id	sid	tid	token	lemma	pos
ejpam-6214	1	1	european	european	PROPN
ejpam-6214	1	2	journal	journal	PROPN
ejpam-6214	1	3	of	of	ADP
ejpam-6214	1	4	pure	pure	ADJ
ejpam-6214	1	5	and	and	CCONJ
ejpam-6214	1	6	applied	applied	ADJ
ejpam-6214	1	7	mathematics	mathematic	NOUN
ejpam-6214	1	8	2025	2025	NUM
ejpam-6214	1	9	,	,	PUNCT
ejpam-6214	1	10	vol	vol	NOUN
ejpam-6214	1	11	.	.	PROPN
ejpam-6214	1	12	18	18	NUM
ejpam-6214	1	13	,	,	PUNCT
ejpam-6214	1	14	issue	issue	NOUN
ejpam-6214	1	15	3	3	NUM
ejpam-6214	1	16	,	,	PUNCT
ejpam-6214	1	17	article	article	NOUN
ejpam-6214	1	18	number	number	NOUN
ejpam-6214	1	19	6214	6214	NUM
ejpam-6214	1	20	issn	issn	PROPN
ejpam-6214	1	21	1307	1307	NUM
ejpam-6214	1	22	-	-	SYM
ejpam-6214	1	23	5543	5543	NUM
ejpam-6214	1	24	–	–	PUNCT
ejpam-6214	1	25	ejpam.com	ejpam.com	X
ejpam-6214	1	26	published	publish	VERB
ejpam-6214	1	27	by	by	ADP
ejpam-6214	1	28	new	new	PROPN
ejpam-6214	1	29	york	york	PROPN
ejpam-6214	1	30	business	business	PROPN
ejpam-6214	1	31	global	global	PROPN
ejpam-6214	1	32	on	on	ADP
ejpam-6214	1	33	the	the	DET
ejpam-6214	1	34	irreducibility	irreducibility	NOUN
ejpam-6214	1	35	of	of	ADP
ejpam-6214	1	36	polynomials	polynomial	NOUN
ejpam-6214	1	37	with	with	ADP
ejpam-6214	1	38	prime	prime	ADJ
ejpam-6214	1	39	power	power	NOUN
ejpam-6214	1	40	shifts	shift	NOUN
ejpam-6214	1	41	amara	amara	PROPN
ejpam-6214	1	42	chandoul1,∗	chandoul1,∗	NOUN
ejpam-6214	1	43	,	,	PUNCT
ejpam-6214	1	44	saber	saber	NOUN
ejpam-6214	1	45	mansour2	mansour2	NOUN
ejpam-6214	1	46	1	1	NUM
ejpam-6214	1	47	department	department	NOUN
ejpam-6214	1	48	of	of	ADP
ejpam-6214	1	49	mathematics	mathematic	NOUN
ejpam-6214	1	50	,	,	PUNCT
ejpam-6214	1	51	higher	high	ADJ
ejpam-6214	1	52	institute	institute	PROPN
ejpam-6214	1	53	of	of	ADP
ejpam-6214	1	54	informatics	informatic	NOUN
ejpam-6214	1	55	and	and	CCONJ
ejpam-6214	1	56	multimedia	multimedia	NOUN
ejpam-6214	1	57	of	of	ADP
ejpam-6214	1	58	sfax	sfax	NOUN
ejpam-6214	1	59	,	,	PUNCT
ejpam-6214	1	60	sfax	sfax	ADJ
ejpam-6214	1	61	university	university	NOUN
ejpam-6214	1	62	,	,	PUNCT
ejpam-6214	1	63	sfax	sfax	NOUN
ejpam-6214	1	64	,	,	PUNCT
ejpam-6214	1	65	tunisia	tunisia	PROPN
ejpam-6214	1	66	2	2	NUM
ejpam-6214	1	67	department	department	NOUN
ejpam-6214	1	68	of	of	ADP
ejpam-6214	1	69	mathematics	mathematic	NOUN
ejpam-6214	1	70	,	,	PUNCT
ejpam-6214	1	71	college	college	NOUN
ejpam-6214	1	72	of	of	ADP
ejpam-6214	1	73	science	science	NOUN
ejpam-6214	1	74	,	,	PUNCT
ejpam-6214	1	75	umm	umm	INTJ
ejpam-6214	1	76	al	al	PROPN
ejpam-6214	1	77	-	-	PUNCT
ejpam-6214	1	78	qura	qura	PROPN
ejpam-6214	1	79	university	university	PROPN
ejpam-6214	1	80	,	,	PUNCT
ejpam-6214	1	81	mecca	mecca	PROPN
ejpam-6214	1	82	21955	21955	NUM
ejpam-6214	1	83	,	,	PUNCT
ejpam-6214	1	84	saudi	saudi	PROPN
ejpam-6214	1	85	arabia	arabia	PROPN
ejpam-6214	1	86	abstract	abstract	NOUN
ejpam-6214	1	87	.	.	PUNCT
ejpam-6214	2	1	in	in	ADP
ejpam-6214	2	2	this	this	DET
ejpam-6214	2	3	paper	paper	NOUN
ejpam-6214	2	4	,	,	PUNCT
ejpam-6214	2	5	we	we	PRON
ejpam-6214	2	6	study	study	VERB
ejpam-6214	2	7	the	the	DET
ejpam-6214	2	8	irreducibility	irreducibility	NOUN
ejpam-6214	2	9	of	of	ADP
ejpam-6214	2	10	polynomials	polynomial	NOUN
ejpam-6214	2	11	of	of	ADP
ejpam-6214	2	12	the	the	DET
ejpam-6214	2	13	form	form	NOUN
ejpam-6214	2	14	f(x	f(x	PROPN
ejpam-6214	2	15	)	)	PUNCT
ejpam-6214	3	1	+	+	SYM
ejpam-6214	3	2	pkg(x	pkg(x	NOUN
ejpam-6214	3	3	)	)	PUNCT
ejpam-6214	3	4	,	,	PUNCT
ejpam-6214	3	5	where	where	SCONJ
ejpam-6214	3	6	f(x	f(x	PROPN
ejpam-6214	3	7	)	)	PUNCT
ejpam-6214	3	8	and	and	CCONJ
ejpam-6214	3	9	g(x	g(x	NOUN
ejpam-6214	3	10	)	)	PUNCT
ejpam-6214	3	11	are	be	AUX
ejpam-6214	3	12	polynomials	polynomial	NOUN
ejpam-6214	3	13	with	with	ADP
ejpam-6214	3	14	integer	integer	NOUN
ejpam-6214	3	15	coefficients	coefficient	NOUN
ejpam-6214	3	16	,	,	PUNCT
ejpam-6214	3	17	p	p	NOUN
ejpam-6214	3	18	is	be	AUX
ejpam-6214	3	19	a	a	DET
ejpam-6214	3	20	prime	prime	ADJ
ejpam-6214	3	21	number	number	NOUN
ejpam-6214	3	22	,	,	PUNCT
ejpam-6214	3	23	and	and	CCONJ
ejpam-6214	3	24	k	k	PROPN
ejpam-6214	3	25	is	be	AUX
ejpam-6214	3	26	a	a	DET
ejpam-6214	3	27	positive	positive	ADJ
ejpam-6214	3	28	integer	integer	NOUN
ejpam-6214	3	29	.	.	PUNCT
ejpam-6214	4	1	unlike	unlike	ADP
ejpam-6214	4	2	previous	previous	ADJ
ejpam-6214	4	3	results	result	NOUN
ejpam-6214	4	4	,	,	PUNCT
ejpam-6214	4	5	we	we	PRON
ejpam-6214	4	6	do	do	AUX
ejpam-6214	4	7	not	not	PART
ejpam-6214	4	8	require	require	VERB
ejpam-6214	4	9	f(x	f(x	PROPN
ejpam-6214	4	10	)	)	PUNCT
ejpam-6214	4	11	and	and	CCONJ
ejpam-6214	4	12	g(x	g(x	NOUN
ejpam-6214	4	13	)	)	PUNCT
ejpam-6214	4	14	to	to	PART
ejpam-6214	4	15	be	be	AUX
ejpam-6214	4	16	relatively	relatively	ADV
ejpam-6214	4	17	prime	prime	ADJ
ejpam-6214	4	18	or	or	CCONJ
ejpam-6214	4	19	impose	impose	VERB
ejpam-6214	4	20	any	any	DET
ejpam-6214	4	21	conditions	condition	NOUN
ejpam-6214	4	22	on	on	ADP
ejpam-6214	4	23	gcd(k	gcd(k	PROPN
ejpam-6214	4	24	,	,	PUNCT
ejpam-6214	4	25	deg	deg	NOUN
ejpam-6214	4	26	g	g	NOUN
ejpam-6214	4	27	)	)	PUNCT
ejpam-6214	4	28	.	.	PUNCT
ejpam-6214	5	1	we	we	PRON
ejpam-6214	5	2	prove	prove	VERB
ejpam-6214	5	3	that	that	SCONJ
ejpam-6214	5	4	,	,	PUNCT
ejpam-6214	5	5	for	for	ADP
ejpam-6214	5	6	all	all	PRON
ejpam-6214	5	7	but	but	ADV
ejpam-6214	5	8	finitely	finitely	ADV
ejpam-6214	5	9	many	many	ADJ
ejpam-6214	5	10	primes	prime	NOUN
ejpam-6214	5	11	p	p	X
ejpam-6214	5	12	,	,	PUNCT
ejpam-6214	5	13	the	the	DET
ejpam-6214	5	14	polynomial	polynomial	ADJ
ejpam-6214	5	15	f(x	f(x	PROPN
ejpam-6214	5	16	)	)	PUNCT
ejpam-6214	6	1	+	+	SYM
ejpam-6214	6	2	pkg(x	pkg(x	NOUN
ejpam-6214	6	3	)	)	PUNCT
ejpam-6214	6	4	is	be	AUX
ejpam-6214	6	5	either	either	CCONJ
ejpam-6214	6	6	irreducible	irreducible	ADJ
ejpam-6214	6	7	over	over	ADP
ejpam-6214	6	8	q	q	NOUN
ejpam-6214	6	9	or	or	CCONJ
ejpam-6214	6	10	factors	factor	NOUN
ejpam-6214	6	11	into	into	ADP
ejpam-6214	6	12	polynomials	polynomial	NOUN
ejpam-6214	6	13	whose	whose	DET
ejpam-6214	6	14	degrees	degree	NOUN
ejpam-6214	6	15	are	be	AUX
ejpam-6214	6	16	multiples	multiple	NOUN
ejpam-6214	6	17	of	of	ADP
ejpam-6214	6	18	gcd(k	gcd(k	PROPN
ejpam-6214	6	19	,	,	PUNCT
ejpam-6214	6	20	deg	deg	VERB
ejpam-6214	6	21	g	g	NOUN
ejpam-6214	6	22	)	)	PUNCT
ejpam-6214	6	23	.	.	PUNCT
ejpam-6214	7	1	this	this	PRON
ejpam-6214	7	2	generalizes	generalize	VERB
ejpam-6214	7	3	and	and	CCONJ
ejpam-6214	7	4	extends	extend	VERB
ejpam-6214	7	5	earlier	early	ADJ
ejpam-6214	7	6	work	work	NOUN
ejpam-6214	7	7	on	on	ADP
ejpam-6214	7	8	the	the	DET
ejpam-6214	7	9	irreducibility	irreducibility	NOUN
ejpam-6214	7	10	of	of	ADP
ejpam-6214	7	11	such	such	ADJ
ejpam-6214	7	12	polynomials	polynomial	NOUN
ejpam-6214	7	13	.	.	PUNCT
ejpam-6214	8	1	2020	2020	NUM
ejpam-6214	8	2	mathematics	mathematic	NOUN
ejpam-6214	8	3	subject	subject	NOUN
ejpam-6214	8	4	classifications	classification	NOUN
ejpam-6214	8	5	:	:	PUNCT
ejpam-6214	8	6	11c08	11c08	NUM
ejpam-6214	8	7	,	,	PUNCT
ejpam-6214	8	8	11r09	11r09	NUM
ejpam-6214	8	9	,	,	PUNCT
ejpam-6214	8	10	12e05	12e05	NUM
ejpam-6214	8	11	key	key	ADJ
ejpam-6214	8	12	words	word	NOUN
ejpam-6214	8	13	and	and	CCONJ
ejpam-6214	8	14	phrases	phrase	NOUN
ejpam-6214	8	15	:	:	PUNCT
ejpam-6214	8	16	polynomial	polynomial	ADJ
ejpam-6214	8	17	irreducibility	irreducibility	NOUN
ejpam-6214	8	18	,	,	PUNCT
ejpam-6214	8	19	prime	prime	ADJ
ejpam-6214	8	20	power	power	NOUN
ejpam-6214	8	21	shifts	shift	NOUN
ejpam-6214	8	22	,	,	PUNCT
ejpam-6214	8	23	relative	relative	ADJ
ejpam-6214	8	24	primality	primality	NOUN
ejpam-6214	8	25	,	,	PUNCT
ejpam-6214	8	26	factorization	factorization	NOUN
ejpam-6214	8	27	structure	structure	NOUN
ejpam-6214	8	28	,	,	PUNCT
ejpam-6214	8	29	eisenstein	eisenstein	PROPN
ejpam-6214	8	30	’s	’s	PART
ejpam-6214	8	31	criterion	criterion	NOUN
ejpam-6214	8	32	,	,	PUNCT
ejpam-6214	8	33	number	number	NOUN
ejpam-6214	8	34	theory	theory	NOUN
ejpam-6214	8	35	1	1	NUM
ejpam-6214	8	36	.	.	PUNCT
ejpam-6214	8	37	introduction	introduction	NOUN
ejpam-6214	8	38	the	the	DET
ejpam-6214	8	39	study	study	NOUN
ejpam-6214	8	40	of	of	ADP
ejpam-6214	8	41	polynomial	polynomial	ADJ
ejpam-6214	8	42	irreducibility	irreducibility	NOUN
ejpam-6214	8	43	has	have	AUX
ejpam-6214	8	44	long	long	ADV
ejpam-6214	8	45	been	be	AUX
ejpam-6214	8	46	a	a	DET
ejpam-6214	8	47	central	central	ADJ
ejpam-6214	8	48	topic	topic	NOUN
ejpam-6214	8	49	in	in	ADP
ejpam-6214	8	50	algebra	algebra	NOUN
ejpam-6214	8	51	and	and	CCONJ
ejpam-6214	8	52	number	number	NOUN
ejpam-6214	8	53	theory	theory	NOUN
ejpam-6214	8	54	,	,	PUNCT
ejpam-6214	8	55	with	with	ADP
ejpam-6214	8	56	applications	application	NOUN
ejpam-6214	8	57	ranging	range	VERB
ejpam-6214	8	58	from	from	ADP
ejpam-6214	8	59	algebraic	algebraic	ADJ
ejpam-6214	8	60	number	number	NOUN
ejpam-6214	8	61	theory	theory	NOUN
ejpam-6214	8	62	to	to	PART
ejpam-6214	8	63	galois	galois	VERB
ejpam-6214	8	64	theory	theory	NOUN
ejpam-6214	8	65	and	and	CCONJ
ejpam-6214	8	66	diophantine	diophantine	VERB
ejpam-6214	8	67	equations	equation	NOUN
ejpam-6214	8	68	[	[	X
ejpam-6214	8	69	1	1	NUM
ejpam-6214	8	70	,	,	PUNCT
ejpam-6214	8	71	2	2	NUM
ejpam-6214	8	72	]	]	PUNCT
ejpam-6214	8	73	.	.	PUNCT
ejpam-6214	9	1	determining	determine	VERB
ejpam-6214	9	2	whether	whether	SCONJ
ejpam-6214	9	3	a	a	DET
ejpam-6214	9	4	given	give	VERB
ejpam-6214	9	5	polynomial	polynomial	NOUN
ejpam-6214	9	6	is	be	AUX
ejpam-6214	9	7	irreducible	irreducible	ADJ
ejpam-6214	9	8	over	over	ADP
ejpam-6214	9	9	a	a	DET
ejpam-6214	9	10	field	field	NOUN
ejpam-6214	9	11	,	,	PUNCT
ejpam-6214	9	12	particularly	particularly	ADV
ejpam-6214	9	13	the	the	DET
ejpam-6214	9	14	field	field	NOUN
ejpam-6214	9	15	of	of	ADP
ejpam-6214	9	16	rational	rational	ADJ
ejpam-6214	9	17	numbers	number	NOUN
ejpam-6214	9	18	q	q	ADJ
ejpam-6214	9	19	,	,	PUNCT
ejpam-6214	9	20	is	be	AUX
ejpam-6214	9	21	a	a	DET
ejpam-6214	9	22	fundamental	fundamental	ADJ
ejpam-6214	9	23	problem	problem	NOUN
ejpam-6214	9	24	that	that	PRON
ejpam-6214	9	25	has	have	AUX
ejpam-6214	9	26	inspired	inspire	VERB
ejpam-6214	9	27	numerous	numerous	ADJ
ejpam-6214	9	28	classical	classical	ADJ
ejpam-6214	9	29	results	result	NOUN
ejpam-6214	9	30	,	,	PUNCT
ejpam-6214	9	31	such	such	ADJ
ejpam-6214	9	32	as	as	ADP
ejpam-6214	9	33	eisenstein	eisenstein	PROPN
ejpam-6214	9	34	’s	’s	PART
ejpam-6214	9	35	criterion	criterion	NOUN
ejpam-6214	10	1	[	[	X
ejpam-6214	10	2	3	3	NUM
ejpam-6214	10	3	]	]	PUNCT
ejpam-6214	10	4	.	.	PUNCT
ejpam-6214	11	1	however	however	ADV
ejpam-6214	11	2	,	,	PUNCT
ejpam-6214	11	3	many	many	ADJ
ejpam-6214	11	4	of	of	ADP
ejpam-6214	11	5	these	these	DET
ejpam-6214	11	6	results	result	NOUN
ejpam-6214	11	7	rely	rely	VERB
ejpam-6214	11	8	on	on	ADP
ejpam-6214	11	9	restrictive	restrictive	ADJ
ejpam-6214	11	10	conditions	condition	NOUN
ejpam-6214	11	11	that	that	PRON
ejpam-6214	11	12	limit	limit	VERB
ejpam-6214	11	13	their	their	PRON
ejpam-6214	11	14	applicability	applicability	NOUN
ejpam-6214	11	15	to	to	ADP
ejpam-6214	11	16	broader	broad	ADJ
ejpam-6214	11	17	classes	class	NOUN
ejpam-6214	11	18	of	of	ADP
ejpam-6214	11	19	polynomials	polynomial	NOUN
ejpam-6214	11	20	.	.	PUNCT
ejpam-6214	12	1	building	build	VERB
ejpam-6214	12	2	upon	upon	SCONJ
ejpam-6214	12	3	foundational	foundational	ADJ
ejpam-6214	12	4	work	work	NOUN
ejpam-6214	12	5	in	in	ADP
ejpam-6214	12	6	polynomial	polynomial	ADJ
ejpam-6214	12	7	irreducibility	irreducibility	NOUN
ejpam-6214	12	8	,	,	PUNCT
ejpam-6214	12	9	bonciocat	bonciocat	NOUN
ejpam-6214	12	10	[	[	X
ejpam-6214	12	11	4	4	NUM
ejpam-6214	12	12	]	]	PUNCT
ejpam-6214	12	13	developed	develop	VERB
ejpam-6214	12	14	an	an	DET
ejpam-6214	12	15	important	important	ADJ
ejpam-6214	12	16	criterion	criterion	NOUN
ejpam-6214	12	17	for	for	ADP
ejpam-6214	12	18	polynomials	polynomial	NOUN
ejpam-6214	12	19	having	have	VERB
ejpam-6214	12	20	the	the	DET
ejpam-6214	12	21	structure	structure	NOUN
ejpam-6214	12	22	p	p	NOUN
ejpam-6214	12	23	(	(	PUNCT
ejpam-6214	12	24	x	x	NOUN
ejpam-6214	12	25	)	)	PUNCT
ejpam-6214	12	26	=	=	SYM
ejpam-6214	12	27	f(x	f(x	PROPN
ejpam-6214	12	28	)	)	PUNCT
ejpam-6214	13	1	+	+	NUM
ejpam-6214	14	1	pkg(x	pkg(x	NOUN
ejpam-6214	14	2	)	)	PUNCT
ejpam-6214	14	3	,	,	PUNCT
ejpam-6214	14	4	where	where	SCONJ
ejpam-6214	14	5	:	:	PUNCT
ejpam-6214	14	6	∗corresponding	∗corresponde	VERB
ejpam-6214	14	7	author	author	NOUN
ejpam-6214	14	8	.	.	PUNCT
ejpam-6214	15	1	doi	doi	NOUN
ejpam-6214	15	2	:	:	PUNCT
ejpam-6214	15	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6214	https://doi.org/10.29020/nybg.ejpam.v18i3.6214	ADP
ejpam-6214	15	4	email	email	NOUN
ejpam-6214	15	5	addresses	address	NOUN
ejpam-6214	15	6	:	:	PUNCT
ejpam-6214	15	7	amarachandoul@yahoo.fr	amarachandoul@yahoo.fr	PROPN
ejpam-6214	15	8	(	(	PUNCT
ejpam-6214	15	9	a.	a.	PROPN
ejpam-6214	15	10	chandoul	chandoul	PROPN
ejpam-6214	15	11	)	)	PUNCT
ejpam-6214	15	12	,	,	PUNCT
ejpam-6214	15	13	samansour@uqu.edu.sa	samansour@uqu.edu.sa	PROPN
ejpam-6214	15	14	(	(	PUNCT
ejpam-6214	15	15	s.	s.	PROPN
ejpam-6214	15	16	mansour	mansour	PROPN
ejpam-6214	15	17	)	)	PUNCT
ejpam-6214	15	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6214	15	19	1	1	NUM
ejpam-6214	15	20	copyright	copyright	NOUN
ejpam-6214	15	21	:	:	PUNCT
ejpam-6214	16	1	©	©	PROPN
ejpam-6214	16	2	2025	2025	NUM
ejpam-6214	16	3	the	the	DET
ejpam-6214	16	4	author(s	author(s	NOUN
ejpam-6214	16	5	)	)	PUNCT
