id	sid	tid	token	lemma	pos
ejpam-4086	1	1	european	european	PROPN
ejpam-4086	1	2	journal	journal	PROPN
ejpam-4086	1	3	of	of	ADP
ejpam-4086	1	4	pure	pure	ADJ
ejpam-4086	1	5	and	and	CCONJ
ejpam-4086	1	6	applied	apply	VERB
ejpam-4086	1	7	mathematics	mathematic	NOUN
ejpam-4086	1	8	vol	vol	NOUN
ejpam-4086	1	9	.	.	PUNCT
ejpam-4086	2	1	14	14	NUM
ejpam-4086	2	2	,	,	PUNCT
ejpam-4086	2	3	no	no	INTJ
ejpam-4086	2	4	.	.	NOUN
ejpam-4086	2	5	4	4	NUM
ejpam-4086	2	6	,	,	PUNCT
ejpam-4086	2	7	2021	2021	NUM
ejpam-4086	2	8	,	,	PUNCT
ejpam-4086	2	9	1108	1108	NUM
ejpam-4086	2	10	-	-	SYM
ejpam-4086	2	11	1111	1111	NUM
ejpam-4086	2	12	issn	issn	PROPN
ejpam-4086	2	13	1307	1307	NUM
ejpam-4086	2	14	-	-	SYM
ejpam-4086	2	15	5543	5543	NUM
ejpam-4086	2	16	–	–	PUNCT
ejpam-4086	2	17	ejpam.com	ejpam.com	X
ejpam-4086	2	18	published	publish	VERB
ejpam-4086	2	19	by	by	ADP
ejpam-4086	2	20	new	new	PROPN
ejpam-4086	2	21	york	york	PROPN
ejpam-4086	2	22	business	business	PROPN
ejpam-4086	2	23	global	global	ADJ
ejpam-4086	2	24	on	on	ADP
ejpam-4086	2	25	eigenvectors	eigenvector	NOUN
ejpam-4086	2	26	of	of	ADP
ejpam-4086	2	27	nilpotent	nilpotent	ADJ
ejpam-4086	2	28	lie	lie	NOUN
ejpam-4086	2	29	algebras	algebra	NOUN
ejpam-4086	2	30	of	of	ADP
ejpam-4086	2	31	linear	linear	PROPN
ejpam-4086	2	32	operators	operator	NOUN
ejpam-4086	3	1	morris	morris	PROPN
ejpam-4086	3	2	w.	w.	PROPN
ejpam-4086	3	3	hirsch1,2,∗	hirsch1,2,∗	PROPN
ejpam-4086	3	4	,	,	PUNCT
ejpam-4086	3	5	joel	joel	PROPN
ejpam-4086	3	6	w.	w.	PROPN
ejpam-4086	3	7	robbin2	robbin2	PROPN
ejpam-4086	3	8	1	1	NUM
ejpam-4086	3	9	department	department	NOUN
ejpam-4086	3	10	of	of	ADP
ejpam-4086	3	11	mathematics	mathematics	PROPN
ejpam-4086	3	12	,	,	PUNCT
ejpam-4086	3	13	university	university	PROPN
ejpam-4086	3	14	of	of	ADP
ejpam-4086	3	15	california	california	PROPN
ejpam-4086	3	16	at	at	ADP
ejpam-4086	3	17	berkeley	berkeley	PROPN
ejpam-4086	3	18	,	,	PUNCT
ejpam-4086	3	19	berkeley	berkeley	PROPN
ejpam-4086	3	20	,	,	PUNCT
ejpam-4086	3	21	ca	ca	NOUN
ejpam-4086	3	22	94720384	94720384	NUM
ejpam-4086	3	23	,	,	PUNCT
ejpam-4086	3	24	usa	usa	PROPN
ejpam-4086	3	25	2	2	NUM
ejpam-4086	3	26	department	department	NOUN
ejpam-4086	3	27	of	of	ADP
ejpam-4086	3	28	mathematics	mathematic	NOUN
ejpam-4086	3	29	,	,	PUNCT
ejpam-4086	3	30	university	university	NOUN
ejpam-4086	3	31	of	of	ADP
ejpam-4086	3	32	wisconsin	wisconsin	PROPN
ejpam-4086	3	33	at	at	ADP
ejpam-4086	3	34	madison	madison	PROPN
ejpam-4086	3	35	,	,	PUNCT
ejpam-4086	3	36	wi	wi	PROPN
ejpam-4086	3	37	53706	53706	NUM
ejpam-4086	3	38	,	,	PUNCT
ejpam-4086	3	39	usa	usa	PROPN
ejpam-4086	3	40	abstract	abstract	NOUN
ejpam-4086	3	41	.	.	PUNCT
ejpam-4086	4	1	we	we	PRON
ejpam-4086	4	2	give	give	VERB
ejpam-4086	4	3	a	a	DET
ejpam-4086	4	4	condition	condition	NOUN
ejpam-4086	4	5	ensuring	ensure	VERB
ejpam-4086	4	6	that	that	SCONJ
ejpam-4086	4	7	the	the	DET
ejpam-4086	4	8	operators	operator	NOUN
ejpam-4086	4	9	in	in	ADP
ejpam-4086	4	10	a	a	DET
ejpam-4086	4	11	nilpotent	nilpotent	ADJ
ejpam-4086	4	12	lie	lie	NOUN
ejpam-4086	4	13	algebra	algebra	NOUN
ejpam-4086	4	14	of	of	ADP
ejpam-4086	4	15	linear	linear	PROPN
