)	PUNCT
ejpam-628	618	5	(	(	PUNCT
ejpam-628	618	6	1969	1969	NUM
ejpam-628	618	7	)	)	PUNCT
ejpam-628	618	8	,	,	PUNCT
ejpam-628	618	9	1ű7	1ű7	NUM
ejpam-628	618	10	.	.	PUNCT
ejpam-628	619	1	[	[	X
ejpam-628	619	2	5	5	NUM
ejpam-628	619	3	]	]	X
ejpam-628	619	4	n.	n.	PROPN
ejpam-628	619	5	ganesan	ganesan	PROPN
ejpam-628	619	6	,	,	PUNCT
ejpam-628	619	7	properties	property	NOUN
ejpam-628	619	8	of	of	ADP
ejpam-628	619	9	rings	ring	NOUN
ejpam-628	619	10	with	with	ADP
ejpam-628	619	11	a	a	DET
ejpam-628	619	12	finite	finite	ADJ
ejpam-628	619	13	number	number	NOUN
ejpam-628	619	14	of	of	ADP
ejpam-628	619	15	zero	zero	NUM
ejpam-628	619	16	-	-	PUNCT
ejpam-628	619	17	divisors	divisor	NOUN
ejpam-628	619	18	,	,	PUNCT
ejpam-628	619	19	math	math	NOUN
ejpam-628	619	20	.	.	PUNCT
ejpam-628	620	1	ann	ann	PROPN
ejpam-628	620	2	.	.	PROPN
ejpam-628	621	1	157	157	NUM
ejpam-628	621	2	(	(	PUNCT
ejpam-628	621	3	1964	1964	NUM
ejpam-628	621	4	)	)	PUNCT
ejpam-628	621	5	,	,	PUNCT
ejpam-628	622	1	215ű218	215ű218	X
ejpam-628	622	2	.	.	PUNCT
ejpam-628	623	1	[	[	X
ejpam-628	623	2	6	6	NUM
ejpam-628	623	3	]	]	PUNCT
ejpam-628	623	4	r.	r.	PROPN
ejpam-628	623	5	gilmer	gilmer	PROPN
ejpam-628	623	6	,	,	PUNCT
ejpam-628	623	7	zero	zero	NUM
ejpam-628	623	8	-	-	PUNCT
ejpam-628	623	9	divisors	divisor	NOUN
ejpam-628	623	10	in	in	ADP
ejpam-628	623	11	commutative	commutative	ADJ
ejpam-628	623	12	rings	ring	NOUN
ejpam-628	623	13	.	.	PUNCT
ejpam-628	624	1	the	the	DET
ejpam-628	624	2	american	american	PROPN
ejpam-628	624	3	mathematical	mathematical	PROPN
ejpam-628	624	4	monthly	monthly	ADV
ejpam-628	624	5	,	,	PUNCT
ejpam-628	624	6	93	93	NUM
ejpam-628	624	7	(	(	PUNCT
ejpam-628	624	8	5	5	NUM
ejpam-628	624	9	)	)	PUNCT
ejpam-628	624	10	(	(	PUNCT
ejpam-628	624	11	1986	1986	NUM
ejpam-628	624	12	)	)	PUNCT
ejpam-628	624	13	,	,	PUNCT
ejpam-628	624	14	382	382	NUM
ejpam-628	624	15	-	-	SYM
ejpam-628	624	16	387	387	NUM
ejpam-628	624	17	.	.	PUNCT
ejpam-628	624	18	references	reference	NOUN
ejpam-628	624	19	316	316	NUM
ejpam-628	625	1	[	[	X
ejpam-628	625	2	7	7	NUM
ejpam-628	625	3	]	]	PUNCT
ejpam-628	625	4	k.	k.	PROPN
ejpam-628	625	5	koh	koh	PROPN
ejpam-628	625	6	,	,	PUNCT
ejpam-628	625	7	on	on	ADP
ejpam-628	625	8	properties	property	NOUN
ejpam-628	625	9	of	of	ADP
ejpam-628	625	10	rings	ring	NOUN
ejpam-628	625	11	with	with	ADP
ejpam-628	625	12	a	a	DET
ejpam-628	625	13	finite	finite	ADJ
ejpam-628	625	14	number	number	NOUN
ejpam-628	625	15	of	of	ADP
ejpam-628	625	16	zero	zero	NUM
ejpam-628	625	17	-	-	PUNCT
ejpam-628	625	18	divisors	divisor	NOUN
ejpam-628	625	19	,	,	PUNCT
ejpam-628	625	20	math	math	NOUN
ejpam-628	625	21	.	.	PUNCT
ejpam-628	626	1	ann	ann	PROPN
ejpam-628	626	2	.	.	PROPN
ejpam-628	627	1	171	171	NUM
ejpam-628	627	2	(	(	PUNCT
ejpam-628	627	3	1967	1967	NUM
ejpam-628	627	4	)	)	PUNCT
ejpam-628	627	5	,	,	PUNCT
ejpam-628	627	6	79	79	NUM
ejpam-628	627	7	-	-	SYM
ejpam-628	627	8	80	80	NUM
ejpam-628	627	9	.	.	PUNCT
ejpam-628	628	1	[	[	X
ejpam-628	628	2	8	8	NUM
ejpam-628	628	3	]	]	X
ejpam-628	628	4	b.r	b.r	PROPN