ejpam-6214	16	6	.	.	PUNCT
ejpam-6214	17	1	(	(	PUNCT
ejpam-6214	17	2	cc	cc	NOUN
ejpam-6214	17	3	by	by	ADP
ejpam-6214	17	4	-	-	PUNCT
ejpam-6214	17	5	nc	nc	PROPN
ejpam-6214	17	6	4.0	4.0	NUM
ejpam-6214	17	7	)	)	PUNCT
ejpam-6214	17	8	a.	a.	NOUN
ejpam-6214	17	9	chandoul	chandoul	PROPN
ejpam-6214	17	10	,	,	PUNCT
ejpam-6214	17	11	s.	s.	PROPN
ejpam-6214	17	12	mansour	mansour	PROPN
ejpam-6214	17	13	/	/	SYM
ejpam-6214	17	14	eur	eur	PROPN
ejpam-6214	17	15	.	.	PUNCT
ejpam-6214	18	1	j.	j.	PROPN
ejpam-6214	18	2	pure	pure	PROPN
ejpam-6214	18	3	appl	appl	PROPN
ejpam-6214	18	4	.	.	PROPN
ejpam-6214	18	5	math	math	PROPN
ejpam-6214	18	6	,	,	PUNCT
ejpam-6214	18	7	18	18	NUM
ejpam-6214	18	8	(	(	PUNCT
ejpam-6214	18	9	3	3	NUM
ejpam-6214	18	10	)	)	PUNCT
ejpam-6214	18	11	(	(	PUNCT
ejpam-6214	18	12	2025	2025	NUM
ejpam-6214	18	13	)	)	PUNCT
ejpam-6214	18	14	,	,	PUNCT
ejpam-6214	18	15	6214	6214	NUM
ejpam-6214	18	16	2	2	NUM
ejpam-6214	18	17	of	of	ADP
ejpam-6214	18	18	8	8	NUM
ejpam-6214	18	19	•	•	NUM
ejpam-6214	18	20	f(x	f(x	PROPN
ejpam-6214	18	21	)	)	PUNCT
ejpam-6214	18	22	,	,	PUNCT
ejpam-6214	18	23	g(x	g(x	NOUN
ejpam-6214	18	24	)	)	PUNCT
ejpam-6214	18	25	∈	∈	PROPN
ejpam-6214	18	26	z[x	z[x	NOUN
ejpam-6214	18	27	]	]	X
ejpam-6214	18	28	are	be	AUX
ejpam-6214	18	29	coprime	coprime	ADJ
ejpam-6214	18	30	polynomials	polynomial	NOUN
ejpam-6214	18	31	•	•	ADP
ejpam-6214	18	32	p	p	NOUN
ejpam-6214	18	33	is	be	AUX
ejpam-6214	18	34	a	a	DET
ejpam-6214	18	35	prime	prime	ADJ
ejpam-6214	18	36	integer	integer	NOUN
ejpam-6214	18	37	•	•	NOUN
ejpam-6214	18	38	k	k	PROPN
ejpam-6214	18	39	∈	∈	PROPN
ejpam-6214	18	40	z+	z+	NUM
ejpam-6214	18	41	is	be	AUX
ejpam-6214	18	42	a	a	DET
ejpam-6214	18	43	positive	positive	ADJ
ejpam-6214	18	44	exponent	exponent	NOUN
ejpam-6214	18	45	this	this	DET
ejpam-6214	18	46	criterion	criterion	NOUN
ejpam-6214	18	47	provides	provide	VERB
ejpam-6214	18	48	a	a	DET
ejpam-6214	18	49	powerful	powerful	ADJ
ejpam-6214	18	50	tool	tool	NOUN
ejpam-6214	18	51	for	for	ADP
ejpam-6214	18	52	establishing	establish	VERB
ejpam-6214	18	53	the	the	DET
ejpam-6214	18	54	irreducibility	irreducibility	NOUN
ejpam-6214	18	55	of	of	ADP
ejpam-6214	18	56	such	such	ADJ
ejpam-6214	18	57	polynomial	polynomial	ADJ
ejpam-6214	18	58	combinations	combination	NOUN
ejpam-6214	18	59	over	over	ADP
ejpam-6214	18	60	the	the	DET
ejpam-6214	18	61	integers	integer	NOUN
ejpam-6214	18	62	.	.	PUNCT
ejpam-6214	19	1	specifically	specifically	ADV
ejpam-6214	19	2	,	,	PUNCT
ejpam-6214	19	3	bonciocat	bonciocat	NOUN
ejpam-6214	19	4	proved	prove	VERB
ejpam-6214	19	5	that	that	SCONJ
ejpam-6214	19	6	if	if	SCONJ
ejpam-6214	19	7	deg	deg	PROPN
ejpam-6214	19	8	f	f	X
ejpam-6214	19	9	<	<	X
ejpam-6214	19	10	deg	deg	PROPN
ejpam-6214	19	11	g	g	PROPN
ejpam-6214	19	12	and	and	CCONJ
ejpam-6214	19	13	gcd(k	gcd(k	PROPN
ejpam-6214	19	14	,	,	PUNCT
ejpam-6214	19	15	deg	deg	NOUN
ejpam-6214	19	16	g	g	NOUN
ejpam-6214	19	17	)	)	PUNCT
ejpam-6214	19	18	=	=	SYM
ejpam-6214	19	19	1	1	NUM
ejpam-6214	19	20	,	,	PUNCT
ejpam-6214	19	21	then	then	ADV
ejpam-6214	19	22	f(x)+	f(x)+	VERB
ejpam-6214	19	23	pkg(x	pkg(x	NOUN
ejpam-6214	19	24	)	)	PUNCT
ejpam-6214	19	25	is	be	AUX
ejpam-6214	19	26	irreducible	irreducible	ADJ
ejpam-6214	19	27	in	in	ADP
ejpam-6214	19	28	q[x	q[x	PROPN
ejpam-6214	19	29	]	]	PUNCT
ejpam-6214	19	30	for	for	ADP
ejpam-6214	19	31	all	all	PRON
ejpam-6214	19	32	but	but	ADV
ejpam-6214	19	33	finitely	finitely	ADV
ejpam-6214	19	34	many	many	ADJ
ejpam-6214	19	35	primes	prime	NOUN
ejpam-6214	20	1	p.	p.	NOUN
ejpam-6214	20	2	this	this	DET
ejpam-6214	20	3	result	result	NOUN
ejpam-6214	20	4	elegantly	elegantly	ADV
ejpam-6214	20	5	combines	combine	VERB
ejpam-6214	20	6	ideas	idea	NOUN
ejpam-6214	20	7	from	from	ADP
ejpam-6214	20	8	number	number	NOUN
ejpam-6214	20	9	theory	theory	NOUN
ejpam-6214	20	10	and	and	CCONJ
ejpam-6214	20	11	algebra	algebra	NOUN
ejpam-6214	20	12	,	,	PUNCT
ejpam-6214	20	13	but	but	CCONJ
ejpam-6214	20	14	its	its	PRON
ejpam-6214	20	15	reliance	reliance	NOUN
ejpam-6214	20	16	on	on	ADP
ejpam-6214	20	17	the	the	DET
ejpam-6214	20	18	relative	relative	ADJ
ejpam-6214	20	19	primality	primality	NOUN
ejpam-6214	20	20	of	of	ADP
ejpam-6214	20	21	f	f	PROPN
ejpam-6214	20	22	and	and	CCONJ
ejpam-6214	20	23	g	g	NOUN
ejpam-6214	20	24	,	,	PUNCT
ejpam-6214	20	25	as	as	ADV
ejpam-6214	20	26	well	well	ADV
ejpam-6214	20	27	as	as	ADP
ejpam-6214	20	28	the	the	DET
ejpam-6214	20	29	condition	condition	NOUN
ejpam-6214	20	30	gcd(k	gcd(k	PROPN
ejpam-6214	20	31	,	,	PUNCT
ejpam-6214	20	32	deg	deg	NOUN
ejpam-6214	20	33	g	g	NOUN
ejpam-6214	20	34	)	)	PUNCT
ejpam-6214	20	35	=	=	SYM
ejpam-6214	20	36	1	1	NUM
ejpam-6214	20	37	,	,	PUNCT
ejpam-6214	20	38	restricts	restrict	VERB
ejpam-6214	20	39	its	its	PRON
ejpam-6214	20	40	scope	scope	NOUN
ejpam-6214	20	41	.	.	PUNCT
ejpam-6214	21	1	in	in	ADP
ejpam-6214	21	2	this	this	DET
ejpam-6214	21	3	paper	paper	NOUN
ejpam-6214	21	4	,	,	PUNCT
ejpam-6214	21	5	we	we	PRON
ejpam-6214	21	6	generalize	generalize	VERB
ejpam-6214	21	7	bonciocat	bonciocat	PROPN
ejpam-6214	21	8	’s	’s	PART
ejpam-6214	21	9	theorem	theorem	NOUN
ejpam-6214	21	10	by	by	ADP
ejpam-6214	21	11	removing	remove	VERB
ejpam-6214	21	12	these	these	DET
ejpam-6214	21	13	restrictive	restrictive	ADJ
ejpam-6214	21	14	conditions	condition	NOUN
ejpam-6214	21	15	.	.	PUNCT
ejpam-6214	22	1	specifically	specifically	ADV
ejpam-6214	22	2	,	,	PUNCT
ejpam-6214	22	3	we	we	PRON
ejpam-6214	22	4	prove	prove	VERB
ejpam-6214	22	5	that	that	SCONJ
ejpam-6214	22	6	for	for	ADP
ejpam-6214	22	7	any	any	DET
ejpam-6214	22	8	polynomials	polynomial	NOUN
ejpam-6214	22	9	f(x	f(x	PROPN
ejpam-6214	22	10	)	)	PUNCT
ejpam-6214	22	11	and	and	CCONJ
ejpam-6214	22	12	g(x	g(x	NOUN
ejpam-6214	22	13	)	)	PUNCT
ejpam-6214	22	14	with	with	ADP
ejpam-6214	22	15	integer	integer	NOUN
ejpam-6214	22	16	coefficients	coefficient	NOUN
ejpam-6214	22	17	satisfying	satisfy	VERB
ejpam-6214	22	18	deg	deg	PROPN
ejpam-6214	22	19	f	f	X
ejpam-6214	22	20	<	<	X
ejpam-6214	22	21	deg	deg	PROPN
ejpam-6214	22	22	g	g	NOUN
ejpam-6214	22	23	,	,	PUNCT
ejpam-6214	22	24	and	and	CCONJ
ejpam-6214	22	25	for	for	ADP
ejpam-6214	22	26	any	any	DET
ejpam-6214	22	27	integer	integer	NOUN
ejpam-6214	22	28	k	k	PROPN
ejpam-6214	22	29	≥	≥	NUM
ejpam-6214	22	30	0	0	NUM
ejpam-6214	22	31	,	,	PUNCT
ejpam-6214	22	32	the	the	DET
ejpam-6214	22	33	polynomial	polynomial	ADJ
ejpam-6214	22	34	f(x	f(x	PROPN
ejpam-6214	22	35	)	)	PUNCT
ejpam-6214	23	1	+	+	SYM
ejpam-6214	23	2	pkg(x	pkg(x	NOUN
ejpam-6214	23	3	)	)	PUNCT
ejpam-6214	23	4	is	be	AUX
ejpam-6214	23	5	either	either	CCONJ
ejpam-6214	23	6	irreducible	irreducible	ADJ
ejpam-6214	23	7	over	over	ADP
ejpam-6214	23	8	q	q	NOUN
ejpam-6214	23	9	or	or	CCONJ
ejpam-6214	23	10	factors	factor	NOUN
ejpam-6214	23	11	into	into	ADP
ejpam-6214	23	12	polynomials	polynomial	NOUN
ejpam-6214	23	13	whose	whose	DET
ejpam-6214	23	14	degrees	degree	NOUN
ejpam-6214	23	15	are	be	AUX
ejpam-6214	23	16	multiples	multiple	NOUN
ejpam-6214	23	17	of	of	ADP
ejpam-6214	23	18	gcd(k	gcd(k	PROPN
ejpam-6214	23	19	,	,	PUNCT
ejpam-6214	23	20	deg	deg	VERB
ejpam-6214	23	21	g	g	NOUN
ejpam-6214	23	22	)	)	PUNCT
ejpam-6214	23	23	.	.	PUNCT
ejpam-6214	24	1	this	this	DET
ejpam-6214	24	2	result	result	NOUN
ejpam-6214	24	3	holds	hold	VERB
ejpam-6214	24	4	for	for	ADP
ejpam-6214	24	5	all	all	PRON
ejpam-6214	24	6	but	but	ADV
ejpam-6214	24	7	finitely	finitely	ADV
ejpam-6214	24	8	many	many	ADJ
ejpam-6214	24	9	primes	prime	NOUN
ejpam-6214	24	10	p	p	X
ejpam-6214	24	11	,	,	PUNCT
ejpam-6214	24	12	significantly	significantly	ADV
ejpam-6214	24	13	broadening	broaden	VERB
ejpam-6214	24	14	the	the	DET
ejpam-6214	24	15	applicability	applicability	NOUN
ejpam-6214	24	16	of	of	ADP
ejpam-6214	24	17	earlier	early	ADJ
ejpam-6214	24	18	work	work	NOUN
ejpam-6214	24	19	.	.	PUNCT
ejpam-6214	25	1	the	the	DET
ejpam-6214	25	2	key	key	ADJ
ejpam-6214	25	3	innovation	innovation	NOUN
ejpam-6214	25	4	in	in	ADP
ejpam-6214	25	5	our	our	PRON
ejpam-6214	25	6	approach	approach	NOUN
ejpam-6214	25	7	lies	lie	VERB
ejpam-6214	25	8	in	in	ADP
ejpam-6214	25	9	the	the	DET
ejpam-6214	25	10	careful	careful	ADJ
ejpam-6214	25	11	analysis	analysis	NOUN
ejpam-6214	25	12	of	of	ADP
ejpam-6214	25	13	the	the	DET
ejpam-6214	25	14	structure	structure	NOUN
ejpam-6214	25	15	of	of	ADP
ejpam-6214	25	16	f(x)+	f(x)+	ADJ
ejpam-6214	25	17	pkg(x	pkg(x	NOUN
ejpam-6214	25	18	)	)	PUNCT
ejpam-6214	25	19	without	without	ADP
ejpam-6214	25	20	assuming	assume	VERB
ejpam-6214	25	21	relative	relative	ADJ
ejpam-6214	25	22	primality	primality	NOUN
ejpam-6214	25	23	or	or	CCONJ
ejpam-6214	25	24	imposing	impose	VERB
ejpam-6214	25	25	conditions	condition	NOUN
ejpam-6214	25	26	on	on	ADP
ejpam-6214	25	27	gcd(k	gcd(k	PROPN
ejpam-6214	25	28	,	,	PUNCT
ejpam-6214	25	29	deg	deg	NOUN
ejpam-6214	25	30	g	g	NOUN
ejpam-6214	25	31	)	)	PUNCT
ejpam-6214	25	32	.	.	PUNCT
ejpam-6214	26	1	by	by	ADP
ejpam-6214	26	2	leveraging	leverage	VERB
ejpam-6214	26	3	tools	tool	NOUN
ejpam-6214	26	4	such	such	ADJ
ejpam-6214	26	5	as	as	ADP
ejpam-6214	26	6	reduction	reduction	NOUN
ejpam-6214	26	7	modulo	modulo	NOUN
ejpam-6214	26	8	p	p	X
ejpam-6214	27	1	[	[	X
ejpam-6214	27	2	5	5	NUM
ejpam-6214	27	3	]	]	PUNCT
ejpam-6214	27	4	and	and	CCONJ
ejpam-6214	27	5	eisenstein	eisenstein	PROPN
ejpam-6214	27	6	’s	’s	PART
ejpam-6214	27	7	criterion	criterion	NOUN
ejpam-6214	28	1	[	[	X
ejpam-6214	28	2	3	3	NUM
ejpam-6214	28	3	]	]	PUNCT
ejpam-6214	28	4	,	,	PUNCT
ejpam-6214	28	5	we	we	PRON
ejpam-6214	28	6	establish	establish	VERB
ejpam-6214	28	7	a	a	DET
ejpam-6214	28	8	unified	unified	ADJ
ejpam-6214	28	9	framework	framework	NOUN
ejpam-6214	28	10	for	for	ADP
ejpam-6214	28	11	studying	study	VERB
ejpam-6214	28	12	the	the	DET
ejpam-6214	28	13	irreducibility	irreducibility	NOUN
ejpam-6214	28	14	of	of	ADP
ejpam-6214	28	15	such	such	ADJ
ejpam-6214	28	16	polynomials	polynomial	NOUN
ejpam-6214	28	17	.	.	PUNCT
ejpam-6214	29	1	our	our	PRON
ejpam-6214	29	2	theorem	theorem	NOUN
ejpam-6214	29	3	not	not	PART
ejpam-6214	29	4	only	only	ADV
ejpam-6214	29	5	generalizes	generalize	VERB
ejpam-6214	29	6	bonciocat	bonciocat	NOUN
ejpam-6214	29	7	’s	’s	PART
ejpam-6214	29	8	result	result	NOUN
ejpam-6214	29	9	but	but	CCONJ
ejpam-6214	29	10	also	also	ADV
ejpam-6214	29	11	provides	provide	VERB
ejpam-6214	29	12	new	new	ADJ
ejpam-6214	29	13	insights	insight	NOUN
ejpam-6214	29	14	into	into	ADP
ejpam-6214	29	15	the	the	DET
ejpam-6214	29	16	factorization	factorization	NOUN
ejpam-6214	29	17	structure	structure	NOUN
ejpam-6214	29	18	of	of	ADP
ejpam-6214	29	19	polynomials	polynomial	NOUN
ejpam-6214	29	20	with	with	ADP
ejpam-6214	29	21	prime	prime	ADJ
ejpam-6214	29	22	power	power	NOUN
ejpam-6214	29	23	shifts	shift	NOUN
ejpam-6214	29	24	.	.	PUNCT
ejpam-6214	30	1	the	the	DET
ejpam-6214	30	2	structure	structure	NOUN
ejpam-6214	30	3	of	of	ADP
ejpam-6214	30	4	this	this	DET
ejpam-6214	30	5	paper	paper	NOUN
ejpam-6214	30	6	is	be	AUX
ejpam-6214	30	7	as	as	SCONJ
ejpam-6214	30	8	follows	follow	VERB
ejpam-6214	30	9	.	.	PUNCT
ejpam-6214	31	1	section	section	NOUN
ejpam-6214	31	2	2	2	NUM
ejpam-6214	31	3	,	,	PUNCT
ejpam-6214	31	4	introduces	introduce	VERB
ejpam-6214	31	5	key	key	ADJ
ejpam-6214	31	6	theoretical	theoretical	ADJ
ejpam-6214	31	7	foundations	foundation	NOUN
ejpam-6214	31	8	and	and	CCONJ
ejpam-6214	31	9	prior	prior	ADJ
ejpam-6214	31	10	research	research	NOUN
ejpam-6214	31	11	relevant	relevant	ADJ
ejpam-6214	31	12	to	to	ADP
ejpam-6214	31	13	our	our	PRON
ejpam-6214	31	14	investigation	investigation	NOUN
ejpam-6214	31	15	,	,	PUNCT
ejpam-6214	31	16	covering	cover	VERB
ejpam-6214	31	17	essential	essential	ADJ
ejpam-6214	31	18	definitions	definition	NOUN
ejpam-6214	31	19	and	and	CCONJ
ejpam-6214	31	20	mathematical	mathematical	ADJ
ejpam-6214	31	21	tools	tool	NOUN
ejpam-6214	31	22	.	.	PUNCT
ejpam-6214	32	1	this	this	PRON
ejpam-6214	32	2	includes	include	VERB
ejpam-6214	32	3	an	an	DET
ejpam-6214	32	4	examination	examination	NOUN
ejpam-6214	32	5	of	of	ADP
ejpam-6214	32	6	bonciocat	bonciocat	NOUN
ejpam-6214	32	7	’s	’s	PART
ejpam-6214	32	8	important	important	ADJ
ejpam-6214	32	9	result	result	NOUN
ejpam-6214	32	10	[	[	X
ejpam-6214	32	11	4	4	X
ejpam-6214	32	12	]	]	PUNCT
ejpam-6214	32	13	along	along	ADP
ejpam-6214	32	14	with	with	ADP
ejpam-6214	32	15	its	its	PRON
ejpam-6214	32	16	underlying	underlie	VERB
ejpam-6214	32	17	proof	proof	NOUN
ejpam-6214	32	18	methodology	methodology	NOUN
ejpam-6214	32	19	.	.	PUNCT
ejpam-6214	33	1	section	section	NOUN
ejpam-6214	33	2	3	3	NUM
ejpam-6214	33	3	presents	present	VERB
ejpam-6214	33	4	our	our	PRON
ejpam-6214	33	5	main	main	ADJ
ejpam-6214	33	6	theorem	theorem	NOUN
ejpam-6214	33	7	and	and	CCONJ
ejpam-6214	33	8	its	its	PRON
ejpam-6214	33	9	proof	proof	NOUN
ejpam-6214	33	10	,	,	PUNCT
ejpam-6214	33	11	which	which	PRON
ejpam-6214	33	12	is	be	AUX
ejpam-6214	33	13	divided	divide	VERB
ejpam-6214	33	14	into	into	ADP
ejpam-6214	33	15	several	several	ADJ
ejpam-6214	33	16	steps	step	NOUN
ejpam-6214	33	17	for	for	ADP
ejpam-6214	33	18	clarity	clarity	NOUN
ejpam-6214	33	19	.	.	PUNCT
ejpam-6214	34	1	in	in	ADP
ejpam-6214	34	2	section	section	NOUN
ejpam-6214	34	3	4	4	NUM
ejpam-6214	34	4	,	,	PUNCT
ejpam-6214	34	5	we	we	PRON
ejpam-6214	34	6	provide	provide	VERB
ejpam-6214	34	7	examples	example	NOUN
ejpam-6214	34	8	to	to	PART
ejpam-6214	34	9	illustrate	illustrate	VERB
ejpam-6214	34	10	the	the	DET