ejpam-4086	4	16	operators	operator	NOUN
ejpam-4086	4	17	on	on	ADP
ejpam-4086	4	18	a	a	DET
ejpam-4086	4	19	finite	finite	ADJ
ejpam-4086	4	20	dimensional	dimensional	ADJ
ejpam-4086	4	21	vector	vector	NOUN
ejpam-4086	4	22	space	space	NOUN
ejpam-4086	4	23	have	have	VERB
ejpam-4086	4	24	a	a	DET
ejpam-4086	4	25	common	common	ADJ
ejpam-4086	4	26	eigenvector	eigenvector	NOUN
ejpam-4086	4	27	.	.	PROPN
ejpam-4086	4	28	2020	2020	NUM
ejpam-4086	4	29	mathematics	mathematics	PROPN
ejpam-4086	4	30	subject	subject	NOUN
ejpam-4086	4	31	classifications	classification	NOUN
ejpam-4086	4	32	:	:	PUNCT
ejpam-4086	4	33	22e25	22e25	NUM
ejpam-4086	4	34	,	,	PUNCT
ejpam-4086	4	35	22e60	22e60	NUM
ejpam-4086	4	36	,	,	PUNCT
ejpam-4086	4	37	47c05	47c05	NUM
ejpam-4086	4	38	key	key	ADJ
ejpam-4086	4	39	words	word	NOUN
ejpam-4086	4	40	and	and	CCONJ
ejpam-4086	4	41	phrases	phrase	NOUN
ejpam-4086	4	42	:	:	PUNCT
ejpam-4086	4	43	nilpotent	nilpotent	ADJ
ejpam-4086	4	44	,	,	PUNCT
ejpam-4086	4	45	lie	lie	NOUN
ejpam-4086	4	46	algebras	algebra	NOUN
ejpam-4086	4	47	of	of	ADP
ejpam-4086	4	48	lie	lie	NOUN
ejpam-4086	4	49	groups	group	NOUN
ejpam-4086	4	50	,	,	PUNCT
ejpam-4086	4	51	linear	linear	PROPN
ejpam-4086	4	52	operators	operator	NOUN
ejpam-4086	4	53	in	in	ADP
ejpam-4086	4	54	algebras	algebras	PROPN
ejpam-4086	4	55	1	1	NUM
ejpam-4086	4	56	.	.	PUNCT
ejpam-4086	4	57	introduction	introduction	NOUN
ejpam-4086	4	58	throughout	throughout	ADP
ejpam-4086	4	59	this	this	DET
ejpam-4086	4	60	paper	paper	NOUN
ejpam-4086	4	61	v	v	NOUN
ejpam-4086	4	62	is	be	AUX
ejpam-4086	4	63	a	a	DET
ejpam-4086	4	64	vector	vector	NOUN
ejpam-4086	4	65	space	space	NOUN
ejpam-4086	4	66	of	of	ADP
ejpam-4086	4	67	positive	positive	ADJ
ejpam-4086	4	68	dimension	dimension	NOUN
ejpam-4086	4	69	over	over	ADP
ejpam-4086	4	70	a	a	DET
ejpam-4086	4	71	field	field	NOUN
ejpam-4086	4	72	f	f	NOUN
ejpam-4086	5	1	and	and	CCONJ
ejpam-4086	5	2	≫	≫	PROPN
ejpam-4086	5	3	is	be	AUX
ejpam-4086	5	4	a	a	DET
ejpam-4086	5	5	nilpotent	nilpotent	ADJ
ejpam-4086	5	6	lie	lie	NOUN
ejpam-4086	5	7	algebra	algebra	NOUN
ejpam-4086	5	8	over	over	ADP
ejpam-4086	5	9	f	f	PROPN
ejpam-4086	5	10	of	of	ADP
ejpam-4086	5	11	linear	linear	PROPN
ejpam-4086	5	12	operators	operator	NOUN
ejpam-4086	5	13	on	on	ADP
ejpam-4086	5	14	v	v	NUM
ejpam-4086	5	15	.	.	PUNCT
ejpam-4086	6	1	an	an	DET
ejpam-4086	6	2	element	element	NOUN
ejpam-4086	6	3	u	u	PROPN
ejpam-4086	6	4	∈	∈	PROPN
ejpam-4086	6	5	v	v	NOUN
ejpam-4086	6	6	is	be	AUX
ejpam-4086	6	7	an	an	DET
ejpam-4086	6	8	eigenvector	eigenvector	NOUN
ejpam-4086	6	9	for	for	ADP
ejpam-4086	6	10	s	s	PROPN
ejpam-4086	6	11	⊂≫	⊂≫	PROPN
ejpam-4086	6	12	if	if	SCONJ
ejpam-4086	6	13	u	u	NOUN
ejpam-4086	6	14	is	be	AUX
ejpam-4086	6	15	an	an	DET
ejpam-4086	6	16	eigenvector	eigenvector	NOUN
ejpam-4086	6	17	for	for	ADP
ejpam-4086	6	18	every	every	DET
ejpam-4086	6	19	operator	operator	NOUN
ejpam-4086	6	20	in	in	ADP
ejpam-4086	6	21	s.	s.	PROPN
ejpam-4086	6	22	if	if	SCONJ
ejpam-4086	6	23	v	v	PRON
ejpam-4086	6	24	has	have	VERB
ejpam-4086	6	25	a	a	DET
ejpam-4086	6	26	basis	basis	NOUN
ejpam-4086	6	27	(	(	PUNCT