ejpam-628	628	5	.	.	PROPN
ejpam-628	628	6	mcdonald	mcdonald	PROPN
ejpam-628	628	7	,	,	PUNCT
ejpam-628	628	8	finite	finite	PROPN
ejpam-628	628	9	rings	ring	NOUN
ejpam-628	628	10	with	with	ADP
ejpam-628	628	11	identity	identity	NOUN
ejpam-628	628	12	(	(	PUNCT
ejpam-628	628	13	marcell	marcell	PROPN
ejpam-628	628	14	dekker	dekker	PROPN
ejpam-628	628	15	,	,	PUNCT
ejpam-628	628	16	new	new	PROPN
ejpam-628	628	17	york	york	PROPN
ejpam-628	628	18	,	,	PUNCT
ejpam-628	628	19	1974	1974	NUM
ejpam-628	628	20	)	)	PUNCT
ejpam-628	628	21	.	.	PUNCT
ejpam-628	629	1	[	[	X
ejpam-628	629	2	9	9	NUM
ejpam-628	629	3	]	]	X
ejpam-628	629	4	r.	r.	NOUN
ejpam-628	629	5	raghavendran	raghavendran	PROPN
ejpam-628	629	6	,	,	PUNCT
ejpam-628	629	7	finite	finite	PROPN
ejpam-628	629	8	associative	associative	ADJ
ejpam-628	629	9	rings	ring	NOUN
ejpam-628	629	10	,	,	PUNCT
ejpam-628	629	11	compositio	compositio	NOUN
ejpam-628	629	12	math	math	NOUN
ejpam-628	629	13	.	.	PUNCT
ejpam-628	630	1	21	21	NUM
ejpam-628	630	2	(	(	PUNCT
ejpam-628	630	3	1969	1969	NUM
ejpam-628	630	4	)	)	PUNCT
ejpam-628	630	5	,	,	PUNCT
ejpam-628	630	6	195	195	NUM
ejpam-628	630	7	-	-	SYM
ejpam-628	630	8	229	229	NUM
ejpam-628	630	9	.	.	PUNCT
ejpam-628	631	1	[	[	X
ejpam-628	631	2	10	10	NUM
ejpam-628	631	3	]	]	X
ejpam-628	631	4	s.p	s.p	PROPN
ejpam-628	631	5	.	.	PROPN
ejpam-628	631	6	redmond	redmond	PROPN
ejpam-628	631	7	,	,	PUNCT
ejpam-628	631	8	an	an	DET
ejpam-628	631	9	ideal	ideal	ADV
ejpam-628	631	10	-	-	PUNCT
ejpam-628	631	11	based	base	VERB
ejpam-628	631	12	zero	zero	NUM
ejpam-628	631	13	-	-	PUNCT
ejpam-628	631	14	divisor	divisor	NOUN
ejpam-628	631	15	graph	graph	NOUN
ejpam-628	631	16	of	of	ADP
ejpam-628	631	17	a	a	DET
ejpam-628	631	18	commutative	commutative	ADJ
ejpam-628	631	19	ring	ring	NOUN
ejpam-628	631	20	,	,	PUNCT
ejpam-628	631	21	comm	comm	NOUN
ejpam-628	631	22	.	.	PUNCT
ejpam-628	632	1	algebra	algebra	NOUN
ejpam-628	632	2	31	31	NUM
ejpam-628	632	3	(	(	PUNCT
ejpam-628	632	4	2003	2003	NUM
ejpam-628	632	5	)	)	PUNCT
ejpam-628	632	6	,	,	PUNCT
ejpam-628	632	7	4425	4425	NUM
ejpam-628	632	8	-	-	SYM
ejpam-628	632	9	4443	4443	NUM
ejpam-628	632	10	.	.	PUNCT
ejpam-628	633	1	[	[	X
ejpam-628	633	2	11	11	NUM
ejpam-628	633	3	]	]	X
ejpam-628	633	4	s.p	s.p	PROPN
ejpam-628	633	5	.	.	PROPN
ejpam-628	633	6	redmond	redmond	PROPN
ejpam-628	633	7	,	,	PUNCT
ejpam-628	633	8	on	on	ADP
ejpam-628	633	9	zero	zero	NUM
ejpam-628	633	10	-	-	PUNCT
ejpam-628	633	11	divisor	divisor	NOUN
ejpam-628	633	12	graphs	graph	NOUN
ejpam-628	633	13	of	of	ADP
ejpam-628	633	14	small	small	ADJ
ejpam-628	633	15	finite	finite	PROPN
ejpam-628	633	16	commutative	commutative	ADJ
ejpam-628	633	17	rings	ring	NOUN
ejpam-628	633	18	.	.	PUNCT
ejpam-628	634	1	discrete	discrete	ADJ
ejpam-628	634	2	math	math	NOUN
ejpam-628	634	3	.	.	PUNCT
ejpam-628	635	1	307	307	NUM
ejpam-628	635	2	(	(	PUNCT
ejpam-628	635	3	2007	2007	NUM
ejpam-628	635	4	)	)	PUNCT
ejpam-628	635	5	,	,	PUNCT
ejpam-628	635	6	1155	1155	NUM
ejpam-628	635	7	-	-	SYM
ejpam-628	635	8	1166	1166	NUM
ejpam-628	635	9	.	.	PUNCT