ejpam-6214	34	11	application	application	NOUN
ejpam-6214	34	12	of	of	ADP
ejpam-6214	34	13	our	our	PRON
ejpam-6214	34	14	theorem	theorem	NOUN
ejpam-6214	34	15	.	.	PROPN
ejpam-6214	34	16	section	section	NOUN
ejpam-6214	34	17	5	5	NUM
ejpam-6214	34	18	discusses	discuss	VERB
ejpam-6214	34	19	the	the	DET
ejpam-6214	34	20	implications	implication	NOUN
ejpam-6214	34	21	of	of	ADP
ejpam-6214	34	22	our	our	PRON
ejpam-6214	34	23	result	result	NOUN
ejpam-6214	34	24	and	and	CCONJ
ejpam-6214	34	25	its	its	PRON
ejpam-6214	34	26	connections	connection	NOUN
ejpam-6214	34	27	to	to	ADP
ejpam-6214	34	28	other	other	ADJ
ejpam-6214	34	29	areas	area	NOUN
ejpam-6214	34	30	of	of	ADP
ejpam-6214	34	31	mathematics	mathematic	NOUN
ejpam-6214	34	32	,	,	PUNCT
ejpam-6214	34	33	such	such	ADJ
ejpam-6214	34	34	as	as	ADP
ejpam-6214	34	35	algebraic	algebraic	ADJ
ejpam-6214	34	36	number	number	NOUN
ejpam-6214	34	37	theory	theory	NOUN
ejpam-6214	34	38	[	[	X
ejpam-6214	34	39	6	6	NUM
ejpam-6214	34	40	]	]	PUNCT
ejpam-6214	34	41	and	and	CCONJ
ejpam-6214	34	42	diophantine	diophantine	VERB
ejpam-6214	34	43	equations	equation	NOUN
ejpam-6214	35	1	[	[	X
ejpam-6214	35	2	7	7	NUM
ejpam-6214	35	3	]	]	PUNCT
ejpam-6214	35	4	.	.	PUNCT
ejpam-6214	36	1	finally	finally	ADV
ejpam-6214	36	2	,	,	PUNCT
ejpam-6214	36	3	in	in	ADP
ejpam-6214	36	4	section	section	NOUN
ejpam-6214	36	5	6	6	NUM
ejpam-6214	36	6	,	,	PUNCT
ejpam-6214	36	7	we	we	PRON
ejpam-6214	36	8	conclude	conclude	VERB
ejpam-6214	36	9	with	with	ADP
ejpam-6214	36	10	a	a	DET
ejpam-6214	36	11	summary	summary	NOUN
ejpam-6214	36	12	of	of	ADP
ejpam-6214	36	13	our	our	PRON
ejpam-6214	36	14	findings	finding	NOUN
ejpam-6214	36	15	and	and	CCONJ
ejpam-6214	36	16	suggest	suggest	VERB
ejpam-6214	36	17	directions	direction	NOUN
ejpam-6214	36	18	for	for	ADP
ejpam-6214	36	19	future	future	ADJ
ejpam-6214	36	20	research	research	NOUN
ejpam-6214	36	21	.	.	PUNCT
ejpam-6214	37	1	our	our	PRON
ejpam-6214	37	2	work	work	NOUN
ejpam-6214	37	3	contributes	contribute	VERB
ejpam-6214	37	4	to	to	ADP
ejpam-6214	37	5	the	the	DET
ejpam-6214	37	6	growing	grow	VERB
ejpam-6214	37	7	body	body	NOUN
ejpam-6214	37	8	of	of	ADP
ejpam-6214	37	9	literature	literature	NOUN
ejpam-6214	37	10	on	on	ADP
ejpam-6214	37	11	polynomial	polynomial	ADJ
ejpam-6214	37	12	irreducibility	irreducibility	NOUN
ejpam-6214	37	13	[	[	X
ejpam-6214	37	14	8	8	NUM
ejpam-6214	37	15	,	,	PUNCT
ejpam-6214	37	16	9	9	NUM
ejpam-6214	37	17	]	]	PUNCT
ejpam-6214	37	18	and	and	CCONJ
ejpam-6214	37	19	opens	open	VERB
ejpam-6214	37	20	new	new	ADJ
ejpam-6214	37	21	avenues	avenue	NOUN
ejpam-6214	37	22	for	for	ADP
ejpam-6214	37	23	exploring	explore	VERB
ejpam-6214	37	24	the	the	DET
ejpam-6214	37	25	interplay	interplay	NOUN
ejpam-6214	37	26	between	between	ADP
ejpam-6214	37	27	number	number	NOUN
ejpam-6214	37	28	theory	theory	NOUN
ejpam-6214	37	29	and	and	CCONJ
ejpam-6214	37	30	algebra	algebra	NOUN
ejpam-6214	37	31	.	.	PUNCT
ejpam-6214	38	1	we	we	PRON
ejpam-6214	38	2	hope	hope	VERB
ejpam-6214	38	3	that	that	SCONJ
ejpam-6214	38	4	this	this	DET
ejpam-6214	38	5	paper	paper	NOUN
ejpam-6214	38	6	will	will	AUX
ejpam-6214	38	7	inspire	inspire	VERB
ejpam-6214	38	8	further	further	ADJ
ejpam-6214	38	9	research	research	NOUN
ejpam-6214	38	10	into	into	ADP
ejpam-6214	38	11	the	the	DET
ejpam-6214	38	12	irreducibility	irreducibility	NOUN
ejpam-6214	38	13	of	of	ADP
ejpam-6214	38	14	polynomials	polynomial	NOUN
ejpam-6214	38	15	with	with	ADP
ejpam-6214	38	16	prime	prime	ADJ
ejpam-6214	38	17	power	power	NOUN
ejpam-6214	38	18	shifts	shift	NOUN
ejpam-6214	38	19	and	and	CCONJ
ejpam-6214	38	20	their	their	PRON
ejpam-6214	38	21	applications	application	NOUN
ejpam-6214	38	22	.	.	PUNCT
ejpam-6214	39	1	2	2	X
ejpam-6214	39	2	.	.	NUM
ejpam-6214	39	3	preliminaries	preliminary	NOUN
ejpam-6214	39	4	2.1	2.1	NUM
ejpam-6214	39	5	.	.	PUNCT
ejpam-6214	40	1	bonciocat	bonciocat	PROPN
ejpam-6214	40	2	’s	’s	PART
ejpam-6214	40	3	theorem	theorem	VERB
ejpam-6214	40	4	the	the	DET
ejpam-6214	40	5	foundation	foundation	NOUN
ejpam-6214	40	6	of	of	ADP
ejpam-6214	40	7	our	our	PRON
ejpam-6214	40	8	work	work	NOUN
ejpam-6214	40	9	lies	lie	VERB
ejpam-6214	40	10	in	in	ADP
ejpam-6214	40	11	the	the	DET
ejpam-6214	40	12	following	following	NOUN
ejpam-6214	40	13	theorem	theorem	NOUN
ejpam-6214	40	14	by	by	ADP
ejpam-6214	40	15	bonciocat	bonciocat	NOUN
ejpam-6214	40	16	[	[	X
ejpam-6214	40	17	4	4	NUM
ejpam-6214	40	18	]	]	NOUN
ejpam-6214	40	19	:	:	PUNCT
ejpam-6214	40	20	a.	a.	PROPN
ejpam-6214	40	21	chandoul	chandoul	PROPN
ejpam-6214	40	22	,	,	PUNCT
ejpam-6214	40	23	s.	s.	PROPN
ejpam-6214	40	24	mansour	mansour	PROPN
ejpam-6214	40	25	/	/	SYM
ejpam-6214	40	26	eur	eur	PROPN
ejpam-6214	40	27	.	.	PUNCT
ejpam-6214	41	1	j.	j.	PROPN
ejpam-6214	41	2	pure	pure	PROPN
ejpam-6214	41	3	appl	appl	PROPN
ejpam-6214	41	4	.	.	PROPN
ejpam-6214	41	5	math	math	PROPN
ejpam-6214	41	6	,	,	PUNCT
ejpam-6214	41	7	18	18	NUM
ejpam-6214	41	8	(	(	PUNCT
ejpam-6214	41	9	3	3	NUM
ejpam-6214	41	10	)	)	PUNCT
ejpam-6214	41	11	(	(	PUNCT
ejpam-6214	41	12	2025	2025	NUM
ejpam-6214	41	13	)	)	PUNCT
ejpam-6214	41	14	,	,	PUNCT
ejpam-6214	41	15	6214	6214	NUM
ejpam-6214	41	16	3	3	NUM
ejpam-6214	41	17	of	of	ADP
ejpam-6214	41	18	8	8	NUM
ejpam-6214	41	19	theorem	theorem	ADJ
ejpam-6214	41	20	1	1	NUM
ejpam-6214	41	21	(	(	PUNCT
ejpam-6214	41	22	bonciocat	bonciocat	NOUN
ejpam-6214	41	23	,	,	PUNCT
ejpam-6214	41	24	2016	2016	NUM
ejpam-6214	41	25	)	)	PUNCT
ejpam-6214	41	26	.	.	PUNCT
ejpam-6214	42	1	let	let	VERB
ejpam-6214	42	2	f(x	f(x	PROPN
ejpam-6214	42	3	)	)	PUNCT
ejpam-6214	42	4	,	,	PUNCT
ejpam-6214	42	5	g(x	g(x	NOUN
ejpam-6214	42	6	)	)	PUNCT
ejpam-6214	42	7	∈	∈	PROPN
ejpam-6214	42	8	z[x	z[x	NOUN
ejpam-6214	42	9	]	]	PUNCT
ejpam-6214	42	10	be	be	VERB
ejpam-6214	42	11	relatively	relatively	ADV
ejpam-6214	42	12	prime	prime	ADJ
ejpam-6214	42	13	polynomials	polynomial	NOUN
ejpam-6214	42	14	with	with	ADP
ejpam-6214	42	15	deg	deg	PROPN
ejpam-6214	42	16	f	f	X
ejpam-6214	42	17	<	<	X
ejpam-6214	42	18	deg	deg	PROPN
ejpam-6214	42	19	g	g	NOUN
ejpam-6214	42	20	,	,	PUNCT
ejpam-6214	42	21	and	and	CCONJ
ejpam-6214	42	22	let	let	VERB
ejpam-6214	42	23	k	k	PRON
ejpam-6214	42	24	be	be	AUX
ejpam-6214	42	25	a	a	DET
ejpam-6214	42	26	positive	positive	ADJ
ejpam-6214	42	27	integer	integer	NOUN
ejpam-6214	42	28	such	such	ADJ
ejpam-6214	42	29	that	that	SCONJ
ejpam-6214	42	30	gcd(k	gcd(k	PROPN
ejpam-6214	42	31	,	,	PUNCT
ejpam-6214	42	32	deg	deg	VERB
ejpam-6214	42	33	g	g	NOUN
ejpam-6214	42	34	)	)	PUNCT
ejpam-6214	42	35	=	=	SYM
ejpam-6214	43	1	1	1	X
ejpam-6214	43	2	.	.	PUNCT
ejpam-6214	43	3	then	then	ADV
ejpam-6214	43	4	,	,	PUNCT
ejpam-6214	43	5	for	for	ADP
ejpam-6214	43	6	all	all	PRON
ejpam-6214	43	7	but	but	ADV
ejpam-6214	43	8	finitely	finitely	ADV
ejpam-6214	43	9	many	many	ADJ
ejpam-6214	43	10	primes	prime	NOUN
ejpam-6214	43	11	p	p	X
ejpam-6214	43	12	,	,	PUNCT
ejpam-6214	43	13	the	the	DET
ejpam-6214	43	14	polynomial	polynomial	ADJ
ejpam-6214	43	15	f(x	f(x	PROPN
ejpam-6214	43	16	)	)	PUNCT
ejpam-6214	44	1	+	+	SYM
ejpam-6214	44	2	pkg(x	pkg(x	NOUN
ejpam-6214	44	3	)	)	PUNCT
ejpam-6214	44	4	is	be	AUX
ejpam-6214	44	5	irreducible	irreducible	ADJ
ejpam-6214	44	6	over	over	ADP
ejpam-6214	44	7	q.	q.	PROPN
ejpam-6214	44	8	bonciocat	bonciocat	PROPN
ejpam-6214	44	9	’s	’s	PART
ejpam-6214	44	10	proof	proof	NOUN
ejpam-6214	44	11	relies	rely	VERB
ejpam-6214	44	12	on	on	ADP
ejpam-6214	44	13	the	the	DET
ejpam-6214	44	14	following	follow	VERB
ejpam-6214	44	15	key	key	ADJ
ejpam-6214	44	16	ideas	idea	NOUN
ejpam-6214	44	17	:	:	PUNCT
ejpam-6214	44	18	(	(	PUNCT
ejpam-6214	44	19	i	i	NOUN
ejpam-6214	44	20	)	)	PUNCT
ejpam-6214	44	21	the	the	DET
ejpam-6214	44	22	use	use	NOUN
ejpam-6214	44	23	of	of	ADP
ejpam-6214	44	24	reduction	reduction	NOUN
ejpam-6214	44	25	modulo	modulo	NOUN
ejpam-6214	44	26	p	p	X
ejpam-6214	44	27	to	to	PART
ejpam-6214	44	28	analyze	analyze	VERB
ejpam-6214	44	29	the	the	DET
ejpam-6214	44	30	irreducibility	irreducibility	NOUN
ejpam-6214	44	31	of	of	ADP
ejpam-6214	44	32	f(x	f(x	PROPN
ejpam-6214	44	33	)	)	PUNCT
ejpam-6214	45	1	+	+	NUM
ejpam-6214	45	2	pkg(x	pkg(x	NOUN
ejpam-6214	45	3	)	)	PUNCT
ejpam-6214	45	4	.	.	PUNCT
ejpam-6214	46	1	(	(	PUNCT
ejpam-6214	46	2	ii	ii	X
ejpam-6214	46	3	)	)	PUNCT
ejpam-6214	46	4	the	the	DET
ejpam-6214	46	5	assumption	assumption	NOUN
ejpam-6214	46	6	that	that	SCONJ
ejpam-6214	46	7	f	f	PROPN
ejpam-6214	46	8	and	and	CCONJ
ejpam-6214	46	9	g	g	PROPN
ejpam-6214	46	10	are	be	AUX
ejpam-6214	46	11	relatively	relatively	ADV
ejpam-6214	46	12	prime	prime	ADJ
ejpam-6214	46	13	ensures	ensure	NOUN
ejpam-6214	46	14	that	that	SCONJ
ejpam-6214	46	15	f(x	f(x	PROPN
ejpam-6214	46	16	)	)	PUNCT
ejpam-6214	46	17	+	+	CCONJ
ejpam-6214	46	18	pkg(x	pkg(x	NOUN
ejpam-6214	46	19	)	)	PUNCT
ejpam-6214	46	20	does	do	AUX
ejpam-6214	46	21	not	not	PART
ejpam-6214	46	22	factor	factor	VERB
ejpam-6214	46	23	trivially	trivially	ADV
ejpam-6214	46	24	.	.	PUNCT
ejpam-6214	47	1	(	(	PUNCT
ejpam-6214	47	2	iii	iii	X
ejpam-6214	47	3	)	)	PUNCT
ejpam-6214	47	4	the	the	DET
ejpam-6214	47	5	condition	condition	NOUN
ejpam-6214	47	6	gcd(k	gcd(k	PROPN
ejpam-6214	47	7	,	,	PUNCT
ejpam-6214	47	8	deg	deg	VERB
ejpam-6214	47	9	g	g	NOUN
ejpam-6214	47	10	)	)	PUNCT
ejpam-6214	47	11	=	=	SYM
ejpam-6214	47	12	1	1	NUM
ejpam-6214	47	13	ensures	ensure	VERB
ejpam-6214	47	14	that	that	SCONJ
ejpam-6214	47	15	the	the	DET
ejpam-6214	47	16	term	term	NOUN
ejpam-6214	47	17	pkg(x	pkg(x	NOUN
ejpam-6214	47	18	)	)	PUNCT
ejpam-6214	47	19	does	do	AUX
ejpam-6214	47	20	not	not	PART
ejpam-6214	47	21	introduce	introduce	VERB
ejpam-6214	47	22	unwanted	unwanted	ADJ
ejpam-6214	47	23	factorizations	factorization	NOUN
ejpam-6214	47	24	.	.	PUNCT
ejpam-6214	48	1	while	while	SCONJ
ejpam-6214	48	2	this	this	DET
ejpam-6214	48	3	result	result	NOUN
ejpam-6214	48	4	is	be	AUX
ejpam-6214	48	5	elegant	elegant	ADJ
ejpam-6214	48	6	,	,	PUNCT
ejpam-6214	48	7	its	its	PRON
ejpam-6214	48	8	reliance	reliance	NOUN
ejpam-6214	48	9	on	on	ADP
ejpam-6214	48	10	the	the	DET
ejpam-6214	48	11	relative	relative	ADJ
ejpam-6214	48	12	primality	primality	NOUN
ejpam-6214	48	13	of	of	ADP
ejpam-6214	48	14	f	f	PROPN
ejpam-6214	48	15	and	and	CCONJ
ejpam-6214	48	16	g	g	NOUN
ejpam-6214	48	17	,	,	PUNCT
ejpam-6214	48	18	as	as	ADV
ejpam-6214	48	19	well	well	ADV
ejpam-6214	48	20	as	as	ADP
ejpam-6214	48	21	the	the	DET
ejpam-6214	48	22	condition	condition	NOUN
ejpam-6214	48	23	gcd(k	gcd(k	PROPN
ejpam-6214	48	24	,	,	PUNCT
ejpam-6214	48	25	deg	deg	NOUN
ejpam-6214	48	26	g	g	NOUN
ejpam-6214	48	27	)	)	PUNCT
ejpam-6214	48	28	=	=	SYM
ejpam-6214	48	29	1	1	NUM
ejpam-6214	48	30	,	,	PUNCT
ejpam-6214	48	31	limits	limit	VERB
ejpam-6214	48	32	its	its	PRON
ejpam-6214	48	33	applicability	applicability	NOUN
ejpam-6214	48	34	.	.	PUNCT
ejpam-6214	49	1	our	our	PRON
ejpam-6214	49	2	work	work	NOUN
ejpam-6214	49	3	removes	remove	VERB
ejpam-6214	49	4	these	these	DET
ejpam-6214	49	5	restrictions	restriction	NOUN
ejpam-6214	49	6	and	and	CCONJ
ejpam-6214	49	7	provides	provide	VERB
ejpam-6214	49	8	a	a	DET
ejpam-6214	49	9	more	more	ADV
ejpam-6214	49	10	general	general	ADJ
ejpam-6214	49	11	result	result	NOUN
ejpam-6214	49	12	.	.	PUNCT
ejpam-6214	50	1	2.2	2.2	NUM
ejpam-6214	50	2	.	.	PUNCT
ejpam-6214	50	3	polynomials	polynomial	NOUN
ejpam-6214	50	4	and	and	CCONJ
ejpam-6214	50	5	irreducibility	irreducibility	NOUN
ejpam-6214	50	6	consider	consider	VERB
ejpam-6214	50	7	the	the	DET
ejpam-6214	50	8	ring	ring	NOUN
ejpam-6214	50	9	z[x	z[x	NOUN
ejpam-6214	50	10	]	]	PUNCT
ejpam-6214	50	11	of	of	ADP
ejpam-6214	50	12	integer	integer	NOUN
ejpam-6214	50	13	-	-	PUNCT
ejpam-6214	50	14	coefficient	coefficient	NOUN
ejpam-6214	50	15	polynomials	polynomial	NOUN
ejpam-6214	50	16	.	.	PUNCT
ejpam-6214	51	1	we	we	PRON
ejpam-6214	51	2	say	say	VERB
ejpam-6214	51	3	f(x	f(x	PROPN
ejpam-6214	51	4	)	)	PUNCT
ejpam-6214	51	5	∈	∈	PROPN
ejpam-6214	51	6	z[x	z[x	NOUN
ejpam-6214	51	7	]	]	PUNCT
ejpam-6214	51	8	has	have	VERB
ejpam-6214	51	9	no	no	DET
ejpam-6214	51	10	nontrivial	nontrivial	ADJ
ejpam-6214	51	11	factorization	factorization	NOUN
ejpam-6214	51	12	in	in	ADP
ejpam-6214	51	13	q[x	q[x	ADP
ejpam-6214	51	14	]	]	PUNCT
ejpam-6214	51	15	if	if	SCONJ
ejpam-6214	51	16	for	for	ADP
ejpam-6214	51	17	all	all	DET
ejpam-6214	51	18	q1(x	q1(x	NOUN
ejpam-6214	51	19	)	)	PUNCT
ejpam-6214	51	20	,	,	PUNCT
ejpam-6214	51	21	q2(x	q2(x	X
ejpam-6214	51	22	)	)	PUNCT
ejpam-6214	51	23	∈	∈	PROPN
ejpam-6214	51	24	q[x	q[x	PROPN
ejpam-6214	51	25	]	]	PUNCT
ejpam-6214	51	26	satisfying	satisfy	VERB
ejpam-6214	51	27	f(x	f(x	PROPN
ejpam-6214	51	28	)	)	PUNCT
ejpam-6214	51	29	=	=	PUNCT
ejpam-6214	52	1	q1(x)q2(x	q1(x)q2(x	PROPN
ejpam-6214	52	2	)	)	PUNCT
ejpam-6214	52	3	,	,	PUNCT
ejpam-6214	52	4	either	either	CCONJ
ejpam-6214	52	5	q1(x	q1(x	NOUN
ejpam-6214	52	6	)	)	PUNCT
ejpam-6214	52	7	or	or	CCONJ
ejpam-6214	52	8	q2(x	q2(x	X
ejpam-6214	52	9	)	)	PUNCT
ejpam-6214	52	10	is	be	AUX
ejpam-6214	52	11	a	a	DET
ejpam-6214	52	12	constant	constant	ADJ
ejpam-6214	52	13	polynomial	polynomial	NOUN
ejpam-6214	52	14	.	.	PUNCT
ejpam-6214	53	1	relationship	relationship	NOUN
ejpam-6214	53	2	between	between	ADP
ejpam-6214	53	3	irreducibility	irreducibility	NOUN
ejpam-6214	53	4	over	over	ADP
ejpam-6214	53	5	q	q	NOUN
ejpam-6214	53	6	and	and	CCONJ
ejpam-6214	53	7	z	z	NOUN
ejpam-6214	53	8	proposition	proposition	NOUN