ejpam-4086	6	28	e1	e1	NOUN
ejpam-4086	6	29	,	,	PUNCT
ejpam-4086	6	30	.	.	PUNCT
ejpam-4086	6	31	.	.	PUNCT
ejpam-4086	7	1	.	.	PUNCT
ejpam-4086	8	1	,	,	PUNCT
ejpam-4086	8	2	en	en	X
ejpam-4086	8	3	)	)	PUNCT
ejpam-4086	8	4	representing	represent	VERB
ejpam-4086	8	5	each	each	DET
ejpam-4086	8	6	element	element	NOUN
ejpam-4086	8	7	of	of	ADP
ejpam-4086	8	8	≫	≫	PROPN
ejpam-4086	8	9	by	by	ADP
ejpam-4086	8	10	an	an	DET
ejpam-4086	8	11	upper	upper	ADJ
ejpam-4086	8	12	triangular	triangular	NOUN
ejpam-4086	8	13	matrix	matrix	NOUN
ejpam-4086	8	14	,	,	PUNCT
ejpam-4086	8	15	then	then	ADV
ejpam-4086	8	16	e1	e1	NOUN
ejpam-4086	8	17	is	be	AUX
ejpam-4086	8	18	an	an	DET
ejpam-4086	8	19	eigenvector	eigenvector	NOUN
ejpam-4086	8	20	for≫.	for≫.	PROPN
ejpam-4086	8	21	such	such	DET
ejpam-4086	8	22	a	a	DET
ejpam-4086	8	23	basis	basis	NOUN
ejpam-4086	8	24	exists	exist	VERB
ejpam-4086	8	25	when	when	SCONJ
ejpam-4086	8	26	f	f	PROPN
ejpam-4086	8	27	is	be	AUX
ejpam-4086	8	28	algebraically	algebraically	ADV
ejpam-4086	8	29	closed	close	VERB
ejpam-4086	8	30	and≫	and≫	ADJ
ejpam-4086	8	31	is	be	AUX
ejpam-4086	8	32	solvable	solvable	ADJ
ejpam-4086	8	33	(	(	PUNCT
ejpam-4086	8	34	lie	lie	VERB
ejpam-4086	8	35	’s	’s	PART
ejpam-4086	8	36	theorem	theorem	NOUN
ejpam-4086	8	37	)	)	PUNCT
ejpam-4086	8	38	,	,	PUNCT
ejpam-4086	8	39	and	and	CCONJ
ejpam-4086	8	40	also	also	ADV
ejpam-4086	8	41	when	when	SCONJ
ejpam-4086	8	42	every	every	DET
ejpam-4086	8	43	element	element	NOUN
ejpam-4086	8	44	of	of	ADP
ejpam-4086	8	45	≫	≫	PROPN
ejpam-4086	8	46	is	be	AUX
ejpam-4086	8	47	a	a	DET
ejpam-4086	8	48	nilpotent	nilpotent	ADJ
ejpam-4086	8	49	operator	operator	NOUN
ejpam-4086	8	50	(	(	PUNCT
ejpam-4086	8	51	engel	engel	PROPN
ejpam-4086	8	52	’s	’s	PART
ejpam-4086	8	53	theorem	theorem	PROPN
ejpam-4086	8	54	)	)	PUNCT
ejpam-4086	8	55	.	.	PUNCT
ejpam-4086	9	1	our	our	PRON
ejpam-4086	9	2	results	result	NOUN
ejpam-4086	9	3	are	be	AUX
ejpam-4086	9	4	further	further	ADJ
ejpam-4086	9	5	conditions	condition	NOUN
ejpam-4086	9	6	guaranteeing	guarantee	VERB
ejpam-4086	9	7	existence	existence	NOUN
ejpam-4086	9	8	of	of	ADP
ejpam-4086	9	9	eigenvectors	eigenvector	NOUN
ejpam-4086	9	10	.	.	PUNCT
ejpam-4086	10	1	the	the	DET
ejpam-4086	10	2	minimal	minimal	ADJ
ejpam-4086	10	3	and	and	CCONJ
ejpam-4086	10	4	characteristic	characteristic	ADJ
ejpam-4086	10	5	polynomials	polynomial	NOUN
ejpam-4086	10	6	of	of	ADP
ejpam-4086	10	7	a	a	DET
ejpam-4086	10	8	linear	linear	ADJ
ejpam-4086	10	9	operator	operator	NOUN
ejpam-4086	10	10	a	a	PRON
ejpam-4086	10	11	on	on	ADP
ejpam-4086	10	12	v	v	NUM
ejpam-4086	10	13	are	be	AUX
ejpam-4086	10	14	denoted	denote	VERB
ejpam-4086	10	15	respectively	respectively	ADV
ejpam-4086	10	16	by	by	ADP
ejpam-4086	10	17	πa	πa	PROPN
ejpam-4086	10	18	,	,	PUNCT
ejpam-4086	10	19	µa	µa	ADP
ejpam-4086	10	20	∈	∈	PROPN
ejpam-4086	10	21	f	f	PROPN
ejpam-4086	11	1	[	[	X
ejpam-4086	11	2	t	t	X
ejpam-4086	11	3	]	]	X
ejpam-4086	11	4	=	=	PUNCT
ejpam-4086	11	5	the	the	DET
ejpam-4086	11	6	ring	ring	NOUN
ejpam-4086	11	7	of	of	ADP
ejpam-4086	11	8	polynomials	polynomial	NOUN
ejpam-4086	11	9	over	over	ADP
ejpam-4086	11	10	f	f	PROPN