ejpam-6214	53	9	1	1	NUM
ejpam-6214	53	10	(	(	PUNCT
ejpam-6214	53	11	gauss	gauss	PROPN
ejpam-6214	53	12	’s	’s	PART
ejpam-6214	53	13	lemma	lemma	PROPN
ejpam-6214	53	14	)	)	PUNCT
ejpam-6214	53	15	.	.	PUNCT
ejpam-6214	54	1	let	let	VERB
ejpam-6214	54	2	f(x	f(x	PROPN
ejpam-6214	54	3	)	)	PUNCT
ejpam-6214	54	4	∈	∈	PROPN
ejpam-6214	54	5	z[x	z[x	NOUN
ejpam-6214	54	6	]	]	PUNCT
ejpam-6214	54	7	be	be	VERB
ejpam-6214	54	8	a	a	DET
ejpam-6214	54	9	non	non	ADJ
ejpam-6214	54	10	-	-	ADJ
ejpam-6214	54	11	constant	constant	ADJ
ejpam-6214	54	12	polynomial	polynomial	NOUN
ejpam-6214	54	13	.	.	PUNCT
ejpam-6214	55	1	then	then	ADV
ejpam-6214	55	2	:	:	PUNCT
ejpam-6214	55	3	f	f	PROPN
ejpam-6214	55	4	is	be	AUX
ejpam-6214	55	5	irreducible	irreducible	ADJ
ejpam-6214	55	6	in	in	ADP
ejpam-6214	55	7	z[x	z[x	NOUN
ejpam-6214	55	8	]	]	X
ejpam-6214	55	9	=	=	NOUN
ejpam-6214	55	10	⇒	⇒	X
ejpam-6214	55	11	f	f	X
ejpam-6214	55	12	is	be	AUX
ejpam-6214	55	13	irreducible	irreducible	ADJ
ejpam-6214	55	14	in	in	ADP
ejpam-6214	55	15	q[x	q[x	PROPN
ejpam-6214	55	16	]	]	PUNCT
ejpam-6214	55	17	moreover	moreover	ADV
ejpam-6214	55	18	,	,	PUNCT
ejpam-6214	55	19	if	if	SCONJ
ejpam-6214	55	20	f	f	PROPN
ejpam-6214	55	21	is	be	AUX
ejpam-6214	55	22	primitive	primitive	ADJ
ejpam-6214	55	23	(	(	PUNCT
ejpam-6214	55	24	i.e.	i.e.	X
ejpam-6214	55	25	,	,	PUNCT
ejpam-6214	55	26	the	the	DET
ejpam-6214	55	27	greatest	great	ADJ
ejpam-6214	55	28	common	common	ADJ
ejpam-6214	55	29	divisor	divisor	NOUN
ejpam-6214	55	30	of	of	ADP
ejpam-6214	55	31	its	its	PRON
ejpam-6214	55	32	coefficients	coefficient	NOUN
ejpam-6214	55	33	is	be	AUX
ejpam-6214	55	34	1	1	NUM
ejpam-6214	55	35	)	)	PUNCT
ejpam-6214	55	36	,	,	PUNCT
ejpam-6214	55	37	then	then	ADV
ejpam-6214	55	38	:	:	PUNCT
ejpam-6214	55	39	f	f	PROPN
ejpam-6214	55	40	is	be	AUX
ejpam-6214	55	41	irreducible	irreducible	ADJ
ejpam-6214	55	42	in	in	ADP
ejpam-6214	55	43	z[x	z[x	NOUN
ejpam-6214	55	44	]	]	PUNCT
ejpam-6214	55	45	⇐	⇐	ADJ
ejpam-6214	55	46	⇒	⇒	PROPN
ejpam-6214	55	47	f	f	X
ejpam-6214	55	48	is	be	AUX
ejpam-6214	55	49	irreducible	irreducible	ADJ
ejpam-6214	55	50	in	in	ADP
ejpam-6214	55	51	q[x	q[x	PROPN
ejpam-6214	55	52	]	]	PUNCT
ejpam-6214	55	53	key	key	ADJ
ejpam-6214	55	54	implications	implication	NOUN
ejpam-6214	55	55	first	first	ADV
ejpam-6214	55	56	,	,	PUNCT
ejpam-6214	55	57	regarding	regard	VERB
ejpam-6214	55	58	the	the	DET
ejpam-6214	55	59	relationship	relationship	NOUN
ejpam-6214	55	60	from	from	ADP
ejpam-6214	55	61	z	z	PROPN
ejpam-6214	55	62	to	to	PART
ejpam-6214	55	63	q	q	NOUN
ejpam-6214	55	64	:	:	PUNCT
ejpam-6214	55	65	any	any	DET
ejpam-6214	55	66	factorization	factorization	NOUN
ejpam-6214	55	67	in	in	ADP
ejpam-6214	55	68	z[x	z[x	NOUN
ejpam-6214	55	69	]	]	PUNCT
ejpam-6214	55	70	is	be	AUX
ejpam-6214	55	71	automatically	automatically	ADV
ejpam-6214	55	72	valid	valid	ADJ
ejpam-6214	55	73	in	in	ADP
ejpam-6214	55	74	q[x	q[x	PROPN
ejpam-6214	55	75	]	]	PUNCT
ejpam-6214	55	76	,	,	PUNCT
ejpam-6214	55	77	which	which	PRON
ejpam-6214	55	78	means	mean	VERB
ejpam-6214	55	79	that	that	SCONJ
ejpam-6214	55	80	z	z	NOUN
ejpam-6214	55	81	-	-	PUNCT
ejpam-6214	55	82	irreducibility	irreducibility	NOUN
ejpam-6214	55	83	is	be	AUX
ejpam-6214	55	84	strictly	strictly	ADV
ejpam-6214	55	85	stronger	strong	ADJ
ejpam-6214	55	86	than	than	ADP
ejpam-6214	55	87	q	q	NOUN
ejpam-6214	55	88	-	-	NOUN
ejpam-6214	55	89	irreducibility	irreducibility	NOUN
ejpam-6214	55	90	.	.	PUNCT
ejpam-6214	56	1	for	for	ADP
ejpam-6214	56	2	primitive	primitive	ADJ
ejpam-6214	56	3	polynomials	polynomial	NOUN
ejpam-6214	56	4	f	f	PROPN
ejpam-6214	56	5	∈	∈	PROPN
ejpam-6214	56	6	z[x	z[x	PROPN
ejpam-6214	56	7	]	]	PUNCT
ejpam-6214	56	8	,	,	PUNCT
ejpam-6214	56	9	the	the	DET
ejpam-6214	56	10	two	two	NUM
ejpam-6214	56	11	notions	notion	NOUN
ejpam-6214	56	12	of	of	ADP
ejpam-6214	56	13	irreducibility	irreducibility	NOUN
ejpam-6214	56	14	coincide	coincide	NOUN
ejpam-6214	56	15	completely	completely	ADV
ejpam-6214	56	16	:	:	PUNCT
ejpam-6214	56	17	irreducibility	irreducibility	NOUN
ejpam-6214	56	18	over	over	ADP
ejpam-6214	56	19	z	z	PROPN
ejpam-6214	56	20	is	be	AUX
ejpam-6214	56	21	equivalent	equivalent	ADJ
ejpam-6214	56	22	to	to	ADP
ejpam-6214	56	23	irreducibility	irreducibility	NOUN
ejpam-6214	56	24	over	over	ADP
ejpam-6214	56	25	q.	q.	PROPN
ejpam-6214	56	26	in	in	ADP
ejpam-6214	56	27	the	the	DET
ejpam-6214	56	28	non	non	ADJ
ejpam-6214	56	29	-	-	ADJ
ejpam-6214	56	30	primitive	primitive	ADJ
ejpam-6214	56	31	case	case	NOUN
ejpam-6214	56	32	,	,	PUNCT
ejpam-6214	56	33	only	only	ADV
ejpam-6214	56	34	one	one	NUM
ejpam-6214	56	35	direction	direction	NOUN
ejpam-6214	56	36	holds	hold	VERB
ejpam-6214	56	37	:	:	PUNCT
ejpam-6214	56	38	while	while	SCONJ
ejpam-6214	56	39	irreducibility	irreducibility	NOUN
ejpam-6214	56	40	over	over	ADP
ejpam-6214	56	41	z	z	NOUN
ejpam-6214	56	42	still	still	ADV
ejpam-6214	56	43	implies	imply	VERB
ejpam-6214	56	44	irreducibility	irreducibility	NOUN
ejpam-6214	56	45	over	over	ADP
ejpam-6214	56	46	q	q	NOUN
ejpam-6214	56	47	,	,	PUNCT
ejpam-6214	56	48	the	the	DET
ejpam-6214	56	49	converse	converse	NOUN
ejpam-6214	56	50	fails	fail	VERB
ejpam-6214	56	51	.	.	PUNCT
ejpam-6214	57	1	a	a	DET
ejpam-6214	57	2	classic	classic	ADJ
ejpam-6214	57	3	example	example	NOUN
ejpam-6214	57	4	is	be	AUX
ejpam-6214	57	5	the	the	DET
ejpam-6214	57	6	polynomial	polynomial	ADJ
ejpam-6214	57	7	2x	2x	NUM
ejpam-6214	57	8	,	,	PUNCT
ejpam-6214	57	9	which	which	PRON
ejpam-6214	57	10	is	be	AUX
ejpam-6214	57	11	irreducible	irreducible	ADJ
ejpam-6214	57	12	over	over	ADP
ejpam-6214	57	13	q	q	NOUN
ejpam-6214	57	14	but	but	CCONJ
ejpam-6214	57	15	reducible	reducible	ADJ
ejpam-6214	57	16	in	in	ADP
ejpam-6214	57	17	z[x	z[x	NOUN
ejpam-6214	57	18	]	]	PUNCT
ejpam-6214	57	19	as	as	SCONJ
ejpam-6214	57	20	it	it	PRON
ejpam-6214	57	21	factors	factor	VERB
ejpam-6214	57	22	into	into	ADP
ejpam-6214	57	23	2	2	NUM
ejpam-6214	57	24	·	·	SYM
ejpam-6214	57	25	x.	x.	NOUN
ejpam-6214	57	26	a.	a.	PROPN
ejpam-6214	57	27	chandoul	chandoul	PROPN
ejpam-6214	57	28	,	,	PUNCT
ejpam-6214	57	29	s.	s.	PROPN
ejpam-6214	57	30	mansour	mansour	PROPN
ejpam-6214	57	31	/	/	SYM
ejpam-6214	57	32	eur	eur	PROPN
ejpam-6214	57	33	.	.	PUNCT
ejpam-6214	58	1	j.	j.	PROPN
ejpam-6214	58	2	pure	pure	PROPN
ejpam-6214	58	3	appl	appl	PROPN
ejpam-6214	58	4	.	.	PROPN
ejpam-6214	58	5	math	math	PROPN
ejpam-6214	58	6	,	,	PUNCT
ejpam-6214	58	7	18	18	NUM
ejpam-6214	58	8	(	(	PUNCT
ejpam-6214	58	9	3	3	NUM
ejpam-6214	58	10	)	)	PUNCT
ejpam-6214	58	11	(	(	PUNCT
ejpam-6214	58	12	2025	2025	NUM
ejpam-6214	58	13	)	)	PUNCT
ejpam-6214	58	14	,	,	PUNCT
ejpam-6214	58	15	6214	6214	NUM
ejpam-6214	58	16	4	4	NUM
ejpam-6214	58	17	of	of	ADP
ejpam-6214	58	18	8	8	NUM
ejpam-6214	58	19	practical	practical	ADJ
ejpam-6214	58	20	test	test	NOUN
ejpam-6214	58	21	to	to	PART
ejpam-6214	58	22	check	check	VERB
ejpam-6214	58	23	q	q	NOUN
ejpam-6214	58	24	-	-	PUNCT
ejpam-6214	58	25	irreducibility	irreducibility	NOUN
ejpam-6214	58	26	of	of	ADP
ejpam-6214	58	27	a	a	DET
ejpam-6214	58	28	polynomial	polynomial	ADJ
ejpam-6214	58	29	f	f	PROPN
ejpam-6214	58	30	∈	∈	PROPN
ejpam-6214	58	31	z[x	z[x	PROPN
ejpam-6214	58	32	]	]	X
ejpam-6214	58	33	,	,	PUNCT
ejpam-6214	58	34	one	one	PRON
ejpam-6214	58	35	should	should	AUX
ejpam-6214	58	36	first	first	ADV
ejpam-6214	58	37	factor	factor	VERB
ejpam-6214	58	38	out	out	ADP
ejpam-6214	58	39	the	the	DET
ejpam-6214	58	40	content	content	NOUN
ejpam-6214	58	41	c(f	c(f	PROPN
ejpam-6214	58	42	)	)	PUNCT
ejpam-6214	58	43	=	=	SYM
ejpam-6214	58	44	gcd(coefficients	gcd(coefficient	NOUN
ejpam-6214	58	45	)	)	PUNCT
ejpam-6214	58	46	,	,	PUNCT
ejpam-6214	58	47	then	then	ADV
ejpam-6214	58	48	apply	apply	VERB
ejpam-6214	58	49	gauss	gauss	PROPN
ejpam-6214	58	50	’s	’s	PART
ejpam-6214	58	51	lemma	lemma	PROPN
ejpam-6214	58	52	to	to	ADP
ejpam-6214	58	53	the	the	DET
ejpam-6214	58	54	primitive	primitive	ADJ
ejpam-6214	58	55	part	part	NOUN
ejpam-6214	58	56	f̃	f̃	PROPN
ejpam-6214	58	57	=	=	SYM
ejpam-6214	58	58	f	f	X
ejpam-6214	58	59	/	/	SYM
ejpam-6214	58	60	c(f	c(f	PROPN
ejpam-6214	58	61	)	)	PUNCT
ejpam-6214	58	62	.	.	PUNCT
ejpam-6214	59	1	the	the	DET
ejpam-6214	59	2	irreducibility	irreducibility	NOUN
ejpam-6214	59	3	of	of	ADP
ejpam-6214	59	4	f̃	f̃	PROPN
ejpam-6214	59	5	over	over	ADP
ejpam-6214	59	6	z	z	PROPN
ejpam-6214	59	7	can	can	AUX
ejpam-6214	59	8	then	then	ADV
ejpam-6214	59	9	be	be	AUX
ejpam-6214	59	10	tested	test	VERB
ejpam-6214	59	11	using	use	VERB
ejpam-6214	59	12	various	various	ADJ
ejpam-6214	59	13	methods	method	NOUN
ejpam-6214	59	14	including	include	VERB
ejpam-6214	59	15	modular	modular	ADJ
ejpam-6214	59	16	reduction	reduction	NOUN
ejpam-6214	59	17	tests	test	NOUN
ejpam-6214	59	18	(	(	PUNCT
ejpam-6214	59	19	mod	mod	PROPN
ejpam-6214	59	20	p	p	X
ejpam-6214	59	21	)	)	PUNCT
ejpam-6214	59	22	,	,	PUNCT
ejpam-6214	59	23	eisenstein	eisenstein	PROPN
ejpam-6214	59	24	’s	’s	PART
ejpam-6214	59	25	criterion	criterion	NOUN
ejpam-6214	59	26	,	,	PUNCT
ejpam-6214	59	27	and	and	CCONJ
ejpam-6214	59	28	analysis	analysis	NOUN
ejpam-6214	59	29	of	of	ADP
ejpam-6214	59	30	the	the	DET
ejpam-6214	59	31	polynomial	polynomial	ADJ
ejpam-6214	59	32	’s	’s	PART
ejpam-6214	59	33	degree	degree	NOUN
ejpam-6214	59	34	.	.	PUNCT
ejpam-6214	60	1	2.3	2.3	NUM
ejpam-6214	60	2	.	.	PUNCT
ejpam-6214	61	1	resultant	resultant	VERB
ejpam-6214	61	2	and	and	CCONJ
ejpam-6214	61	3	relative	relative	ADJ
ejpam-6214	61	4	primality	primality	NOUN
ejpam-6214	61	5	let	let	VERB
ejpam-6214	61	6	p(x	p(x	PROPN
ejpam-6214	61	7	)	)	PUNCT
ejpam-6214	61	8	,	,	PUNCT
ejpam-6214	61	9	q(x	q(x	PROPN
ejpam-6214	61	10	)	)	PUNCT
ejpam-6214	61	11	∈	∈	PROPN
ejpam-6214	61	12	z[x	z[x	NOUN
ejpam-6214	61	13	]	]	PUNCT
ejpam-6214	61	14	be	be	VERB
ejpam-6214	61	15	two	two	NUM
ejpam-6214	61	16	integer	integer	NOUN
ejpam-6214	61	17	polynomials	polynomial	NOUN
ejpam-6214	61	18	with	with	ADP
ejpam-6214	61	19	deg(p	deg(p	PROPN
ejpam-6214	61	20	)	)	PUNCT
ejpam-6214	62	1	=	=	SYM
ejpam-6214	62	2	d1	d1	NOUN
ejpam-6214	62	3	and	and	CCONJ
ejpam-6214	62	4	deg(q	deg(q	NOUN
ejpam-6214	62	5	)	)	PUNCT
ejpam-6214	62	6	=	=	SYM
ejpam-6214	62	7	d2	d2	PROPN
ejpam-6214	62	8	.	.	PUNCT
ejpam-6214	63	1	the	the	DET
ejpam-6214	63	2	resultant	resultant	NOUN
ejpam-6214	63	3	of	of	ADP
ejpam-6214	63	4	p	p	PROPN
ejpam-6214	63	5	and	and	CCONJ
ejpam-6214	63	6	q	q	NOUN
ejpam-6214	63	7	,	,	PUNCT
ejpam-6214	63	8	written	write	VERB
ejpam-6214	63	9	as	as	ADP
ejpam-6214	63	10	r(p	r(p	PROPN
ejpam-6214	63	11	,	,	PUNCT
ejpam-6214	63	12	q	q	NOUN
ejpam-6214	63	13	)	)	PUNCT
ejpam-6214	63	14	,	,	PUNCT
ejpam-6214	63	15	is	be	AUX
ejpam-6214	63	16	an	an	DET
ejpam-6214	63	17	integer	integer	NOUN
ejpam-6214	63	18	polynomial	polynomial	ADJ
ejpam-6214	63	19	expression	expression	NOUN
ejpam-6214	63	20	in	in	ADP
ejpam-6214	63	21	their	their	PRON
ejpam-6214	63	22	coefficients	coefficient	NOUN
ejpam-6214	63	23	that	that	PRON
ejpam-6214	63	24	equals	equal	VERB
ejpam-6214	63	25	zero	zero	NUM
ejpam-6214	63	26	precisely	precisely	ADV
ejpam-6214	63	27	when	when	SCONJ
ejpam-6214	63	28	p	p	NOUN
ejpam-6214	63	29	and	and	CCONJ
ejpam-6214	63	30	q	q	NOUN
ejpam-6214	63	31	possess	possess	VERB
ejpam-6214	63	32	a	a	DET
ejpam-6214	63	33	common	common	ADJ
ejpam-6214	63	34	zero	zero	NUM
ejpam-6214	63	35	.	.	PUNCT
ejpam-6214	64	1	a	a	DET
ejpam-6214	64	2	key	key	ADJ
ejpam-6214	64	3	consequence	consequence	NOUN
ejpam-6214	64	4	is	be	AUX
ejpam-6214	64	5	that	that	SCONJ
ejpam-6214	64	6	when	when	SCONJ
ejpam-6214	64	7	r(p	r(p	NOUN
ejpam-6214	64	8	,	,	PUNCT
ejpam-6214	64	9	q	q	NOUN
ejpam-6214	64	10	)	)	PUNCT
ejpam-6214	64	11	̸=	̸=	PROPN
ejpam-6214	64	12	0	0	NUM
ejpam-6214	64	13	,	,	PUNCT
ejpam-6214	64	14	the	the	DET
ejpam-6214	64	15	polynomials	polynomial	NOUN
ejpam-6214	64	16	p	p	NOUN
ejpam-6214	64	17	and	and	CCONJ
ejpam-6214	64	18	q	q	NOUN
ejpam-6214	64	19	must	must	AUX
ejpam-6214	64	20	be	be	AUX
ejpam-6214	64	21	coprime	coprime	ADJ
ejpam-6214	64	22	.	.	PUNCT
ejpam-6214	65	1	2.4	2.4	NUM
ejpam-6214	65	2	.	.	PUNCT
ejpam-6214	66	1	eisenstein	eisenstein	PROPN
ejpam-6214	66	2	’s	’s	PART
ejpam-6214	66	3	criterion	criterion	NOUN
ejpam-6214	66	4	a	a	DET
ejpam-6214	66	5	polynomial	polynomial	ADJ
ejpam-6214	66	6	f(x	f(x	PROPN
ejpam-6214	66	7	)	)	PUNCT
ejpam-6214	67	1	=	=	SYM
ejpam-6214	67	2	anx	anx	ADJ
ejpam-6214	67	3	n	n	PROPN
ejpam-6214	67	4	+	+	CCONJ
ejpam-6214	67	5	·	·	PUNCT
ejpam-6214	67	6	·	·	PUNCT
ejpam-6214	67	7	·	·	PUNCT
ejpam-6214	68	1	+	+	CCONJ
ejpam-6214	68	2	a0	a0	PROPN
ejpam-6214	68	3	∈	∈	PROPN
ejpam-6214	68	4	z[x	z[x	NOUN
ejpam-6214	68	5	]	]	PUNCT
ejpam-6214	68	6	is	be	AUX
ejpam-6214	68	7	irreducible	irreducible	ADJ
ejpam-6214	68	8	over	over	ADP
ejpam-6214	68	9	q	q	PROPN
ejpam-6214	68	10	if	if	SCONJ
ejpam-6214	68	11	there	there	PRON
ejpam-6214	68	12	exists	exist	VERB
ejpam-6214	68	13	a	a	DET
ejpam-6214	68	14	prime	prime	NOUN
ejpam-6214	68	15	p	p	NOUN
ejpam-6214	68	16	such	such	ADJ
ejpam-6214	68	17	that	that	PRON
ejpam-6214	68	18	:	:	PUNCT
ejpam-6214	68	19	(	(	PUNCT
ejpam-6214	68	20	i	i	NOUN
ejpam-6214	68	21	)	)	PUNCT
ejpam-6214	68	22	p	p	NOUN
ejpam-6214	68	23	divides	divide	NOUN
ejpam-6214	68	24	ai	ai	VERB
ejpam-6214	68	25	for	for	ADP
ejpam-6214	68	26	all	all	PRON
ejpam-6214	68	27	i	i	PRON
ejpam-6214	68	28	=	=	NOUN
ejpam-6214	68	29	0	0	NUM
ejpam-6214	68	30	,	,	PUNCT
ejpam-6214	68	31	.	.	PUNCT
ejpam-6214	68	32	.	.	PUNCT
ejpam-6214	69	1	.	.	PUNCT
ejpam-6214	70	1	,	,	PUNCT
ejpam-6214	70	2	n−	n−	NOUN
ejpam-6214	70	3	1	1	NUM
ejpam-6214	70	4	,	,	PUNCT
ejpam-6214	70	5	(	(	PUNCT
ejpam-6214	70	6	ii	ii	NOUN
ejpam-6214	70	7	)	)	PUNCT
ejpam-6214	70	8	p	p	NOUN
ejpam-6214	70	9	does	do	AUX
ejpam-6214	70	10	not	not	PART
ejpam-6214	70	11	divide	divide	VERB
ejpam-6214	70	12	an	an	DET
ejpam-6214	70	13	,	,	PUNCT
ejpam-6214	70	14	(	(	PUNCT
ejpam-6214	70	15	iii	iii	NOUN
ejpam-6214	70	16	)	)	PUNCT
ejpam-6214	70	17	p2	p2	PROPN
ejpam-6214	70	18	does	do	AUX
ejpam-6214	70	19	not	not	PART
ejpam-6214	70	20	divide	divide	VERB
ejpam-6214	70	21	a0	a0	NOUN
ejpam-6214	70	22	.	.	PUNCT
ejpam-6214	71	1	this	this	DET
ejpam-6214	71	2	criterion	criterion	NOUN
ejpam-6214	71	3	is	be	AUX
ejpam-6214	71	4	a	a	DET
ejpam-6214	71	5	powerful	powerful	ADJ
ejpam-6214	71	6	tool	tool	NOUN
ejpam-6214	71	7	for	for	ADP
ejpam-6214	71	8	proving	prove	VERB
ejpam-6214	71	9	irreducibility	irreducibility	NOUN
ejpam-6214	71	10	,	,	PUNCT
ejpam-6214	71	11	but	but	CCONJ
ejpam-6214	71	12	it	it	PRON
ejpam-6214	71	13	requires	require	VERB
ejpam-6214	71	14	specific	specific	ADJ
ejpam-6214	71	15	divisibility	divisibility	NOUN
ejpam-6214	71	16	conditions	condition	NOUN
ejpam-6214	71	17	that	that	PRON
ejpam-6214	71	18	are	be	AUX
ejpam-6214	71	19	not	not	PART
ejpam-6214	71	20	always	always	ADV
ejpam-6214	71	21	satisfied	satisfied	ADJ
ejpam-6214	71	22	.	.	PUNCT
ejpam-6214	72	1	2.5	2.5	NUM
ejpam-6214	72	2	.	.	PUNCT
ejpam-6214	72	3	reduction	reduction	NOUN
ejpam-6214	72	4	modulo	modulo	NOUN
ejpam-6214	72	5	p	p	NOUN
ejpam-6214	72	6	let	let	VERB
ejpam-6214	72	7	f(x	f(x	PROPN
ejpam-6214	72	8	)	)	PUNCT
ejpam-6214	72	9	∈	∈	PROPN
ejpam-6214	72	10	z[x	z[x	NOUN
ejpam-6214	72	11	]	]	X
ejpam-6214	72	12	and	and	CCONJ
ejpam-6214	72	13	p	p	NOUN
ejpam-6214	72	14	be	be	AUX
ejpam-6214	72	15	a	a	DET
ejpam-6214	72	16	prime	prime	NOUN
ejpam-6214	72	17	.	.	PUNCT
ejpam-6214	73	1	the	the	DET
ejpam-6214	73	2	reduction	reduction	NOUN
ejpam-6214	73	3	of	of	ADP
ejpam-6214	73	4	f	f	PROPN
ejpam-6214	73	5	modulo	modulo	PROPN
ejpam-6214	73	6	p	p	X
ejpam-6214	73	7	,	,	PUNCT
ejpam-6214	73	8	denoted	denote	VERB
ejpam-6214	73	9	f(x	f(x	PROPN
ejpam-6214	73	10	)	)	PUNCT
ejpam-6214	73	11	,	,	PUNCT
ejpam-6214	73	12	is	be	AUX
ejpam-6214	73	13	the	the	DET
ejpam-6214	73	14	polynomial	polynomial	NOUN
ejpam-6214	73	15	obtained	obtain	VERB
ejpam-6214	73	16	by	by	ADP
ejpam-6214	73	17	reducing	reduce	VERB
ejpam-6214	73	18	each	each	DET
ejpam-6214	73	19	coefficient	coefficient	NOUN
ejpam-6214	73	20	of	of	ADP
ejpam-6214	73	21	f	f	PROPN
ejpam-6214	73	22	modulo	modulo	PROPN
ejpam-6214	73	23	p.	p.	NOUN
ejpam-6214	73	24	if	if	SCONJ
ejpam-6214	73	25	f(x	f(x	PROPN
ejpam-6214	73	26	)	)	PUNCT
ejpam-6214	73	27	is	be	AUX
ejpam-6214	73	28	irreducible	irreducible	ADJ
ejpam-6214	73	29	over	over	ADP
ejpam-6214	73	30	fp	fp	PROPN
ejpam-6214	73	31	(	(	PUNCT
ejpam-6214	73	32	the	the	DET
ejpam-6214	73	33	finite	finite	ADJ
ejpam-6214	73	34	field	field	NOUN
ejpam-6214	73	35	with	with	ADP
ejpam-6214	73	36	p	p	NOUN
ejpam-6214	73	37	elements	element	NOUN
ejpam-6214	73	38	)	)	PUNCT
ejpam-6214	73	39	,	,	PUNCT
ejpam-6214	73	40	then	then	ADV
ejpam-6214	73	41	f(x	f(x	PROPN
ejpam-6214	73	42	)	)	PUNCT
ejpam-6214	73	43	is	be	AUX
ejpam-6214	73	44	irreducible	irreducible	ADJ
ejpam-6214	73	45	over	over	ADP
ejpam-6214	73	46	q.	q.	PROPN
ejpam-6214	73	47	2.6	2.6	NUM
ejpam-6214	73	48	.	.	PUNCT
ejpam-6214	74	1	greatest	great	ADJ
ejpam-6214	74	2	common	common	ADJ
ejpam-6214	74	3	divisor	divisor	NOUN
ejpam-6214	74	4	for	for	ADP
ejpam-6214	74	5	integers	integer	NOUN
ejpam-6214	74	6	a	a	PRON
ejpam-6214	74	7	and	and	CCONJ
ejpam-6214	74	8	b	b	NOUN
ejpam-6214	74	9	,	,	PUNCT
ejpam-6214	74	10	the	the	DET
ejpam-6214	74	11	greatest	great	ADJ
ejpam-6214	74	12	common	common	ADJ
ejpam-6214	74	13	divisor	divisor	NOUN
ejpam-6214	74	14	gcd(a	gcd(a	PROPN
ejpam-6214	74	15	,	,	PUNCT
ejpam-6214	74	16	b	b	NOUN
ejpam-6214	74	17	)	)	PUNCT
ejpam-6214	74	18	is	be	AUX
ejpam-6214	74	19	the	the	DET
ejpam-6214	74	20	largest	large	ADJ
ejpam-6214	74	21	integer	integer	NOUN
ejpam-6214	74	22	that	that	PRON
ejpam-6214	74	23	divides	divide	VERB
ejpam-6214	74	24	both	both	DET
ejpam-6214	74	25	a	a	PRON
ejpam-6214	74	26	and	and	CCONJ
ejpam-6214	74	27	b.	b.	PROPN
ejpam-6214	74	28	in	in	ADP
ejpam-6214	74	29	the	the	DET
ejpam-6214	74	30	context	context	NOUN
ejpam-6214	74	31	of	of	ADP
ejpam-6214	74	32	polynomials	polynomial	NOUN
ejpam-6214	74	33	,	,	PUNCT
ejpam-6214	74	34	gcd(k	gcd(k	PROPN
ejpam-6214	74	35	,	,	PUNCT
ejpam-6214	74	36	deg	deg	NOUN
ejpam-6214	74	37	g	g	NOUN
ejpam-6214	74	38	)	)	PUNCT
ejpam-6214	74	39	plays	play	VERB
ejpam-6214	74	40	a	a	DET
ejpam-6214	74	41	key	key	ADJ
ejpam-6214	74	42	role	role	NOUN
ejpam-6214	74	43	in	in	ADP
ejpam-6214	74	44	determining	determine	VERB
ejpam-6214	74	45	the	the	DET
ejpam-6214	74	46	degrees	degree	NOUN
ejpam-6214	74	47	of	of	ADP
ejpam-6214	74	48	factors	factor	NOUN
ejpam-6214	74	49	of	of	ADP
ejpam-6214	74	50	f(x	f(x	PROPN
ejpam-6214	74	51	)	)	PUNCT
ejpam-6214	75	1	+	+	NUM
ejpam-6214	75	2	pkg(x	pkg(x	NOUN
ejpam-6214	75	3	)	)	PUNCT
ejpam-6214	75	4	.	.	PUNCT
ejpam-6214	76	1	2.7	2.7	NUM
ejpam-6214	76	2	.	.	PUNCT
ejpam-6214	76	3	notations	notation	NOUN
ejpam-6214	76	4	and	and	CCONJ
ejpam-6214	76	5	conventions	convention	NOUN
ejpam-6214	76	6	throughout	throughout	ADP
ejpam-6214	76	7	this	this	DET
ejpam-6214	76	8	paper	paper	NOUN
ejpam-6214	76	9	,	,	PUNCT
ejpam-6214	76	10	deg	deg	PROPN
ejpam-6214	76	11	f	f	PROPN
ejpam-6214	76	12	denotes	denote	VERB
ejpam-6214	76	13	the	the	DET
ejpam-6214	76	14	degree	degree	NOUN
ejpam-6214	76	15	of	of	ADP
ejpam-6214	76	16	a	a	DET
ejpam-6214	76	17	polynomial	polynomial	ADJ
ejpam-6214	76	18	f(x	f(x	PROPN
ejpam-6214	76	19	)	)	PUNCT
ejpam-6214	76	20	,	,	PUNCT
ejpam-6214	76	21	and	and	CCONJ
ejpam-6214	76	22	q	q	PROPN
ejpam-6214	76	23	denotes	denote	VERB
ejpam-6214	76	24	the	the	DET
ejpam-6214	76	25	field	field	NOUN
ejpam-6214	76	26	of	of	ADP
ejpam-6214	76	27	rational	rational	ADJ
ejpam-6214	76	28	numbers	number	NOUN
ejpam-6214	76	29	.	.	PUNCT
ejpam-6214	77	1	all	all	DET
ejpam-6214	77	2	polynomials	polynomial	NOUN
ejpam-6214	77	3	are	be	AUX
ejpam-6214	77	4	assumed	assume	VERB
ejpam-6214	77	5	to	to	PART
ejpam-6214	77	6	have	have	VERB
ejpam-6214	77	7	integer	integer	NOUN
ejpam-6214	77	8	coefficients	coefficient	NOUN
ejpam-6214	77	9	unless	unless	SCONJ
ejpam-6214	77	10	stated	state	VERB
ejpam-6214	77	11	otherwise	otherwise	ADV
ejpam-6214	77	12	.	.	PUNCT
ejpam-6214	78	1	a.	a.	PROPN
ejpam-6214	78	2	chandoul	chandoul	PROPN
ejpam-6214	78	3	,	,	PUNCT
ejpam-6214	78	4	s.	s.	PROPN
ejpam-6214	78	5	mansour	mansour	PROPN
ejpam-6214	78	6	/	/	SYM
ejpam-6214	78	7	eur	eur	PROPN
ejpam-6214	78	8	.	.	PUNCT
ejpam-6214	79	1	j.	j.	PROPN
ejpam-6214	79	2	pure	pure	PROPN
ejpam-6214	79	3	appl	appl	PROPN
ejpam-6214	79	4	.	.	PROPN
ejpam-6214	79	5	math	math	PROPN
ejpam-6214	79	6	,	,	PUNCT
ejpam-6214	79	7	18	18	NUM
ejpam-6214	79	8	(	(	PUNCT
ejpam-6214	79	9	3	3	NUM
ejpam-6214	79	10	)	)	PUNCT
ejpam-6214	79	11	(	(	PUNCT
ejpam-6214	79	12	2025	2025	NUM
ejpam-6214	79	13	)	)	PUNCT
ejpam-6214	79	14	,	,	PUNCT
ejpam-6214	79	15	6214	6214	NUM
ejpam-6214	79	16	5	5	NUM
ejpam-6214	79	17	of	of	ADP
ejpam-6214	79	18	8	8	NUM
ejpam-6214	79	19	3	3	NUM
ejpam-6214	79	20	.	.	PUNCT
ejpam-6214	79	21	main	main	ADJ
ejpam-6214	79	22	theorem	theorem	NOUN
ejpam-6214	79	23	theorem	theorem	NOUN
ejpam-6214	79	24	2	2	NUM
ejpam-6214	79	25	(	(	PUNCT
ejpam-6214	79	26	irreducibility	irreducibility	NOUN
ejpam-6214	79	27	criterion	criterion	NOUN
ejpam-6214	79	28	for	for	ADP
ejpam-6214	79	29	prime	prime	ADJ
ejpam-6214	79	30	power	power	NOUN
ejpam-6214	79	31	shifts	shift	NOUN
ejpam-6214	79	32	)	)	PUNCT
ejpam-6214	79	33	.	.	PUNCT
ejpam-6214	80	1	let	let	VERB
ejpam-6214	80	2	f(x	f(x	PROPN
ejpam-6214	80	3	)	)	PUNCT
ejpam-6214	80	4	,	,	PUNCT
ejpam-6214	80	5	g(x	g(x	NOUN
ejpam-6214	80	6	)	)	PUNCT
ejpam-6214	80	7	∈	∈	PROPN
ejpam-6214	80	8	z[x	z[x	NOUN
ejpam-6214	80	9	]	]	PUNCT
ejpam-6214	80	10	be	be	AUX
ejpam-6214	80	11	polynomials	polynomial	NOUN
ejpam-6214	80	12	such	such	ADJ
ejpam-6214	80	13	that	that	PRON
ejpam-6214	80	14	:	:	PUNCT
ejpam-6214	80	15	(	(	PUNCT
ejpam-6214	80	16	i	i	NOUN
ejpam-6214	81	1	)	)	PUNCT
ejpam-6214	81	2	deg	deg	PROPN
ejpam-6214	82	1	f	f	X
ejpam-6214	82	2	<	<	X
ejpam-6214	82	3	deg	deg	PROPN
ejpam-6214	82	4	g	g	PROPN
ejpam-6214	82	5	,	,	PUNCT
ejpam-6214	82	6	(	(	PUNCT
ejpam-6214	82	7	ii	ii	NOUN
ejpam-6214	82	8	)	)	PUNCT
ejpam-6214	82	9	f	f	PROPN
ejpam-6214	82	10	and	and	CCONJ
ejpam-6214	82	11	g	g	PROPN
ejpam-6214	82	12	are	be	AUX
ejpam-6214	82	13	relatively	relatively	ADV
ejpam-6214	82	14	prime	prime	ADJ
ejpam-6214	82	15	,	,	PUNCT
ejpam-6214	82	16	(	(	PUNCT
ejpam-6214	82	17	iii	iii	X
ejpam-6214	82	18	)	)	PUNCT
ejpam-6214	82	19	k	k	NOUN
ejpam-6214	82	20	is	be	AUX
ejpam-6214	82	21	a	a	DET
ejpam-6214	82	22	positive	positive	ADJ
ejpam-6214	82	23	integer	integer	NOUN
ejpam-6214	82	24	with	with	ADP
ejpam-6214	82	25	gcd(k	gcd(k	PROPN
ejpam-6214	82	26	,	,	PUNCT
ejpam-6214	82	27	deg	deg	VERB
ejpam-6214	82	28	g	g	NOUN
ejpam-6214	82	29	)	)	PUNCT
ejpam-6214	82	30	=	=	SYM
ejpam-6214	82	31	1	1	NUM
ejpam-6214	82	32	,	,	PUNCT
ejpam-6214	82	33	(	(	PUNCT
ejpam-6214	82	34	iv	iv	X
ejpam-6214	82	35	)	)	PUNCT
ejpam-6214	82	36	the	the	DET
ejpam-6214	82	37	leading	lead	VERB
ejpam-6214	82	38	coefficient	coefficient	NOUN
ejpam-6214	82	39	of	of	ADP
ejpam-6214	82	40	g	g	PROPN
ejpam-6214	82	41	is	be	AUX
ejpam-6214	82	42	not	not	PART
ejpam-6214	82	43	divisible	divisible	ADJ
ejpam-6214	82	44	by	by	ADP
ejpam-6214	82	45	p.	p.	NOUN
ejpam-6214	82	46	then	then	ADV
ejpam-6214	82	47	,	,	PUNCT
ejpam-6214	82	48	for	for	ADP
ejpam-6214	82	49	all	all	PRON
ejpam-6214	82	50	but	but	ADV
ejpam-6214	82	51	finitely	finitely	ADV
ejpam-6214	82	52	many	many	ADJ
ejpam-6214	82	53	primes	prime	NOUN
ejpam-6214	83	1	p	p	X
ejpam-6214	83	2	,	,	PUNCT
ejpam-6214	83	3	the	the	DET
ejpam-6214	83	4	polynomial	polynomial	ADJ
ejpam-6214	83	5	f(x	f(x	PROPN
ejpam-6214	83	6	)	)	PUNCT
ejpam-6214	84	1	+	+	SYM
ejpam-6214	84	2	pkg(x	pkg(x	NOUN
ejpam-6214	84	3	)	)	PUNCT
ejpam-6214	84	4	is	be	AUX
ejpam-6214	84	5	irreducible	irreducible	ADJ
ejpam-6214	84	6	over	over	ADP
ejpam-6214	84	7	q.	q.	NOUN
ejpam-6214	84	8	proof	proof	NOUN
ejpam-6214	84	9	.	.	PUNCT
ejpam-6214	85	1	the	the	DET
ejpam-6214	85	2	proof	proof	NOUN
ejpam-6214	85	3	follows	follow	VERB
ejpam-6214	85	4	bonciocat	bonciocat	PROPN
ejpam-6214	85	5	’s	’s	PART
ejpam-6214	85	6	approach	approach	NOUN
ejpam-6214	85	7	,	,	PUNCT
ejpam-6214	85	8	combined	combine	VERB
ejpam-6214	85	9	with	with	ADP
ejpam-6214	85	10	eisenstein	eisenstein	PROPN
ejpam-6214	85	11	’s	’s	PART
ejpam-6214	85	12	criterion	criterion	NOUN
ejpam-6214	85	13	and	and	CCONJ
ejpam-6214	85	14	reduction	reduction	NOUN
ejpam-6214	85	15	modulo	modulo	VERB
ejpam-6214	86	1	p	p	X
ejpam-6214	86	2	:	:	PUNCT
ejpam-6214	86	3	•	•	ADP
ejpam-6214	86	4	since	since	SCONJ
ejpam-6214	86	5	f	f	PROPN
ejpam-6214	86	6	and	and	CCONJ
ejpam-6214	86	7	g	g	PROPN
ejpam-6214	86	8	are	be	AUX
ejpam-6214	86	9	relatively	relatively	ADV
ejpam-6214	86	10	prime	prime	ADJ
ejpam-6214	86	11	,	,	PUNCT
ejpam-6214	86	12	res(f	res(f	PROPN
ejpam-6214	86	13	,	,	PUNCT
ejpam-6214	86	14	g	g	NOUN
ejpam-6214	86	15	)	)	PUNCT
ejpam-6214	86	16	̸=	̸=	PROPN
ejpam-6214	86	17	0	0	NUM
ejpam-6214	86	18	,	,	PUNCT
ejpam-6214	86	19	so	so	ADV
ejpam-6214	86	20	only	only	ADV
ejpam-6214	86	21	finitely	finitely	ADV
ejpam-6214	86	22	many	many	ADJ
ejpam-6214	86	23	primes	prime	NOUN
ejpam-6214	86	24	divide	divide	VERB
ejpam-6214	86	25	the	the	DET
ejpam-6214	86	26	resultant	resultant	NOUN
ejpam-6214	86	27	.	.	PUNCT
ejpam-6214	87	1	•	•	NOUN
ejpam-6214	87	2	for	for	ADP
ejpam-6214	87	3	all	all	DET
ejpam-6214	87	4	sufficiently	sufficiently	ADV
ejpam-6214	87	5	large	large	ADJ
ejpam-6214	87	6	primes	prime	NOUN
ejpam-6214	87	7	p	p	NOUN
ejpam-6214	87	8	not	not	PART
ejpam-6214	87	9	dividing	divide	VERB
ejpam-6214	87	10	the	the	DET
ejpam-6214	87	11	resultant	resultant	NOUN
ejpam-6214	87	12	or	or	CCONJ
ejpam-6214	87	13	the	the	DET
ejpam-6214	87	14	leading	leading	ADJ
ejpam-6214	87	15	coefficient	coefficient	NOUN
ejpam-6214	87	16	of	of	ADP
ejpam-6214	87	17	g	g	NOUN
ejpam-6214	87	18	,	,	PUNCT
ejpam-6214	87	19	consider	consider	VERB
ejpam-6214	87	20	f(x	f(x	PROPN
ejpam-6214	87	21	)	)	PUNCT
ejpam-6214	88	1	+	+	SYM
ejpam-6214	88	2	pkg(x	pkg(x	NOUN
ejpam-6214	88	3	)	)	PUNCT
ejpam-6214	88	4	≡	≡	PROPN
ejpam-6214	88	5	f(x	f(x	PROPN
ejpam-6214	88	6	)	)	PUNCT
ejpam-6214	88	7	(	(	PUNCT
ejpam-6214	88	8	mod	mod	PROPN
ejpam-6214	88	9	p	p	X
ejpam-6214	88	10	)	)	PUNCT
ejpam-6214	88	11	.	.	PUNCT
ejpam-6214	89	1	since	since	SCONJ
ejpam-6214	89	2	f	f	PROPN
ejpam-6214	89	3	is	be	AUX
ejpam-6214	89	4	fixed	fix	VERB
ejpam-6214	89	5	and	and	CCONJ
ejpam-6214	89	6	irreducibility	irreducibility	NOUN
ejpam-6214	89	7	over	over	ADP
ejpam-6214	89	8	finite	finite	ADJ
ejpam-6214	89	9	fields	field	NOUN
ejpam-6214	89	10	is	be	AUX
ejpam-6214	89	11	rare	rare	ADJ
ejpam-6214	89	12	to	to	PART
ejpam-6214	89	13	fail	fail	VERB
ejpam-6214	89	14	infinitely	infinitely	ADV
ejpam-6214	89	15	often	often	ADV
ejpam-6214	89	16	,	,	PUNCT
ejpam-6214	89	17	for	for	SCONJ
ejpam-6214	89	18	all	all	PRON
ejpam-6214	89	19	but	but	ADV
ejpam-6214	89	20	finitely	finitely	ADV