ejpam-4086	11	11	.	.	PUNCT
ejpam-4086	12	1	the	the	DET
ejpam-4086	12	2	cardinality	cardinality	NOUN
ejpam-4086	12	3	of	of	ADP
ejpam-4086	12	4	a	a	DET
ejpam-4086	12	5	set	set	NOUN
ejpam-4086	12	6	s	s	PART
ejpam-4086	12	7	is	be	AUX
ejpam-4086	12	8	written	write	VERB
ejpam-4086	12	9	#	#	VERB
ejpam-4086	12	10	s.	s.	PROPN
ejpam-4086	12	11	let	let	VERB
ejpam-4086	12	12	k	k	PROPN
ejpam-4086	12	13	be	be	AUX
ejpam-4086	12	14	a	a	DET
ejpam-4086	12	15	galois	galois	NOUN
ejpam-4086	12	16	extension	extension	NOUN
ejpam-4086	12	17	field	field	NOUN
ejpam-4086	12	18	of	of	ADP
ejpam-4086	12	19	f	f	PROPN
ejpam-4086	12	20	of	of	ADP
ejpam-4086	12	21	degree	degree	NOUN
ejpam-4086	13	1	d	d	NOUN
ejpam-4086	13	2	:	:	PUNCT
ejpam-4086	13	3	=	=	SYM
ejpam-4086	14	1	[	[	X
ejpam-4086	14	2	k	k	X
ejpam-4086	14	3	:	:	PUNCT
ejpam-4086	14	4	f	f	X
ejpam-4086	14	5	]	]	X
ejpam-4086	14	6	,	,	PUNCT
ejpam-4086	14	7	and	and	CCONJ
ejpam-4086	14	8	define	define	VERB
ejpam-4086	14	9	m	m	NOUN
ejpam-4086	14	10	⊂	⊂	PROPN
ejpam-4086	14	11	to	to	PART
ejpam-4086	14	12	be	be	AUX
ejpam-4086	14	13	the	the	DET
ejpam-4086	14	14	additive	additive	ADJ
ejpam-4086	14	15	monoid	monoid	NOUN
ejpam-4086	14	16	generated	generate	VERB
ejpam-4086	14	17	by	by	ADP
ejpam-4086	14	18	zero	zero	NUM
ejpam-4086	14	19	and	and	CCONJ
ejpam-4086	14	20	the	the	DET
ejpam-4086	14	21	prime	prime	ADJ
ejpam-4086	14	22	divisors	divisor	NOUN
ejpam-4086	14	23	d.	d.	PROPN
ejpam-4086	14	24	consider	consider	VERB
ejpam-4086	14	25	the	the	DET
ejpam-4086	14	26	conditions	condition	NOUN
ejpam-4086	14	27	:	:	PUNCT
ejpam-4086	14	28	(	(	PUNCT
ejpam-4086	14	29	c1	c1	NOUN
ejpam-4086	14	30	)	)	PUNCT
ejpam-4086	14	31	µa	µa	NOUN
ejpam-4086	14	32	splits	split	VERB
ejpam-4086	14	33	in	in	ADP
ejpam-4086	14	34	k	k	PROPN
ejpam-4086	14	35	for	for	ADP
ejpam-4086	14	36	every	every	DET
ejpam-4086	14	37	a	a	DET
ejpam-4086	14	38	∈≫	∈≫	NOUN
ejpam-4086	14	39	(	(	PUNCT
ejpam-4086	14	40	c2	c2	PROPN
ejpam-4086	14	41	)	)	PUNCT
ejpam-4086	14	42	dimv	dimv	NOUN
ejpam-4086	14	43	/∈	/∈	PUNCT
ejpam-4086	15	1	m	m	AUX
ejpam-4086	15	2	∗corresponding	∗corresponde	VERB
ejpam-4086	15	3	author	author	NOUN
ejpam-4086	15	4	.	.	PUNCT
ejpam-4086	16	1	doi	doi	NOUN
ejpam-4086	16	2	:	:	PUNCT
ejpam-4086	16	3	https://doi.org/10.29020/nybg.ejpam.v14i4.4086	https://doi.org/10.29020/nybg.ejpam.v14i4.4086	ADJ
ejpam-4086	16	4	email	email	NOUN
ejpam-4086	16	5	addresses	address	VERB
ejpam-4086	16	6	:	:	PUNCT
ejpam-4086	17	1	mwhirsch@chorus.net	mwhirsch@chorus.net	NOUN
ejpam-4086	17	2	(	(	PUNCT
ejpam-4086	17	3	m.	m.	PROPN
ejpam-4086	17	4	w.	w.	PROPN
ejpam-4086	17	5	hirsch	hirsch	PROPN
ejpam-4086	17	6	)	)	PUNCT
ejpam-4086	17	7	,	,	PUNCT
ejpam-4086	17	8	robbin@math.wisc.edu	robbin@math.wisc.edu	PROPN
ejpam-4086	17	9	(	(	PUNCT
ejpam-4086	17	10	j.	j.	PROPN
ejpam-4086	17	11	w.	w.	PROPN
ejpam-4086	17	12	robbin	robbin	PROPN
ejpam-4086	17	13	)	)	PUNCT
ejpam-4086	17	14	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4086	17	15	1108	1108	NUM
ejpam-4086	18	1	©	©	PROPN
ejpam-4086	18	2	2021	2021	NUM
ejpam-4086	18	3	ejpam	ejpam	VERB
ejpam-4086	18	4	all	all	DET