ejpam-6214	89	21	many	many	ADJ
ejpam-6214	89	22	p	p	NOUN
ejpam-6214	89	23	,	,	PUNCT
ejpam-6214	89	24	reduction	reduction	NOUN
ejpam-6214	89	25	modulo	modulo	NOUN
ejpam-6214	89	26	p	p	NOUN
ejpam-6214	89	27	does	do	AUX
ejpam-6214	89	28	not	not	PART
ejpam-6214	89	29	trivialize	trivialize	VERB
ejpam-6214	89	30	.	.	PUNCT
ejpam-6214	90	1	•	•	PROPN
ejpam-6214	90	2	eisenstein	eisenstein	PROPN
ejpam-6214	90	3	’s	’s	PART
ejpam-6214	90	4	criterion	criterion	NOUN
ejpam-6214	90	5	applies	apply	VERB
ejpam-6214	90	6	when	when	SCONJ
ejpam-6214	90	7	the	the	DET
ejpam-6214	90	8	leading	lead	VERB
ejpam-6214	90	9	coefficient	coefficient	NOUN
ejpam-6214	90	10	of	of	ADP
ejpam-6214	90	11	f	f	PROPN
ejpam-6214	90	12	+	+	CCONJ
ejpam-6214	90	13	pkg	pkg	NOUN
ejpam-6214	90	14	is	be	AUX
ejpam-6214	90	15	divisible	divisible	ADJ
ejpam-6214	90	16	by	by	ADP
ejpam-6214	90	17	p	p	NOUN
ejpam-6214	90	18	but	but	CCONJ
ejpam-6214	90	19	not	not	PART
ejpam-6214	90	20	p2	p2	NOUN
ejpam-6214	90	21	,	,	PUNCT
ejpam-6214	90	22	and	and	CCONJ
ejpam-6214	90	23	the	the	DET
ejpam-6214	90	24	constant	constant	ADJ
ejpam-6214	90	25	term	term	NOUN
ejpam-6214	90	26	is	be	AUX
ejpam-6214	90	27	not	not	PART
ejpam-6214	90	28	divisible	divisible	ADJ
ejpam-6214	90	29	by	by	ADP
ejpam-6214	90	30	p.	p.	NOUN
ejpam-6214	90	31	this	this	PRON
ejpam-6214	90	32	typically	typically	ADV
ejpam-6214	90	33	holds	hold	VERB
ejpam-6214	90	34	when	when	SCONJ
ejpam-6214	90	35	k	k	PROPN
ejpam-6214	90	36	=	=	SYM
ejpam-6214	90	37	1	1	NUM
ejpam-6214	90	38	and	and	CCONJ
ejpam-6214	90	39	p	p	NOUN
ejpam-6214	90	40	∤	∤	PROPN
ejpam-6214	90	41	f(0	f(0	PROPN
ejpam-6214	90	42	)	)	PUNCT
ejpam-6214	90	43	,	,	PUNCT
ejpam-6214	90	44	ensuring	ensure	VERB
ejpam-6214	90	45	irreducibility	irreducibility	NOUN
ejpam-6214	90	46	.	.	PUNCT
ejpam-6214	91	1	•	•	NUM
ejpam-6214	91	2	the	the	DET
ejpam-6214	91	3	condition	condition	NOUN
ejpam-6214	91	4	gcd(k	gcd(k	PROPN
ejpam-6214	91	5	,	,	PUNCT
ejpam-6214	91	6	deg	deg	VERB
ejpam-6214	91	7	g	g	NOUN
ejpam-6214	91	8	)	)	PUNCT
ejpam-6214	91	9	=	=	SYM
ejpam-6214	91	10	1	1	NUM
ejpam-6214	91	11	is	be	AUX
ejpam-6214	91	12	crucial	crucial	ADJ
ejpam-6214	91	13	to	to	PART
ejpam-6214	91	14	prevent	prevent	VERB
ejpam-6214	91	15	factor	factor	NOUN
ejpam-6214	91	16	degrees	degree	NOUN
ejpam-6214	91	17	from	from	ADP
ejpam-6214	91	18	contradicting	contradict	VERB
ejpam-6214	91	19	irreducibility	irreducibility	NOUN
ejpam-6214	91	20	.	.	PUNCT
ejpam-6214	92	1	remark	remark	PROPN
ejpam-6214	92	2	1	1	NUM
ejpam-6214	92	3	.	.	PUNCT
ejpam-6214	93	1	our	our	PRON
ejpam-6214	93	2	theorem	theorem	NOUN
ejpam-6214	93	3	generalizes	generalize	VERB
ejpam-6214	93	4	previous	previous	ADJ
ejpam-6214	93	5	results	result	NOUN
ejpam-6214	93	6	by	by	ADP
ejpam-6214	93	7	removing	remove	VERB
ejpam-6214	93	8	the	the	DET
ejpam-6214	93	9	requirement	requirement	NOUN
ejpam-6214	93	10	that	that	SCONJ
ejpam-6214	93	11	f	f	PROPN
ejpam-6214	93	12	and	and	CCONJ
ejpam-6214	93	13	g	g	PROPN
ejpam-6214	93	14	be	be	VERB
ejpam-6214	93	15	relatively	relatively	ADV
ejpam-6214	93	16	prime	prime	ADJ
ejpam-6214	93	17	and	and	CCONJ
ejpam-6214	93	18	the	the	DET
ejpam-6214	93	19	condition	condition	NOUN
ejpam-6214	93	20	gcd(k	gcd(k	PROPN
ejpam-6214	93	21	,	,	PUNCT
ejpam-6214	93	22	deg	deg	NOUN
ejpam-6214	93	23	g	g	NOUN
ejpam-6214	93	24	)	)	PUNCT
ejpam-6214	93	25	=	=	SYM
ejpam-6214	94	1	1	1	X
ejpam-6214	94	2	.	.	PUNCT
ejpam-6214	94	3	this	this	PRON
ejpam-6214	94	4	significantly	significantly	ADV
ejpam-6214	94	5	broadens	broaden	VERB
ejpam-6214	94	6	the	the	DET
ejpam-6214	94	7	applicability	applicability	NOUN
ejpam-6214	94	8	of	of	ADP
ejpam-6214	94	9	the	the	DET
ejpam-6214	94	10	result	result	NOUN
ejpam-6214	94	11	.	.	PUNCT
ejpam-6214	95	1	a.	a.	PROPN
ejpam-6214	95	2	chandoul	chandoul	PROPN
ejpam-6214	95	3	,	,	PUNCT
ejpam-6214	95	4	s.	s.	PROPN
ejpam-6214	95	5	mansour	mansour	PROPN
ejpam-6214	95	6	/	/	SYM
ejpam-6214	95	7	eur	eur	PROPN
ejpam-6214	95	8	.	.	PUNCT
ejpam-6214	96	1	j.	j.	PROPN
ejpam-6214	96	2	pure	pure	PROPN
ejpam-6214	96	3	appl	appl	PROPN
ejpam-6214	96	4	.	.	PROPN
ejpam-6214	96	5	math	math	PROPN
ejpam-6214	96	6	,	,	PUNCT
ejpam-6214	96	7	18	18	NUM
ejpam-6214	96	8	(	(	PUNCT
ejpam-6214	96	9	3	3	NUM
ejpam-6214	96	10	)	)	PUNCT
ejpam-6214	96	11	(	(	PUNCT
ejpam-6214	96	12	2025	2025	NUM
ejpam-6214	96	13	)	)	PUNCT
ejpam-6214	96	14	,	,	PUNCT
ejpam-6214	96	15	6214	6214	NUM
ejpam-6214	96	16	6	6	NUM
ejpam-6214	96	17	of	of	ADP
ejpam-6214	96	18	8	8	NUM
ejpam-6214	96	19	4	4	NUM
ejpam-6214	96	20	.	.	PUNCT
ejpam-6214	97	1	examples	example	NOUN
ejpam-6214	97	2	example	example	VERB
ejpam-6214	97	3	1	1	NUM
ejpam-6214	97	4	:	:	PUNCT
ejpam-6214	97	5	relatively	relatively	ADV
ejpam-6214	97	6	prime	prime	ADJ
ejpam-6214	97	7	polynomials	polynomial	NOUN
ejpam-6214	97	8	with	with	ADP
ejpam-6214	97	9	gcd(k	gcd(k	PROPN
ejpam-6214	97	10	,	,	PUNCT
ejpam-6214	97	11	deg	deg	NOUN
ejpam-6214	97	12	g	g	NOUN
ejpam-6214	97	13	)	)	PUNCT
ejpam-6214	97	14	=	=	SYM
ejpam-6214	98	1	1	1	NUM
ejpam-6214	98	2	consider	consider	VERB
ejpam-6214	98	3	f(x	f(x	NOUN
ejpam-6214	98	4	)	)	PUNCT
ejpam-6214	99	1	=	=	SYM
ejpam-6214	100	1	x2	x2	INTJ
ejpam-6214	101	1	+	+	NOUN
ejpam-6214	101	2	x	x	X
ejpam-6214	101	3	+	+	ADJ
ejpam-6214	101	4	1	1	NUM
ejpam-6214	101	5	,	,	PUNCT
ejpam-6214	101	6	g(x	g(x	NOUN
ejpam-6214	101	7	)	)	PUNCT
ejpam-6214	101	8	=	=	PUNCT
ejpam-6214	102	1	x3	x3	VERB
ejpam-6214	103	1	+	+	NOUN
ejpam-6214	103	2	x	x	X
ejpam-6214	103	3	+	+	NOUN
ejpam-6214	103	4	1	1	X
ejpam-6214	103	5	.	.	PUNCT
ejpam-6214	104	1	these	these	DET
ejpam-6214	104	2	polynomials	polynomial	NOUN
ejpam-6214	104	3	are	be	AUX
ejpam-6214	104	4	relatively	relatively	ADV
ejpam-6214	104	5	prime	prime	ADJ
ejpam-6214	104	6	(	(	PUNCT
ejpam-6214	104	7	their	their	PRON
ejpam-6214	104	8	resultant	resultant	NOUN
ejpam-6214	104	9	is	be	AUX
ejpam-6214	104	10	nonzero	nonzero	NOUN
ejpam-6214	104	11	)	)	PUNCT
ejpam-6214	104	12	.	.	PUNCT
ejpam-6214	105	1	for	for	ADP
ejpam-6214	105	2	k	k	PROPN
ejpam-6214	105	3	=	=	SYM
ejpam-6214	105	4	1	1	NUM
ejpam-6214	105	5	,	,	PUNCT
ejpam-6214	105	6	gcd(1	gcd(1	ADJ
ejpam-6214	105	7	,	,	PUNCT
ejpam-6214	105	8	3	3	X
ejpam-6214	105	9	)	)	PUNCT
ejpam-6214	105	10	=	=	SYM
ejpam-6214	105	11	1	1	NUM
ejpam-6214	105	12	,	,	PUNCT
ejpam-6214	105	13	so	so	ADV
ejpam-6214	105	14	by	by	ADP
ejpam-6214	105	15	theorem	theorem	ADJ
ejpam-6214	105	16	3.1	3.1	NUM
ejpam-6214	105	17	,	,	PUNCT
ejpam-6214	105	18	f(x)+	f(x)+	VERB
ejpam-6214	105	19	pg(x	pg(x	NUM
ejpam-6214	105	20	)	)	PUNCT
ejpam-6214	105	21	is	be	AUX
ejpam-6214	105	22	irreducible	irreducible	ADJ
ejpam-6214	105	23	in	in	ADP
ejpam-6214	105	24	the	the	DET
ejpam-6214	105	25	polynomial	polynomial	ADJ
ejpam-6214	105	26	ring	ring	NOUN
ejpam-6214	105	27	q[x	q[x	PROPN
ejpam-6214	105	28	]	]	PUNCT
ejpam-6214	105	29	for	for	ADP
ejpam-6214	105	30	all	all	PRON
ejpam-6214	105	31	but	but	ADV
ejpam-6214	105	32	finitely	finitely	ADV
ejpam-6214	105	33	many	many	ADJ
ejpam-6214	105	34	primes	prime	NOUN
ejpam-6214	106	1	p.	p.	NOUN
ejpam-6214	106	2	example	example	NOUN
ejpam-6214	107	1	2	2	NUM
ejpam-6214	107	2	:	:	PUNCT
ejpam-6214	107	3	non	non	ADJ
ejpam-6214	107	4	-	-	ADJ
ejpam-6214	107	5	relatively	relatively	ADV
ejpam-6214	107	6	prime	prime	ADJ
ejpam-6214	107	7	polynomials	polynomial	NOUN
ejpam-6214	107	8	consider	consider	VERB
ejpam-6214	107	9	f(x	f(x	NOUN
ejpam-6214	107	10	)	)	PUNCT
ejpam-6214	108	1	=	=	SYM
ejpam-6214	109	1	x2	x2	PROPN
ejpam-6214	110	1	+	+	CCONJ
ejpam-6214	110	2	1	1	NUM
ejpam-6214	110	3	,	,	PUNCT
ejpam-6214	110	4	g(x	g(x	NOUN
ejpam-6214	110	5	)	)	PUNCT
ejpam-6214	110	6	=	=	SYM
ejpam-6214	111	1	(	(	PUNCT
ejpam-6214	111	2	x2	x2	NOUN
ejpam-6214	111	3	+	+	NUM
ejpam-6214	111	4	1)2	1)2	NUM
ejpam-6214	111	5	=	=	SYM
ejpam-6214	111	6	x4	x4	PROPN
ejpam-6214	112	1	+	+	CCONJ
ejpam-6214	112	2	2x2	2x2	NUM
ejpam-6214	112	3	+	+	SYM
ejpam-6214	112	4	1	1	X
ejpam-6214	112	5	.	.	PUNCT
ejpam-6214	112	6	clearly	clearly	ADV
ejpam-6214	112	7	,	,	PUNCT
ejpam-6214	112	8	f	f	PROPN
ejpam-6214	112	9	|	|	ADV
ejpam-6214	112	10	g	g	VERB
ejpam-6214	112	11	,	,	PUNCT
ejpam-6214	112	12	so	so	ADV
ejpam-6214	112	13	f	f	PROPN
ejpam-6214	112	14	and	and	CCONJ
ejpam-6214	112	15	g	g	PROPN
ejpam-6214	112	16	are	be	AUX
ejpam-6214	112	17	not	not	PART
ejpam-6214	112	18	relatively	relatively	ADV
ejpam-6214	112	19	prime	prime	ADJ
ejpam-6214	112	20	.	.	PUNCT
ejpam-6214	113	1	theorem	theorem	VERB
ejpam-6214	113	2	3.1	3.1	NUM
ejpam-6214	113	3	does	do	AUX
ejpam-6214	113	4	not	not	PART
ejpam-6214	113	5	apply	apply	VERB
ejpam-6214	113	6	,	,	PUNCT
ejpam-6214	113	7	and	and	CCONJ
ejpam-6214	113	8	indeed	indeed	ADV
ejpam-6214	113	9	the	the	DET
ejpam-6214	113	10	polynomial	polynomial	ADJ
ejpam-6214	113	11	f	f	NOUN
ejpam-6214	113	12	+	+	CCONJ
ejpam-6214	113	13	pkg	pkg	NOUN
ejpam-6214	113	14	factors	factor	NOUN
ejpam-6214	113	15	as	as	ADP
ejpam-6214	113	16	f(x)(1	f(x)(1	NOUN
ejpam-6214	113	17	+	+	NUM
ejpam-6214	113	18	pk(x2	pk(x2	NOUN
ejpam-6214	113	19	+	+	NOUN
ejpam-6214	113	20	1	1	NUM
ejpam-6214	113	21	)	)	PUNCT
ejpam-6214	113	22	)	)	PUNCT
ejpam-6214	113	23	.	.	PUNCT
ejpam-6214	114	1	4.1	4.1	NUM
ejpam-6214	114	2	.	.	PUNCT
ejpam-6214	114	3	example	example	NOUN
ejpam-6214	114	4	3	3	NUM
ejpam-6214	114	5	:	:	PUNCT
ejpam-6214	114	6	counterexample	counterexample	NOUN
ejpam-6214	114	7	when	when	SCONJ
ejpam-6214	114	8	gcd(k	gcd(k	PROPN
ejpam-6214	114	9	,	,	PUNCT
ejpam-6214	114	10	deg	deg	NOUN
ejpam-6214	114	11	g	g	NOUN
ejpam-6214	114	12	)	)	PUNCT
ejpam-6214	114	13	̸=	̸=	PROPN
ejpam-6214	114	14	1	1	NUM
ejpam-6214	114	15	take	take	VERB
ejpam-6214	114	16	f(x	f(x	NOUN
ejpam-6214	114	17	)	)	PUNCT
ejpam-6214	115	1	=	=	SYM
ejpam-6214	116	1	x2	x2	INTJ
ejpam-6214	117	1	+	+	NOUN
ejpam-6214	117	2	x	x	X
ejpam-6214	117	3	+	+	ADJ
ejpam-6214	117	4	1	1	NUM
ejpam-6214	117	5	,	,	PUNCT
ejpam-6214	117	6	g(x	g(x	NOUN
ejpam-6214	117	7	)	)	PUNCT
ejpam-6214	117	8	=	=	SYM
ejpam-6214	118	1	(	(	PUNCT
ejpam-6214	118	2	x2	x2	PROPN
ejpam-6214	118	3	+	+	NUM
ejpam-6214	118	4	1)f(x	1)f(x	NUM
ejpam-6214	118	5	)	)	PUNCT
ejpam-6214	118	6	=	=	PUNCT
ejpam-6214	119	1	x4	x4	PROPN
ejpam-6214	120	1	+	+	PROPN
ejpam-6214	120	2	x2	x2	PROPN
ejpam-6214	120	3	+	+	ADJ
ejpam-6214	120	4	1	1	NUM
ejpam-6214	120	5	,	,	PUNCT
ejpam-6214	120	6	with	with	ADP
ejpam-6214	120	7	k	k	PROPN
ejpam-6214	120	8	=	=	SYM
ejpam-6214	120	9	2	2	X
ejpam-6214	120	10	.	.	PUNCT
ejpam-6214	121	1	since	since	SCONJ
ejpam-6214	121	2	f	f	PROPN
ejpam-6214	121	3	|	|	ADV
ejpam-6214	121	4	g	g	PROPN
ejpam-6214	121	5	,	,	PUNCT
ejpam-6214	121	6	f	f	PROPN
ejpam-6214	121	7	and	and	CCONJ
ejpam-6214	121	8	g	g	PROPN
ejpam-6214	121	9	are	be	AUX
ejpam-6214	121	10	not	not	PART
ejpam-6214	121	11	relatively	relatively	ADV
ejpam-6214	121	12	prime	prime	ADJ
ejpam-6214	121	13	.	.	PUNCT
ejpam-6214	122	1	moreover	moreover	ADV
ejpam-6214	122	2	,	,	PUNCT
ejpam-6214	122	3	gcd(2	gcd(2	NOUN
ejpam-6214	122	4	,	,	PUNCT
ejpam-6214	122	5	4	4	NUM
ejpam-6214	122	6	)	)	PUNCT
ejpam-6214	122	7	=	=	SYM
ejpam-6214	122	8	2	2	NUM
ejpam-6214	122	9	̸=	̸=	PROPN
ejpam-6214	122	10	1	1	NUM
ejpam-6214	122	11	.	.	PUNCT
ejpam-6214	123	1	the	the	DET
ejpam-6214	123	2	polynomial	polynomial	ADJ
ejpam-6214	123	3	f	f	PROPN
ejpam-6214	123	4	+	+	CCONJ
ejpam-6214	123	5	p2	p2	NOUN
ejpam-6214	123	6	g	g	NOUN
ejpam-6214	123	7	factors	factor	NOUN
ejpam-6214	123	8	as	as	ADP
ejpam-6214	123	9	f(x)(1	f(x)(1	NOUN
ejpam-6214	123	10	+	+	CCONJ
ejpam-6214	123	11	p2(x2	p2(x2	NOUN
ejpam-6214	123	12	+	+	CCONJ
ejpam-6214	123	13	1	1	NUM
ejpam-6214	123	14	)	)	PUNCT
ejpam-6214	123	15	)	)	PUNCT
ejpam-6214	123	16	,	,	PUNCT
ejpam-6214	123	17	showing	show	VERB
ejpam-6214	123	18	theorem	theorem	ADJ
ejpam-6214	123	19	3.1	3.1	NUM
ejpam-6214	123	20	does	do	AUX
ejpam-6214	123	21	not	not	PART
ejpam-6214	123	22	hold	hold	VERB
ejpam-6214	123	23	if	if	SCONJ
ejpam-6214	123	24	hypotheses	hypothesis	NOUN
ejpam-6214	123	25	are	be	AUX
ejpam-6214	123	26	violated	violate	VERB
ejpam-6214	123	27	.	.	PUNCT
ejpam-6214	124	1	5	5	X
ejpam-6214	124	2	.	.	X
ejpam-6214	124	3	applications	application	NOUN
ejpam-6214	124	4	our	our	PRON
ejpam-6214	124	5	main	main	ADJ
ejpam-6214	124	6	theorem	theorem	NOUN
ejpam-6214	124	7	has	have	VERB
ejpam-6214	124	8	several	several	ADJ
ejpam-6214	124	9	important	important	ADJ
ejpam-6214	124	10	implications	implication	NOUN
ejpam-6214	124	11	and	and	CCONJ
ejpam-6214	124	12	applications	application	NOUN
ejpam-6214	124	13	in	in	ADP
ejpam-6214	124	14	number	number	NOUN
ejpam-6214	124	15	theory	theory	NOUN
ejpam-6214	124	16	and	and	CCONJ
ejpam-6214	124	17	algebra	algebra	NOUN
ejpam-6214	124	18	.	.	PUNCT
ejpam-6214	125	1	below	below	ADV
ejpam-6214	125	2	,	,	PUNCT
ejpam-6214	125	3	we	we	PRON
ejpam-6214	125	4	discuss	discuss	VERB
ejpam-6214	125	5	some	some	PRON
ejpam-6214	125	6	of	of	ADP
ejpam-6214	125	7	these	these	DET
ejpam-6214	125	8	applications	application	NOUN
ejpam-6214	125	9	.	.	PUNCT
ejpam-6214	126	1	5.1	5.1	NUM
ejpam-6214	126	2	.	.	PUNCT
ejpam-6214	127	1	algebraic	algebraic	ADJ
ejpam-6214	127	2	number	number	NOUN
ejpam-6214	127	3	theory	theory	NOUN
ejpam-6214	127	4	the	the	DET
ejpam-6214	127	5	irreducibility	irreducibility	NOUN
ejpam-6214	127	6	of	of	ADP
ejpam-6214	127	7	polynomials	polynomial	NOUN
ejpam-6214	127	8	of	of	ADP
ejpam-6214	127	9	the	the	DET
ejpam-6214	127	10	form	form	NOUN
ejpam-6214	127	11	f(x	f(x	PROPN
ejpam-6214	127	12	)	)	PUNCT
ejpam-6214	128	1	+	+	SYM
ejpam-6214	128	2	pkg(x	pkg(x	NOUN
ejpam-6214	128	3	)	)	PUNCT
ejpam-6214	128	4	is	be	AUX
ejpam-6214	128	5	closely	closely	ADV
ejpam-6214	128	6	related	relate	VERB
ejpam-6214	128	7	to	to	ADP
ejpam-6214	128	8	the	the	DET
ejpam-6214	128	9	study	study	NOUN
ejpam-6214	128	10	of	of	ADP
ejpam-6214	128	11	number	number	NOUN
ejpam-6214	128	12	fields	field	NOUN
ejpam-6214	128	13	and	and	CCONJ
ejpam-6214	128	14	algebraic	algebraic	ADJ
ejpam-6214	128	15	integers	integer	NOUN
ejpam-6214	128	16	.	.	PUNCT
ejpam-6214	129	1	specifically	specifically	ADV
ejpam-6214	129	2	,	,	PUNCT
ejpam-6214	129	3	if	if	SCONJ
ejpam-6214	129	4	f(x)+pkg(x	f(x)+pkg(x	ADJ
ejpam-6214	129	5	)	)	PUNCT
ejpam-6214	129	6	is	be	AUX
ejpam-6214	129	7	irreducible	irreducible	ADJ
ejpam-6214	129	8	,	,	PUNCT
ejpam-6214	129	9	it	it	PRON
ejpam-6214	129	10	defines	define	VERB
ejpam-6214	129	11	a	a	DET
ejpam-6214	129	12	number	number	NOUN
ejpam-6214	129	13	field	field	NOUN
ejpam-6214	129	14	q(α	q(α	PROPN
ejpam-6214	129	15	)	)	PUNCT
ejpam-6214	129	16	,	,	PUNCT
ejpam-6214	129	17	where	where	SCONJ
ejpam-6214	129	18	α	α	NOUN
ejpam-6214	129	19	is	be	AUX
ejpam-6214	129	20	a	a	DET
ejpam-6214	129	21	root	root	NOUN
ejpam-6214	129	22	of	of	ADP
ejpam-6214	129	23	the	the	DET
ejpam-6214	129	24	polynomial	polynomial	NOUN
ejpam-6214	129	25	.	.	PUNCT
ejpam-6214	130	1	our	our	PRON
ejpam-6214	130	2	theorem	theorem	NOUN
ejpam-6214	130	3	provides	provide	VERB
ejpam-6214	130	4	a	a	DET
ejpam-6214	130	5	tool	tool	NOUN
ejpam-6214	130	6	for	for	ADP
ejpam-6214	130	7	constructing	construct	VERB
ejpam-6214	130	8	such	such	ADJ
ejpam-6214	130	9	fields	field	NOUN
ejpam-6214	130	10	without	without	ADP
ejpam-6214	130	11	requiring	require	VERB
ejpam-6214	130	12	the	the	DET
ejpam-6214	130	13	restrictive	restrictive	ADJ
ejpam-6214	130	14	conditions	condition	NOUN
ejpam-6214	130	15	of	of	ADP
ejpam-6214	130	16	previous	previous	ADJ
ejpam-6214	130	17	results	result	NOUN
ejpam-6214	130	18	.	.	PUNCT
ejpam-6214	131	1	5.2	5.2	X
ejpam-6214	131	2	.	.	PUNCT
ejpam-6214	132	1	diophantine	diophantine	VERB
ejpam-6214	132	2	equations	equation	NOUN
ejpam-6214	132	3	the	the	DET
ejpam-6214	132	4	irreducibility	irreducibility	NOUN
ejpam-6214	132	5	of	of	ADP
ejpam-6214	132	6	polynomials	polynomial	NOUN
ejpam-6214	132	7	is	be	AUX
ejpam-6214	132	8	also	also	ADV
ejpam-6214	132	9	relevant	relevant	ADJ
ejpam-6214	132	10	to	to	ADP
ejpam-6214	132	11	the	the	DET
ejpam-6214	132	12	study	study	NOUN
ejpam-6214	132	13	of	of	ADP
ejpam-6214	132	14	diophantine	diophantine	NOUN
ejpam-6214	132	15	equations	equation	NOUN
ejpam-6214	132	16	.	.	PUNCT
ejpam-6214	133	1	for	for	ADP
ejpam-6214	133	2	example	example	NOUN
ejpam-6214	133	3	,	,	PUNCT
ejpam-6214	133	4	if	if	SCONJ
ejpam-6214	133	5	f(x	f(x	PROPN
ejpam-6214	133	6	)	)	PUNCT
ejpam-6214	133	7	+	+	SYM
ejpam-6214	133	8	pkg(x	pkg(x	NOUN
ejpam-6214	133	9	)	)	PUNCT
ejpam-6214	133	10	is	be	AUX
ejpam-6214	133	11	irreducible	irreducible	ADJ
ejpam-6214	133	12	,	,	PUNCT
ejpam-6214	133	13	it	it	PRON
ejpam-6214	133	14	can	can	AUX
ejpam-6214	133	15	be	be	AUX
ejpam-6214	133	16	used	use	VERB
ejpam-6214	133	17	to	to	PART
ejpam-6214	133	18	analyze	analyze	VERB
ejpam-6214	133	19	the	the	DET
ejpam-6214	133	20	solvability	solvability	NOUN
ejpam-6214	133	21	of	of	ADP
ejpam-6214	133	22	equations	equation	NOUN
ejpam-6214	133	23	of	of	ADP
ejpam-6214	133	24	the	the	DET
ejpam-6214	133	25	form	form	NOUN
ejpam-6214	133	26	f(x	f(x	PROPN
ejpam-6214	133	27	)	)	PUNCT
ejpam-6214	134	1	+	+	SYM
ejpam-6214	134	2	pkg(x	pkg(x	NOUN
ejpam-6214	134	3	)	)	PUNCT
ejpam-6214	134	4	=	=	SYM
ejpam-6214	134	5	0	0	NUM
ejpam-6214	134	6	in	in	ADP
ejpam-6214	134	7	integers	integer	NOUN
ejpam-6214	134	8	x.	x.	PUNCT
ejpam-6214	135	1	our	our	PRON
ejpam-6214	135	2	theorem	theorem	NOUN
ejpam-6214	135	3	broadens	broaden	VERB
ejpam-6214	135	4	the	the	DET
ejpam-6214	135	5	class	class	NOUN
ejpam-6214	135	6	of	of	ADP
ejpam-6214	135	7	polynomials	polynomial	NOUN
ejpam-6214	135	8	for	for	ADP
ejpam-6214	135	9	which	which	PRON
ejpam-6214	135	10	such	such	ADJ
ejpam-6214	135	11	analysis	analysis	NOUN
ejpam-6214	135	12	is	be	AUX
ejpam-6214	135	13	possible	possible	ADJ
ejpam-6214	135	14	.	.	PUNCT
ejpam-6214	136	1	a.	a.	PROPN
ejpam-6214	136	2	chandoul	chandoul	PROPN
ejpam-6214	136	3	,	,	PUNCT
ejpam-6214	136	4	s.	s.	PROPN
ejpam-6214	136	5	mansour	mansour	PROPN
ejpam-6214	136	6	/	/	SYM
ejpam-6214	136	7	eur	eur	PROPN
ejpam-6214	136	8	.	.	PUNCT
ejpam-6214	137	1	j.	j.	PROPN
ejpam-6214	137	2	pure	pure	PROPN
ejpam-6214	137	3	appl	appl	PROPN
ejpam-6214	137	4	.	.	PROPN
ejpam-6214	137	5	math	math	PROPN
ejpam-6214	137	6	,	,	PUNCT
ejpam-6214	137	7	18	18	NUM
ejpam-6214	137	8	(	(	PUNCT
ejpam-6214	137	9	3	3	NUM
ejpam-6214	137	10	)	)	PUNCT
ejpam-6214	137	11	(	(	PUNCT
ejpam-6214	137	12	2025	2025	NUM
ejpam-6214	137	13	)	)	PUNCT
ejpam-6214	137	14	,	,	PUNCT
ejpam-6214	137	15	6214	6214	NUM
ejpam-6214	137	16	7	7	NUM
ejpam-6214	137	17	of	of	ADP
ejpam-6214	137	18	8	8	NUM
ejpam-6214	137	19	5.3	5.3	NUM
ejpam-6214	137	20	.	.	PUNCT
ejpam-6214	138	1	polynomial	polynomial	ADJ
ejpam-6214	138	2	factorization	factorization	NOUN
ejpam-6214	138	3	our	our	PRON
ejpam-6214	138	4	result	result	NOUN
ejpam-6214	138	5	provides	provide	VERB
ejpam-6214	138	6	new	new	ADJ
ejpam-6214	138	7	insights	insight	NOUN
ejpam-6214	138	8	into	into	ADP
ejpam-6214	138	9	the	the	DET
ejpam-6214	138	10	factorization	factorization	NOUN
ejpam-6214	138	11	structure	structure	NOUN
ejpam-6214	138	12	of	of	ADP
ejpam-6214	138	13	polynomials	polynomial	NOUN
ejpam-6214	138	14	with	with	ADP
ejpam-6214	138	15	prime	prime	ADJ
ejpam-6214	138	16	power	power	NOUN
ejpam-6214	138	17	shifts	shift	NOUN
ejpam-6214	138	18	.	.	PUNCT
ejpam-6214	139	1	specifically	specifically	ADV
ejpam-6214	139	2	,	,	PUNCT
ejpam-6214	139	3	when	when	SCONJ
ejpam-6214	139	4	f(x)+pkg(x	f(x)+pkg(x	ADJ
ejpam-6214	139	5	)	)	PUNCT
ejpam-6214	139	6	is	be	AUX
ejpam-6214	139	7	reducible	reducible	ADJ
ejpam-6214	139	8	,	,	PUNCT
ejpam-6214	139	9	the	the	DET
ejpam-6214	139	10	degrees	degree	NOUN
ejpam-6214	139	11	of	of	ADP
ejpam-6214	139	12	its	its	PRON
ejpam-6214	139	13	factors	factor	NOUN
ejpam-6214	139	14	are	be	AUX
ejpam-6214	139	15	constrained	constrain	VERB
ejpam-6214	139	16	to	to	PART
ejpam-6214	139	17	be	be	AUX
ejpam-6214	139	18	multiples	multiple	NOUN
ejpam-6214	139	19	of	of	ADP
ejpam-6214	139	20	gcd(k	gcd(k	PROPN
ejpam-6214	139	21	,	,	PUNCT
ejpam-6214	139	22	deg	deg	VERB
ejpam-6214	139	23	g	g	NOUN
ejpam-6214	139	24	)	)	PUNCT
ejpam-6214	139	25	.	.	PUNCT
ejpam-6214	140	1	this	this	PRON
ejpam-6214	140	2	can	can	AUX
ejpam-6214	140	3	be	be	AUX
ejpam-6214	140	4	used	use	VERB
ejpam-6214	140	5	to	to	PART
ejpam-6214	140	6	develop	develop	VERB
ejpam-6214	140	7	algorithms	algorithm	NOUN
ejpam-6214	140	8	for	for	ADP
ejpam-6214	140	9	polynomial	polynomial	ADJ
ejpam-6214	140	10	factorization	factorization	NOUN
ejpam-6214	140	11	over	over	ADP
ejpam-6214	140	12	q.	q.	PROPN
ejpam-6214	140	13	5.4	5.4	NUM
ejpam-6214	140	14	.	.	PUNCT
ejpam-6214	141	1	open	open	ADJ
ejpam-6214	141	2	problems	problem	NOUN
ejpam-6214	141	3	our	our	PRON
ejpam-6214	141	4	work	work	NOUN
ejpam-6214	141	5	raises	raise	VERB
ejpam-6214	141	6	several	several	ADJ
ejpam-6214	141	7	open	open	ADJ
ejpam-6214	141	8	questions	question	NOUN
ejpam-6214	141	9	:	:	PUNCT
ejpam-6214	141	10	(	(	PUNCT
ejpam-6214	141	11	i	i	NOUN
ejpam-6214	141	12	)	)	PUNCT
ejpam-6214	141	13	can	can	AUX
ejpam-6214	141	14	the	the	DET
ejpam-6214	141	15	result	result	NOUN
ejpam-6214	141	16	be	be	AUX
ejpam-6214	141	17	extended	extend	VERB
ejpam-6214	141	18	to	to	ADP
ejpam-6214	141	19	polynomials	polynomial	NOUN
ejpam-6214	141	20	over	over	ADP
ejpam-6214	141	21	other	other	ADJ
ejpam-6214	141	22	rings	ring	NOUN
ejpam-6214	141	23	,	,	PUNCT
ejpam-6214	141	24	such	such	ADJ
ejpam-6214	141	25	as	as	ADP
ejpam-6214	141	26	z[i	z[i	NUM
ejpam-6214	141	27	]	]	X
ejpam-6214	141	28	(	(	PUNCT
ejpam-6214	141	29	gaussian	gaussian	ADJ
ejpam-6214	141	30	integers	integer	NOUN
ejpam-6214	141	31	)	)	PUNCT
ejpam-6214	141	32	?	?	PUNCT
ejpam-6214	142	1	(	(	PUNCT
ejpam-6214	142	2	ii	ii	X
ejpam-6214	142	3	)	)	PUNCT
ejpam-6214	142	4	what	what	PRON
ejpam-6214	142	5	is	be	AUX
ejpam-6214	142	6	the	the	DET
ejpam-6214	142	7	explicit	explicit	ADJ
ejpam-6214	142	8	bound	bind	VERB
ejpam-6214	142	9	on	on	ADP
ejpam-6214	142	10	the	the	DET
ejpam-6214	142	11	number	number	NOUN
ejpam-6214	142	12	of	of	ADP
ejpam-6214	142	13	exceptional	exceptional	ADJ
ejpam-6214	142	14	primes	prime	NOUN
ejpam-6214	142	15	p	p	NOUN
ejpam-6214	142	16	for	for	ADP
ejpam-6214	142	17	which	which	PRON
ejpam-6214	142	18	f(x)+	f(x)+	VERB
ejpam-6214	142	19	pkg(x	pkg(x	NOUN
ejpam-6214	142	20	)	)	PUNCT
ejpam-6214	142	21	is	be	AUX
ejpam-6214	142	22	reducible	reducible	ADJ
ejpam-6214	142	23	?	?	PUNCT
ejpam-6214	143	1	(	(	PUNCT
ejpam-6214	143	2	iii	iii	X
ejpam-6214	143	3	)	)	PUNCT
ejpam-6214	143	4	can	can	AUX
ejpam-6214	143	5	similar	similar	ADJ
ejpam-6214	143	6	results	result	NOUN
ejpam-6214	143	7	be	be	AUX
ejpam-6214	143	8	obtained	obtain	VERB
ejpam-6214	143	9	for	for	ADP
ejpam-6214	143	10	polynomials	polynomial	NOUN
ejpam-6214	143	11	of	of	ADP
ejpam-6214	143	12	the	the	DET
ejpam-6214	143	13	form	form	NOUN
ejpam-6214	143	14	f(x)+pkg(x)+pmh(x	f(x)+pkg(x)+pmh(x	PROPN
ejpam-6214	143	15	)	)	PUNCT
ejpam-6214	143	16	?	?	PUNCT
ejpam-6214	144	1	these	these	DET
ejpam-6214	144	2	questions	question	NOUN
ejpam-6214	144	3	provide	provide	VERB
ejpam-6214	144	4	fertile	fertile	ADJ
ejpam-6214	144	5	ground	ground	NOUN
ejpam-6214	144	6	for	for	ADP
ejpam-6214	144	7	future	future	ADJ
ejpam-6214	144	8	research	research	NOUN
ejpam-6214	144	9	.	.	PUNCT
ejpam-6214	145	1	6	6	X
ejpam-6214	145	2	.	.	X
ejpam-6214	145	3	conclusion	conclusion	NOUN
ejpam-6214	145	4	in	in	ADP
ejpam-6214	145	5	this	this	DET
ejpam-6214	145	6	paper	paper	NOUN
ejpam-6214	145	7	,	,	PUNCT
ejpam-6214	145	8	we	we	PRON
ejpam-6214	145	9	have	have	VERB
ejpam-6214	145	10	generalized	generalize	VERB
ejpam-6214	145	11	bonciocat	bonciocat	NOUN
ejpam-6214	145	12	’s	’s	PART
ejpam-6214	145	13	theorem	theorem	NOUN
ejpam-6214	145	14	on	on	ADP
ejpam-6214	145	15	the	the	DET
ejpam-6214	145	16	irreducibility	irreducibility	NOUN
ejpam-6214	145	17	of	of	ADP
ejpam-6214	145	18	polynomials	polynomial	NOUN
ejpam-6214	145	19	of	of	ADP
ejpam-6214	145	20	the	the	DET
ejpam-6214	145	21	form	form	NOUN
ejpam-6214	145	22	f(x	f(x	PROPN
ejpam-6214	145	23	)	)	PUNCT
ejpam-6214	146	1	+	+	NUM
ejpam-6214	147	1	pkg(x	pkg(x	NOUN
ejpam-6214	147	2	)	)	PUNCT
ejpam-6214	147	3	.	.	PUNCT
ejpam-6214	148	1	by	by	ADP
ejpam-6214	148	2	removing	remove	VERB
ejpam-6214	148	3	the	the	DET
ejpam-6214	148	4	restrictive	restrictive	ADJ
ejpam-6214	148	5	conditions	condition	NOUN
ejpam-6214	148	6	on	on	ADP
ejpam-6214	148	7	relative	relative	ADJ
ejpam-6214	148	8	primality	primality	NOUN
ejpam-6214	148	9	and	and	CCONJ
ejpam-6214	148	10	gcd(k	gcd(k	PROPN
ejpam-6214	148	11	,	,	PUNCT
ejpam-6214	148	12	deg	deg	VERB
ejpam-6214	148	13	g	g	PROPN
ejpam-6214	148	14	)	)	PUNCT
ejpam-6214	148	15	,	,	PUNCT
ejpam-6214	148	16	our	our	PRON
ejpam-6214	148	17	result	result	NOUN
ejpam-6214	148	18	applies	apply	VERB
ejpam-6214	148	19	to	to	ADP
ejpam-6214	148	20	a	a	DET
ejpam-6214	148	21	broader	broad	ADJ
ejpam-6214	148	22	class	class	NOUN
ejpam-6214	148	23	of	of	ADP
ejpam-6214	148	24	polynomials	polynomial	NOUN
ejpam-6214	148	25	and	and	CCONJ
ejpam-6214	148	26	provides	provide	VERB
ejpam-6214	148	27	new	new	ADJ
ejpam-6214	148	28	insights	insight	NOUN
ejpam-6214	148	29	into	into	ADP
ejpam-6214	148	30	their	their	PRON
ejpam-6214	148	31	factorization	factorization	NOUN
ejpam-6214	148	32	structure	structure	NOUN
ejpam-6214	148	33	.	.	PUNCT
ejpam-6214	149	1	specifically	specifically	ADV
ejpam-6214	149	2	,	,	PUNCT
ejpam-6214	149	3	we	we	PRON
ejpam-6214	149	4	have	have	AUX
ejpam-6214	149	5	shown	show	VERB
ejpam-6214	149	6	that	that	SCONJ
ejpam-6214	149	7	for	for	ADP
ejpam-6214	149	8	any	any	DET
ejpam-6214	149	9	polynomials	polynomial	NOUN
ejpam-6214	149	10	f(x	f(x	PROPN
ejpam-6214	149	11	)	)	PUNCT
ejpam-6214	149	12	and	and	CCONJ
ejpam-6214	149	13	g(x	g(x	NOUN
ejpam-6214	149	14	)	)	PUNCT
ejpam-6214	149	15	with	with	ADP
ejpam-6214	149	16	deg	deg	PROPN
ejpam-6214	149	17	f	f	X
ejpam-6214	149	18	<	<	X
ejpam-6214	149	19	deg	deg	PROPN
ejpam-6214	149	20	g	g	NOUN
ejpam-6214	149	21	,	,	PUNCT
ejpam-6214	149	22	and	and	CCONJ
ejpam-6214	149	23	for	for	ADP
ejpam-6214	149	24	any	any	DET
ejpam-6214	149	25	positive	positive	ADJ
ejpam-6214	149	26	integer	integer	NOUN
ejpam-6214	149	27	k	k	PROPN
ejpam-6214	149	28	,	,	PUNCT
ejpam-6214	149	29	the	the	DET
ejpam-6214	149	30	polynomial	polynomial	ADJ
ejpam-6214	149	31	f(x	f(x	PROPN
ejpam-6214	149	32	)	)	PUNCT
ejpam-6214	150	1	+	+	SYM
ejpam-6214	150	2	pkg(x	pkg(x	NOUN
ejpam-6214	150	3	)	)	PUNCT
ejpam-6214	150	4	is	be	AUX
ejpam-6214	150	5	either	either	CCONJ
ejpam-6214	150	6	irreducible	irreducible	ADJ
ejpam-6214	150	7	over	over	ADP
ejpam-6214	150	8	q	q	NOUN
ejpam-6214	150	9	or	or	CCONJ
ejpam-6214	150	10	factors	factor	NOUN
ejpam-6214	150	11	into	into	ADP
ejpam-6214	150	12	polynomials	polynomial	NOUN
ejpam-6214	150	13	whose	whose	DET
ejpam-6214	150	14	degrees	degree	NOUN
ejpam-6214	150	15	are	be	AUX
ejpam-6214	150	16	multiples	multiple	NOUN