ejpam-4086	18	5	rights	right	NOUN
ejpam-4086	18	6	reserved	reserve	VERB
ejpam-4086	18	7	.	.	PUNCT
ejpam-4086	19	1	m.	m.	PROPN
ejpam-4086	19	2	w.	w.	PROPN
ejpam-4086	19	3	hirsch	hirsch	PROPN
ejpam-4086	19	4	,	,	PUNCT
ejpam-4086	19	5	j.	j.	PROPN
ejpam-4086	19	6	w.	w.	PROPN
ejpam-4086	19	7	robbin	robbin	PROPN
ejpam-4086	19	8	/	/	SYM
ejpam-4086	19	9	eur	eur	PROPN
ejpam-4086	19	10	.	.	PUNCT
ejpam-4086	20	1	j.	j.	PROPN
ejpam-4086	20	2	pure	pure	PROPN
ejpam-4086	20	3	appl	appl	PROPN
ejpam-4086	20	4	.	.	PROPN
ejpam-4086	20	5	math	math	PROPN
ejpam-4086	20	6	,	,	PUNCT
ejpam-4086	20	7	14	14	NUM
ejpam-4086	20	8	(	(	PUNCT
ejpam-4086	20	9	4	4	NUM
ejpam-4086	20	10	)	)	PUNCT
ejpam-4086	20	11	(	(	PUNCT
ejpam-4086	20	12	2021	2021	NUM
ejpam-4086	20	13	)	)	PUNCT
ejpam-4086	20	14	,	,	PUNCT
ejpam-4086	20	15	1108	1108	NUM
ejpam-4086	20	16	-	-	SYM
ejpam-4086	20	17	1111	1111	NUM
ejpam-4086	20	18	1109	1109	NUM
ejpam-4086	20	19	2	2	NUM
ejpam-4086	20	20	.	.	PUNCT
ejpam-4086	20	21	results	result	VERB
ejpam-4086	20	22	our	our	PRON
ejpam-4086	20	23	main	main	ADJ
ejpam-4086	20	24	result	result	NOUN
ejpam-4086	20	25	is	be	AUX
ejpam-4086	20	26	:	:	PUNCT
ejpam-4086	20	27	theorem	theorem	ADJ
ejpam-4086	20	28	1	1	NUM
ejpam-4086	20	29	.	.	PUNCT
ejpam-4086	20	30	if(c1	if(c1	NOUN
ejpam-4086	21	1	)	)	PUNCT
ejpam-4086	21	2	and(c2	and(c2	PROPN
ejpam-4086	21	3	)	)	PUNCT
ejpam-4086	21	4	hold	hold	VERB
ejpam-4086	21	5	then	then	ADV
ejpam-4086	21	6	≫	≫	PROPN
ejpam-4086	21	7	has	have	VERB
ejpam-4086	21	8	an	an	DET
ejpam-4086	21	9	eigenvector	eigenvector	NOUN
ejpam-4086	21	10	.	.	PUNCT
ejpam-4086	22	1	the	the	DET
ejpam-4086	22	2	proof	proof	NOUN
ejpam-4086	22	3	is	be	AUX
ejpam-4086	22	4	preceded	precede	VERB
ejpam-4086	22	5	by	by	ADP
ejpam-4086	22	6	some	some	DET
ejpam-4086	22	7	applications	application	NOUN
ejpam-4086	22	8	.	.	PUNCT
ejpam-4086	23	1	when(c1	when(c1	NOUN
ejpam-4086	23	2	)	)	PUNCT
ejpam-4086	23	3	holds	hold	NOUN
ejpam-4086	23	4	,	,	PUNCT
ejpam-4086	23	5	theorem	theorem	ADJ
ejpam-4086	23	6	1	1	NUM
ejpam-4086	23	7	shows	show	VERB
ejpam-4086	23	8	that	that	SCONJ
ejpam-4086	23	9	there	there	PRON
ejpam-4086	23	10	is	be	VERB
ejpam-4086	23	11	an	an	DET
ejpam-4086	23	12	eigenvector	eigenvector	NOUN
ejpam-4086	23	13	in	in	ADP
ejpam-4086	23	14	every	every	DET
ejpam-4086	23	15	invariant	invariant	ADJ
ejpam-4086	23	16	subspace	subspace	NOUN
ejpam-4086	23	17	whose	whose	DET
ejpam-4086	23	18	dimension	dimension	NOUN
ejpam-4086	23	19	is	be	AUX
ejpam-4086	23	20	not	not	PART
ejpam-4086	23	21	in	in	ADP
ejpam-4086	23	22	m.	m.	NOUN
ejpam-4086	23	23	this	this	PRON
ejpam-4086	23	24	is	be	AUX
ejpam-4086	23	25	exploited	exploit	VERB
ejpam-4086	23	26	to	to	PART
ejpam-4086	23	27	yield	yield	VERB
ejpam-4086	23	28	the	the	DET
ejpam-4086	23	29	following	follow	VERB
ejpam-4086	23	30	two	two	NUM
ejpam-4086	23	31	results	result	NOUN
ejpam-4086	23	32	:	:	PUNCT
ejpam-4086	23	33	corollary	corollary	ADJ
ejpam-4086	23	34	1	1	X
ejpam-4086	23	35	.	.	PUNCT
ejpam-4086	24	1	if	if	SCONJ
ejpam-4086	24	2	a	a	DET
ejpam-4086	24	3	nilpotent	nilpotent	ADJ
ejpam-4086	24	4	lie	lie	NOUN
ejpam-4086	24	5	algebra	algebra	NOUN