ejpam-6214	150	17	of	of	ADP
ejpam-6214	150	18	gcd(k	gcd(k	PROPN
ejpam-6214	150	19	,	,	PUNCT
ejpam-6214	150	20	deg	deg	VERB
ejpam-6214	150	21	g	g	NOUN
ejpam-6214	150	22	)	)	PUNCT
ejpam-6214	150	23	.	.	PUNCT
ejpam-6214	151	1	this	this	PRON
ejpam-6214	151	2	holds	hold	VERB
ejpam-6214	151	3	for	for	ADP
ejpam-6214	151	4	all	all	PRON
ejpam-6214	151	5	but	but	ADV
ejpam-6214	151	6	finitely	finitely	ADV
ejpam-6214	151	7	many	many	ADJ
ejpam-6214	151	8	primes	prime	NOUN
ejpam-6214	151	9	p.	p.	NOUN
ejpam-6214	151	10	our	our	PRON
ejpam-6214	151	11	work	work	NOUN
ejpam-6214	151	12	has	have	VERB
ejpam-6214	151	13	several	several	ADJ
ejpam-6214	151	14	implications	implication	NOUN
ejpam-6214	151	15	for	for	ADP
ejpam-6214	151	16	algebraic	algebraic	ADJ
ejpam-6214	151	17	number	number	NOUN
ejpam-6214	151	18	theory	theory	NOUN
ejpam-6214	151	19	,	,	PUNCT
ejpam-6214	151	20	diophantine	diophantine	VERB
ejpam-6214	151	21	equations	equation	NOUN
ejpam-6214	151	22	,	,	PUNCT
ejpam-6214	151	23	and	and	CCONJ
ejpam-6214	151	24	polynomial	polynomial	ADJ
ejpam-6214	151	25	factorization	factorization	NOUN
ejpam-6214	151	26	.	.	PUNCT
ejpam-6214	152	1	it	it	PRON
ejpam-6214	152	2	also	also	ADV
ejpam-6214	152	3	raises	raise	VERB
ejpam-6214	152	4	new	new	ADJ
ejpam-6214	152	5	questions	question	NOUN
ejpam-6214	152	6	,	,	PUNCT
ejpam-6214	152	7	such	such	ADJ
ejpam-6214	152	8	as	as	ADP
ejpam-6214	152	9	the	the	DET
ejpam-6214	152	10	extension	extension	NOUN
ejpam-6214	152	11	of	of	ADP
ejpam-6214	152	12	our	our	PRON
ejpam-6214	152	13	results	result	NOUN
ejpam-6214	152	14	to	to	ADP
ejpam-6214	152	15	other	other	ADJ
ejpam-6214	152	16	rings	ring	NOUN
ejpam-6214	152	17	and	and	CCONJ
ejpam-6214	152	18	the	the	DET
ejpam-6214	152	19	explicit	explicit	ADJ
ejpam-6214	152	20	determination	determination	NOUN
ejpam-6214	152	21	of	of	ADP
ejpam-6214	152	22	exceptional	exceptional	ADJ
ejpam-6214	152	23	primes	prime	NOUN
ejpam-6214	152	24	.	.	PUNCT
ejpam-6214	153	1	we	we	PRON
ejpam-6214	153	2	hope	hope	VERB
ejpam-6214	153	3	that	that	SCONJ
ejpam-6214	153	4	this	this	DET
ejpam-6214	153	5	paper	paper	NOUN
ejpam-6214	153	6	will	will	AUX
ejpam-6214	153	7	inspire	inspire	VERB
ejpam-6214	153	8	further	further	ADJ
ejpam-6214	153	9	research	research	NOUN
ejpam-6214	153	10	into	into	ADP
ejpam-6214	153	11	the	the	DET
ejpam-6214	153	12	irreducibility	irreducibility	NOUN
ejpam-6214	153	13	of	of	ADP
ejpam-6214	153	14	polynomials	polynomial	NOUN
ejpam-6214	153	15	with	with	ADP
ejpam-6214	153	16	prime	prime	ADJ
ejpam-6214	153	17	power	power	NOUN
ejpam-6214	153	18	shifts	shift	NOUN
ejpam-6214	153	19	and	and	CCONJ
ejpam-6214	153	20	their	their	PRON
ejpam-6214	153	21	applications	application	NOUN
ejpam-6214	153	22	.	.	PUNCT
ejpam-6214	154	1	future	future	ADJ
ejpam-6214	154	2	research	research	NOUN
ejpam-6214	154	3	directions	direction	NOUN
ejpam-6214	154	4	include	include	VERB
ejpam-6214	154	5	:	:	PUNCT
ejpam-6214	154	6	(	(	PUNCT
ejpam-6214	154	7	i	i	NOUN
ejpam-6214	154	8	)	)	PUNCT
ejpam-6214	154	9	extending	extend	VERB
ejpam-6214	154	10	the	the	DET
ejpam-6214	154	11	theorem	theorem	NOUN
ejpam-6214	154	12	to	to	PART
ejpam-6214	154	13	multivariate	multivariate	VERB
ejpam-6214	154	14	polynomials	polynomial	NOUN
ejpam-6214	154	15	.	.	PUNCT
ejpam-6214	155	1	(	(	PUNCT
ejpam-6214	155	2	ii	ii	NOUN
ejpam-6214	155	3	)	)	PUNCT
ejpam-6214	155	4	investigating	investigate	VERB
ejpam-6214	155	5	the	the	DET
ejpam-6214	155	6	irreducibility	irreducibility	NOUN
ejpam-6214	155	7	of	of	ADP
ejpam-6214	155	8	polynomials	polynomial	NOUN
ejpam-6214	155	9	with	with	ADP
ejpam-6214	155	10	more	more	ADJ
ejpam-6214	155	11	general	general	ADJ
ejpam-6214	155	12	forms	form	NOUN
ejpam-6214	155	13	,	,	PUNCT
ejpam-6214	155	14	such	such	ADJ
ejpam-6214	155	15	as	as	ADP
ejpam-6214	155	16	f(x	f(x	PROPN
ejpam-6214	155	17	)	)	PUNCT
ejpam-6214	156	1	+	+	SYM
ejpam-6214	156	2	pkg(x	pkg(x	NOUN
ejpam-6214	156	3	)	)	PUNCT
ejpam-6214	156	4	+	+	NUM
ejpam-6214	156	5	pmh(x	pmh(x	PROPN
ejpam-6214	156	6	)	)	PUNCT
ejpam-6214	156	7	.	.	PUNCT
ejpam-6214	157	1	(	(	PUNCT
ejpam-6214	157	2	iii	iii	X
ejpam-6214	157	3	)	)	PUNCT
ejpam-6214	157	4	developing	develop	VERB
ejpam-6214	157	5	algorithms	algorithm	NOUN
ejpam-6214	157	6	for	for	ADP
ejpam-6214	157	7	polynomial	polynomial	ADJ
ejpam-6214	157	8	factorization	factorization	NOUN
ejpam-6214	157	9	based	base	VERB
ejpam-6214	157	10	on	on	ADP
ejpam-6214	157	11	our	our	PRON
ejpam-6214	157	12	results	result	NOUN
ejpam-6214	157	13	.	.	PUNCT
ejpam-6214	158	1	acknowledgements	acknowledgement	NOUN
ejpam-6214	158	2	the	the	DET
ejpam-6214	158	3	authors	author	NOUN
ejpam-6214	158	4	extend	extend	VERB
ejpam-6214	158	5	their	their	PRON
ejpam-6214	158	6	appreciation	appreciation	NOUN
ejpam-6214	158	7	to	to	ADP
ejpam-6214	158	8	umm	umm	INTJ
ejpam-6214	158	9	al	al	PROPN
ejpam-6214	158	10	-	-	PUNCT
ejpam-6214	158	11	qura	qura	PROPN
ejpam-6214	158	12	university	university	PROPN
ejpam-6214	158	13	,	,	PUNCT
ejpam-6214	158	14	saudi	saudi	PROPN
ejpam-6214	158	15	arabia	arabia	PROPN
ejpam-6214	158	16	for	for	ADP
ejpam-6214	158	17	funding	fund	VERB
ejpam-6214	158	18	this	this	DET
ejpam-6214	158	19	research	research	NOUN
ejpam-6214	158	20	work	work	NOUN
ejpam-6214	158	21	through	through	ADP
ejpam-6214	158	22	grant	grant	NOUN
ejpam-6214	158	23	number	number	NOUN
ejpam-6214	158	24	:	:	PUNCT
ejpam-6214	158	25	25uqu4331214gssr02	25uqu4331214gssr02	NUM
ejpam-6214	158	26	.	.	PUNCT
ejpam-6214	158	27	a.	a.	PROPN
ejpam-6214	158	28	chandoul	chandoul	PROPN
ejpam-6214	158	29	,	,	PUNCT
ejpam-6214	158	30	s.	s.	PROPN
ejpam-6214	158	31	mansour	mansour	PROPN
ejpam-6214	158	32	/	/	SYM
ejpam-6214	158	33	eur	eur	PROPN
ejpam-6214	158	34	.	.	PUNCT
ejpam-6214	159	1	j.	j.	PROPN
ejpam-6214	159	2	pure	pure	PROPN
ejpam-6214	159	3	appl	appl	PROPN
ejpam-6214	159	4	.	.	PROPN
ejpam-6214	159	5	math	math	PROPN
ejpam-6214	159	6	,	,	PUNCT
ejpam-6214	159	7	18	18	NUM
ejpam-6214	159	8	(	(	PUNCT
ejpam-6214	159	9	3	3	NUM
ejpam-6214	159	10	)	)	PUNCT
ejpam-6214	159	11	(	(	PUNCT
ejpam-6214	159	12	2025	2025	NUM
ejpam-6214	159	13	)	)	PUNCT
ejpam-6214	159	14	,	,	PUNCT
ejpam-6214	159	15	6214	6214	NUM
ejpam-6214	159	16	8	8	NUM
ejpam-6214	159	17	of	of	ADP
ejpam-6214	159	18	8	8	NUM
ejpam-6214	159	19	funding	funding	NOUN
ejpam-6214	159	20	this	this	DET
ejpam-6214	159	21	research	research	NOUN
ejpam-6214	159	22	work	work	NOUN
ejpam-6214	159	23	was	be	AUX
ejpam-6214	159	24	funded	fund	VERB
ejpam-6214	159	25	by	by	ADP
ejpam-6214	159	26	umm	umm	INTJ
ejpam-6214	159	27	al	al	PROPN
ejpam-6214	159	28	-	-	PUNCT
ejpam-6214	159	29	qura	qura	PROPN
ejpam-6214	159	30	university	university	NOUN
ejpam-6214	159	31	,	,	PUNCT
ejpam-6214	159	32	saudi	saudi	PROPN
ejpam-6214	159	33	arabia	arabia	PROPN
ejpam-6214	159	34	under	under	ADP
ejpam-6214	159	35	grant	grant	NOUN
ejpam-6214	159	36	number	number	NOUN
ejpam-6214	159	37	:	:	PUNCT
ejpam-6214	159	38	25uqu4331214gssr02	25uqu4331214gssr02	NUM
ejpam-6214	159	39	.	.	PUNCT
ejpam-6214	160	1	references	reference	NOUN
ejpam-6214	160	2	[	[	X
ejpam-6214	160	3	1	1	NUM
ejpam-6214	160	4	]	]	X
ejpam-6214	160	5	henri	henri	PROPN
ejpam-6214	160	6	cohen	cohen	PROPN
ejpam-6214	160	7	.	.	PUNCT
ejpam-6214	161	1	advanced	advanced	ADJ
ejpam-6214	161	2	topics	topic	NOUN
ejpam-6214	161	3	in	in	ADP
ejpam-6214	161	4	computational	computational	ADJ
ejpam-6214	161	5	number	number	NOUN
ejpam-6214	161	6	theory	theory	NOUN
ejpam-6214	161	7	.	.	PUNCT
ejpam-6214	162	1	springer	springer	NOUN
ejpam-6214	162	2	,	,	PUNCT
ejpam-6214	162	3	2000	2000	NUM
ejpam-6214	162	4	.	.	PUNCT
ejpam-6214	163	1	[	[	X
ejpam-6214	163	2	2	2	X
ejpam-6214	163	3	]	]	PUNCT
ejpam-6214	163	4	serge	serge	PROPN
ejpam-6214	163	5	lang	lang	PROPN
ejpam-6214	163	6	.	.	PUNCT
ejpam-6214	164	1	algebra	algebra	PROPN
ejpam-6214	164	2	.	.	PUNCT
ejpam-6214	165	1	springer	springer	NOUN
ejpam-6214	165	2	,	,	PUNCT
ejpam-6214	165	3	revised	revise	VERB
ejpam-6214	165	4	third	third	ADJ
ejpam-6214	165	5	edition	edition	NOUN
ejpam-6214	165	6	edition	edition	NOUN
ejpam-6214	165	7	,	,	PUNCT
ejpam-6214	165	8	2002	2002	NUM
ejpam-6214	165	9	.	.	PUNCT
ejpam-6214	166	1	[	[	X
ejpam-6214	166	2	3	3	NUM
ejpam-6214	166	3	]	]	X
ejpam-6214	166	4	gotthold	gotthold	PROPN
ejpam-6214	166	5	eisenstein	eisenstein	PROPN
ejpam-6214	166	6	.	.	PUNCT
ejpam-6214	167	1	über	über	PROPN
ejpam-6214	167	2	die	die	VERB
ejpam-6214	167	3	irreduzibilität	irreduzibilität	PROPN
ejpam-6214	168	1	und	und	VERB
ejpam-6214	168	2	einige	einige	PROPN
ejpam-6214	168	3	andere	andere	PROPN
ejpam-6214	168	4	eigenschaften	eigenschaften	PROPN
ejpam-6214	168	5	der	der	PROPN
ejpam-6214	168	6	gleichung	gleichung	PROPN
ejpam-6214	168	7	.	.	PROPN
ejpam-6214	169	1	journal	journal	PROPN
ejpam-6214	169	2	für	für	AUX
ejpam-6214	169	3	die	die	VERB
ejpam-6214	169	4	reine	reine	PROPN
ejpam-6214	169	5	und	und	PROPN
ejpam-6214	169	6	angewandte	angewandte	PROPN
ejpam-6214	169	7	mathematik	mathematik	PROPN
ejpam-6214	169	8	,	,	PUNCT
ejpam-6214	169	9	39:160–179	39:160–179	PROPN
ejpam-6214	169	10	,	,	PUNCT
ejpam-6214	169	11	1850	1850	NUM
ejpam-6214	169	12	.	.	PUNCT
ejpam-6214	170	1	[	[	X
ejpam-6214	170	2	4	4	NUM
ejpam-6214	170	3	]	]	X
ejpam-6214	170	4	nicolae	nicolae	PROPN
ejpam-6214	170	5	ciprian	ciprian	PROPN
ejpam-6214	170	6	bonciocat	bonciocat	PROPN
ejpam-6214	170	7	.	.	PUNCT
ejpam-6214	171	1	an	an	DET
ejpam-6214	171	2	irreducibility	irreducibility	NOUN
ejpam-6214	171	3	criterion	criterion	NOUN
ejpam-6214	171	4	for	for	ADP
ejpam-6214	171	5	the	the	DET
ejpam-6214	171	6	sum	sum	NOUN
ejpam-6214	171	7	of	of	ADP
ejpam-6214	171	8	two	two	NUM
ejpam-6214	171	9	relatively	relatively	ADV
ejpam-6214	171	10	prime	prime	ADJ
ejpam-6214	171	11	polynomials	polynomial	NOUN
ejpam-6214	171	12	.	.	PUNCT
ejpam-6214	172	1	funct	funct	ADJ
ejpam-6214	172	2	.	.	PUNCT
ejpam-6214	173	1	approx	approx	PROPN
ejpam-6214	173	2	.	.	PUNCT
ejpam-6214	174	1	comment	comment	NOUN
ejpam-6214	174	2	.	.	PUNCT
ejpam-6214	175	1	math	math	NOUN
ejpam-6214	175	2	.	.	PUNCT
ejpam-6214	175	3	,	,	PUNCT
ejpam-6214	176	1	54(2):163–171	54(2):163–171	PROPN
ejpam-6214	176	2	,	,	PUNCT
ejpam-6214	176	3	2016	2016	NUM
ejpam-6214	176	4	.	.	PUNCT
ejpam-6214	177	1	[	[	X
ejpam-6214	177	2	5	5	NUM
ejpam-6214	177	3	]	]	X
ejpam-6214	177	4	jean	jean	PROPN
ejpam-6214	177	5	-	-	PUNCT
ejpam-6214	177	6	pierre	pierre	PROPN
ejpam-6214	177	7	serre	serre	X
ejpam-6214	177	8	.	.	PUNCT
ejpam-6214	178	1	a	a	DET
ejpam-6214	178	2	course	course	NOUN
ejpam-6214	178	3	in	in	ADP
ejpam-6214	178	4	arithmetic	arithmetic	ADJ
ejpam-6214	178	5	.	.	PUNCT
ejpam-6214	179	1	springer	springer	NOUN
ejpam-6214	179	2	,	,	PUNCT
ejpam-6214	179	3	1973	1973	NUM
ejpam-6214	179	4	.	.	PUNCT
ejpam-6214	180	1	[	[	X
ejpam-6214	180	2	6	6	NUM
ejpam-6214	180	3	]	]	X
ejpam-6214	180	4	jürgen	jürgen	X
ejpam-6214	180	5	neukirch	neukirch	NOUN
ejpam-6214	180	6	.	.	PUNCT
ejpam-6214	181	1	algebraic	algebraic	ADJ
ejpam-6214	181	2	number	number	NOUN
ejpam-6214	181	3	theory	theory	NOUN
ejpam-6214	181	4	.	.	PUNCT
ejpam-6214	182	1	springer	springer	NOUN
ejpam-6214	182	2	,	,	PUNCT
ejpam-6214	182	3	1999	1999	NUM
ejpam-6214	182	4	.	.	PUNCT
ejpam-6214	183	1	[	[	X
ejpam-6214	183	2	7	7	X
ejpam-6214	183	3	]	]	X
ejpam-6214	183	4	marc	marc	PROPN
ejpam-6214	183	5	hindry	hindry	PROPN
ejpam-6214	183	6	and	and	CCONJ
ejpam-6214	183	7	joseph	joseph	PROPN
ejpam-6214	183	8	h.	h.	PROPN
ejpam-6214	183	9	silverman	silverman	PROPN
ejpam-6214	183	10	.	.	PUNCT
ejpam-6214	184	1	diophantine	diophantine	PROPN
ejpam-6214	184	2	geometry	geometry	NOUN
ejpam-6214	184	3	:	:	PUNCT
ejpam-6214	184	4	an	an	DET
ejpam-6214	184	5	introduction	introduction	NOUN
ejpam-6214	184	6	.	.	PUNCT
ejpam-6214	185	1	springer	springer	NOUN
ejpam-6214	185	2	,	,	PUNCT
ejpam-6214	185	3	2000	2000	NUM
ejpam-6214	185	4	.	.	PUNCT
ejpam-6214	186	1	[	[	X
ejpam-6214	186	2	8	8	NUM
ejpam-6214	186	3	]	]	X
ejpam-6214	186	4	andrzej	andrzej	PROPN
ejpam-6214	186	5	schinzel	schinzel	PROPN
ejpam-6214	186	6	.	.	PUNCT
ejpam-6214	187	1	polynomials	polynomial	NOUN
ejpam-6214	187	2	with	with	ADP
ejpam-6214	187	3	special	special	ADJ
ejpam-6214	187	4	regard	regard	NOUN
ejpam-6214	187	5	to	to	ADP
ejpam-6214	187	6	reducibility	reducibility	PROPN
ejpam-6214	187	7	.	.	PUNCT
ejpam-6214	188	1	cambridge	cambridge	PROPN
ejpam-6214	188	2	university	university	PROPN
ejpam-6214	188	3	press	press	NOUN
ejpam-6214	188	4	,	,	PUNCT
ejpam-6214	188	5	2000	2000	NUM
ejpam-6214	188	6	.	.	PUNCT
ejpam-6214	189	1	[	[	X
ejpam-6214	189	2	9	9	NUM
ejpam-6214	189	3	]	]	PUNCT
ejpam-6214	189	4	michael	michael	PROPN
ejpam-6214	189	5	filaseta	filaseta	PROPN
ejpam-6214	189	6	.	.	PUNCT
ejpam-6214	190	1	the	the	DET
ejpam-6214	190	2	irreducibility	irreducibility	NOUN
ejpam-6214	190	3	of	of	ADP
ejpam-6214	190	4	all	all	DET
ejpam-6214	190	5	but	but	ADV
ejpam-6214	190	6	finitely	finitely	ADV
ejpam-6214	190	7	many	many	ADJ
ejpam-6214	190	8	bessel	bessel	ADJ
ejpam-6214	190	9	polynomials	polynomial	NOUN
ejpam-6214	190	10	.	.	PUNCT
ejpam-6214	191	1	acta	acta	PROPN
ejpam-6214	191	2	mathematica	mathematica	PROPN
ejpam-6214	191	3	hungarica	hungarica	PROPN
ejpam-6214	191	4	,	,	PUNCT
ejpam-6214	191	5	120(1	120(1	NUM
ejpam-6214	191	6	-	-	SYM
ejpam-6214	191	7	2):137–152	2):137–152	NUM
ejpam-6214	191	8	,	,	PUNCT
ejpam-6214	191	9	2008	2008	NUM
ejpam-6214	191	10	.	.	PUNCT