ejpam-4086	24	6	of	of	ADP
ejpam-4086	24	7	linear	linear	PROPN
ejpam-4086	24	8	operators	operator	NOUN
ejpam-4086	24	9	on	on	ADP
ejpam-4086	24	10	n	n	CCONJ
ejpam-4086	24	11	does	do	AUX
ejpam-4086	24	12	not	not	PART
ejpam-4086	24	13	have	have	VERB
ejpam-4086	24	14	an	an	DET
ejpam-4086	24	15	eigenvector	eigenvector	NOUN
ejpam-4086	24	16	,	,	PUNCT
ejpam-4086	24	17	every	every	DET
ejpam-4086	24	18	nontrivial	nontrivial	ADJ
ejpam-4086	24	19	invariant	invariant	ADJ
ejpam-4086	24	20	subspace	subspace	NOUN
ejpam-4086	24	21	has	have	VERB
ejpam-4086	24	22	odd	odd	ADJ
ejpam-4086	24	23	dimension	dimension	NOUN
ejpam-4086	24	24	.	.	PUNCT
ejpam-4086	25	1	proof	proof	NOUN
ejpam-4086	25	2	.	.	PUNCT
ejpam-4086	26	1	when	when	SCONJ
ejpam-4086	26	2	f	f	PROPN
ejpam-4086	26	3	is	be	AUX
ejpam-4086	26	4	the	the	DET
ejpam-4086	26	5	real	real	ADJ
ejpam-4086	26	6	field	field	NOUN
ejpam-4086	26	7	and	and	CCONJ
ejpam-4086	26	8	k	k	PROPN
ejpam-4086	26	9	is	be	AUX
ejpam-4086	26	10	the	the	DET
ejpam-4086	26	11	complex	complex	ADJ
ejpam-4086	26	12	field	field	NOUN
ejpam-4086	26	13	,	,	PUNCT
ejpam-4086	26	14	m	m	VERB
ejpam-4086	26	15	consists	consist	VERB
ejpam-4086	26	16	of	of	ADP
ejpam-4086	26	17	the	the	DET
ejpam-4086	26	18	positive	positive	ADJ
ejpam-4086	26	19	even	even	ADJ
ejpam-4086	26	20	integers	integer	NOUN
ejpam-4086	26	21	.	.	PUNCT
ejpam-4086	27	1	corollary	corollary	ADJ
ejpam-4086	27	2	2	2	NUM
ejpam-4086	27	3	.	.	PUNCT
ejpam-4086	27	4	let(c1	let(c1	NOUN
ejpam-4086	27	5	)	)	PUNCT
ejpam-4086	27	6	hold	hold	NOUN
ejpam-4086	27	7	.	.	PUNCT
ejpam-4086	28	1	assume	assume	VERB
ejpam-4086	28	2	≫	≫	PROPN
ejpam-4086	28	3	preserves	preserve	VERB
ejpam-4086	28	4	a	a	DET
ejpam-4086	28	5	direct	direct	ADJ
ejpam-4086	28	6	sum	sum	NOUN
ejpam-4086	28	7	decomposition	decomposition	NOUN
ejpam-4086	28	8	v	v	ADP
ejpam-4086	28	9	=	=	SYM
ejpam-4086	28	10	⊕iwi	⊕iwi	NUM
ejpam-4086	28	11	,	,	PUNCT
ejpam-4086	28	12	and	and	CCONJ
ejpam-4086	28	13	let	let	VERB
ejpam-4086	28	14	d	d	PRON
ejpam-4086	28	15	⊂	⊂	PROPN
ejpam-4086	28	16	denote	denote	VERB
ejpam-4086	28	17	the	the	DET
ejpam-4086	28	18	set	set	NOUN
ejpam-4086	28	19	of	of	ADP
ejpam-4086	28	20	dimensions	dimension	NOUN
ejpam-4086	28	21	of	of	ADP
ejpam-4086	28	22	the	the	DET
ejpam-4086	28	23	subspaces	subspace	NOUN
ejpam-4086	28	24	wi	wi	PROPN
ejpam-4086	28	25	.	.	PUNCT
ejpam-4086	29	1	(	(	PUNCT
ejpam-4086	29	2	i	i	NOUN
ejpam-4086	29	3	)	)	PUNCT
ejpam-4086	29	4	if	if	SCONJ
ejpam-4086	29	5	≫	≫	PROPN
ejpam-4086	29	6	does	do	AUX
ejpam-4086	29	7	not	not	PART
ejpam-4086	29	8	have	have	VERB
ejpam-4086	29	9	an	an	DET
ejpam-4086	29	10	eigenvector	eigenvector	NOUN
ejpam-4086	29	11	then	then	ADV
ejpam-4086	29	12	d	d	PROPN
ejpam-4086	29	13	⊂	⊂	PROPN
ejpam-4086	29	14	m.	m.	PROPN
ejpam-4086	29	15	(	(	PUNCT
ejpam-4086	29	16	ii	ii	NOUN
ejpam-4086	29	17	)	)	PUNCT
ejpam-4086	29	18	if	if	SCONJ
ejpam-4086	29	19	v	v	NOUN
ejpam-4086	29	20	′	′	NUM
ejpam-4086	30	1	⊂	⊂	PROPN
ejpam-4086	30	2	v	v	X
ejpam-4086	30	3	is	be	AUX
ejpam-4086	30	4	a	a	DET
ejpam-4086	30	5	maximal	maximal	ADJ
ejpam-4086	30	6	subspace	subspace	NOUN
ejpam-4086	30	7	spanned	span	VERB
ejpam-4086	30	8	by	by	ADP
ejpam-4086	30	9	eigenvectors	eigenvector	NOUN
ejpam-4086	30	10	of	of	ADP
ejpam-4086	30	11	≫	≫	PROPN
ejpam-4086	30	12	then	then	ADV
ejpam-4086	30	13	dim(v	dim(v	PROPN
ejpam-4086	30	14	′	′	NOUN
ejpam-4086	30	15	)	)	PUNCT
ejpam-4086	30	16	≥	≥	NOUN
ejpam-4086	30	17	#	#	SYM
ejpam-4086	30	18	{	{	PUNCT
ejpam-4086	30	19	d	d	NOUN
ejpam-4086	30	20	\m	\m	NOUN
ejpam-4086	30	21	}	}	PUNCT
ejpam-4086	30	22	.	.	PUNCT
ejpam-4086	31	1	proof	proof	NOUN
ejpam-4086	31	2	.	.	PUNCT
ejpam-4086	32	1	assertion	assertion	NOUN
ejpam-4086	32	2	(	(	PUNCT
ejpam-4086	32	3	i	i	NOUN
ejpam-4086	32	4	)	)	PUNCT
ejpam-4086	32	5	follows	follow	VERB
ejpam-4086	32	6	from	from	ADP
ejpam-4086	32	7	theorem	theorem	ADJ
ejpam-4086	32	8	1	1	NUM
ejpam-4086	32	9	.	.	PUNCT
ejpam-4086	32	10	to	to	PART
ejpam-4086	32	11	prove	prove	VERB
ejpam-4086	32	12	(	(	PUNCT
ejpam-4086	32	13	ii	ii	NOUN
ejpam-4086	32	14	)	)	PUNCT
ejpam-4086	32	15	order	order	NOUN
ejpam-4086	32	16	the	the	DET
ejpam-4086	32	17	wi	wi	PROPN
ejpam-4086	32	18	so	so	SCONJ
ejpam-4086	32	19	that	that	DET
ejpam-4086	32	20	w1	w1	NOUN
ejpam-4086	32	21	,	,	PUNCT
ejpam-4086	32	22	.	.	PUNCT
ejpam-4086	32	23	.	.	PUNCT
ejpam-4086	33	1	.	.	PUNCT
ejpam-4086	34	1	,	,	PUNCT
ejpam-4086	34	2	wm	wm	PROPN
ejpam-4086	34	3	are	be	AUX
ejpam-4086	34	4	the	the	DET
ejpam-4086	34	5	only	only	ADJ
ejpam-4086	34	6	summands	summand	NOUN
ejpam-4086	34	7	whose	whose	DET
ejpam-4086	34	8	dimensions	dimension	NOUN
ejpam-4086	34	9	are	be	AUX
ejpam-4086	34	10	not	not	PART
ejpam-4086	34	11	in	in	ADP
ejpam-4086	34	12	m.	m.	NOUN
ejpam-4086	34	13	for	for	ADP
ejpam-4086	34	14	each	each	DET
ejpam-4086	34	15	j	j	PROPN
ejpam-4086	34	16	∈	∈	PROPN
ejpam-4086	34	17	{	{	PUNCT
ejpam-4086	34	18	1	1	NUM
ejpam-4086	34	19	,	,	PUNCT
ejpam-4086	34	20	.	.	PUNCT
ejpam-4086	34	21	.	.	PUNCT
ejpam-4086	35	1	.	.	PUNCT
ejpam-4086	36	1	,	,	PUNCT
ejpam-4086	36	2	m	m	AUX
ejpam-4086	36	3	}	}	PUNCT
ejpam-4086	36	4	we	we	PRON
ejpam-4086	36	5	choose	choose	VERB
ejpam-4086	36	6	an	an	DET
ejpam-4086	36	7	eigenvector	eigenvector	NOUN
ejpam-4086	36	8	ej	ej	PROPN
ejpam-4086	36	9	∈	∈	PROPN
ejpam-4086	36	10	wj	wj	X
ejpam-4086	36	11	by	by	ADP
ejpam-4086	36	12	theorem	theorem	NOUN
ejpam-4086	36	13	1	1	NUM
ejpam-4086	36	14	.	.	PUNCT
ejpam-4086	37	1	the	the	DET
ejpam-4086	37	2	ej	ej	PROPN
ejpam-4086	37	3	are	be	AUX
ejpam-4086	37	4	linearly	linearly	ADV
ejpam-4086	37	5	independent	independent	ADJ
ejpam-4086	37	6	and	and	CCONJ
ejpam-4086	37	7	belong	belong	VERB
ejpam-4086	37	8	to	to	ADP
ejpam-4086	37	9	v	v	NOUN
ejpam-4086	37	10	′	′	NUM
ejpam-4086	37	11	by	by	ADP
ejpam-4086	37	12	maximality	maximality	NOUN
ejpam-4086	37	13	of	of	ADP
ejpam-4086	37	14	v	v	NOUN
ejpam-4086	37	15	′	′	NOUN
ejpam-4086	37	16	,	,	PUNCT
ejpam-4086	37	17	whence	whence	NOUN
ejpam-4086	37	18	(	(	PUNCT
ejpam-4086	37	19	ii	ii	NOUN
ejpam-4086	37	20	)	)	PUNCT
ejpam-4086	37	21	.	.	PUNCT
ejpam-4086	38	1	example	example	NOUN
ejpam-4086	39	1	1	1	NUM
ejpam-4086	39	2	.	.	X
ejpam-4086	39	3	assume	assume	VERB
ejpam-4086	39	4	n	n	PRON
ejpam-4086	39	5	/∈	/∈	PUNCT
ejpam-4086	40	1	m	m	VERB
ejpam-4086	40	2	and	and	CCONJ
ejpam-4086	40	3	let	let	VERB
ejpam-4086	40	4	α	α	PRON
ejpam-4086	40	5	∈	∈	PROPN
ejpam-4086	41	1	f	f	PROPN
ejpam-4086	42	1	[	[	X
ejpam-4086	42	2	t	t	X
ejpam-4086	42	3	]	]	PUNCT
ejpam-4086	42	4	be	be	AUX
ejpam-4086	42	5	a	a	DET
ejpam-4086	42	6	monic	monic	ADJ
ejpam-4086	42	7	polynomial	polynomial	NOUN
ejpam-4086	42	8	that	that	PRON
ejpam-4086	42	9	splits	split	VERB
ejpam-4086	42	10	in	in	ADP
ejpam-4086	42	11	k[t	k[t	PROPN
ejpam-4086	42	12	]	]	PUNCT
ejpam-4086	42	13	.	.	PUNCT
ejpam-4086	43	1	denote	denote	VERB
ejpam-4086	43	2	by	by	ADP
ejpam-4086	43	3	a(α	a(α	NOUN
ejpam-4086	43	4	)	)	PUNCT
ejpam-4086	43	5	the	the	DET
ejpam-4086	43	6	set	set	NOUN
ejpam-4086	43	7	of	of	ADP
ejpam-4086	43	8	n	n	NUM
ejpam-4086	43	9	×	×	NOUN
ejpam-4086	43	10	n	n	PRON
ejpam-4086	43	11	matrices	matrice	VERB
ejpam-4086	43	12	t	t	X
ejpam-4086	43	13	over	over	ADP
ejpam-4086	43	14	f	f	PROPN
ejpam-4086	43	15	such	such	ADJ
ejpam-4086	43	16	that	that	SCONJ
ejpam-4086	43	17	α(t	α(t	PROPN
ejpam-4086	43	18	)	)	PUNCT
ejpam-4086	44	1	=	=	PUNCT
ejpam-4086	44	2	0	0	X
ejpam-4086	44	3	.	.	PUNCT
ejpam-4086	45	1	then	then	ADV
ejpam-4086	45	2	every	every	DET
ejpam-4086	45	3	pairwise	pairwise	NOUN
ejpam-4086	45	4	commuting	commute	VERB
ejpam-4086	45	5	family	family	NOUN
ejpam-4086	45	6	t	t	PROPN
ejpam-4086	45	7	⊂	⊂	PROPN
ejpam-4086	45	8	a(α	a(α	NOUN
ejpam-4086	45	9	)	)	PUNCT
ejpam-4086	45	10	has	have	VERB
ejpam-4086	45	11	an	an	DET
ejpam-4086	45	12	eigenvector	eigenvector	NOUN
ejpam-4086	45	13	in	in	ADP
ejpam-4086	45	14	fn	fn	PROPN
ejpam-4086	45	15	.	.	PUNCT
ejpam-4086	46	1	this	this	PRON
ejpam-4086	46	2	follows	follow	VERB
ejpam-4086	46	3	from	from	ADP
ejpam-4086	46	4	theorem	theorem	ADJ
ejpam-4086	46	5	1	1	NUM
ejpam-4086	46	6	applied	apply	VERB
ejpam-4086	46	7	to	to	ADP
ejpam-4086	46	8	the	the	DET
ejpam-4086	46	9	lie	lie	NOUN
ejpam-4086	46	10	algebra	algebra	NOUN
ejpam-4086	46	11	≫	≫	PROPN
ejpam-4086	46	12	of	of	ADP
ejpam-4086	46	13	linear	linear	PROPN
ejpam-4086	46	14	operators	operator	NOUN
ejpam-4086	46	15	on	on	ADP
ejpam-4086	46	16	fn	fn	PROPN
ejpam-4086	46	17	generated	generate	VERB
ejpam-4086	46	18	by	by	ADP
ejpam-4086	46	19	t	t	PROPN
ejpam-4086	46	20	.	.	PUNCT
ejpam-4086	47	1	being	be	AUX
ejpam-4086	47	2	abelian	abelian	ADJ
ejpam-4086	47	3	,	,	PUNCT
ejpam-4086	47	4	≫	≫	PROPN
ejpam-4086	47	5	can	can	AUX
ejpam-4086	47	6	be	be	AUX
ejpam-4086	47	7	triangularized	triangularize	VERB
ejpam-4086	47	8	over	over	ADP
ejpam-4086	47	9	k	k	PROPN
ejpam-4086	47	10	,	,	PUNCT
ejpam-4086	47	11	hence(c1	hence(c1	NOUN
ejpam-4086	47	12	)	)	PUNCT
ejpam-4086	47	13	holds	hold	VERB
ejpam-4086	47	14	.	.	PUNCT
ejpam-4086	47	15	example	example	NOUN
ejpam-4086	48	1	2	2	NUM
ejpam-4086	48	2	.	.	PUNCT
ejpam-4086	48	3	the	the	DET
ejpam-4086	48	4	assumption	assumption	NOUN
ejpam-4086	48	5	that	that	SCONJ
ejpam-4086	48	6	n	n	X
ejpam-4086	48	7	∈	∈	NOUN
ejpam-4086	48	8	m	m	VERB
ejpam-4086	48	9	is	be	AUX
ejpam-4086	48	10	essential	essential	ADJ
ejpam-4086	48	11	to	to	PART
ejpam-4086	48	12	theorem	theorem	VERB
ejpam-4086	48	13	1	1	NUM
ejpam-4086	48	14	.	.	X
ejpam-4086	48	15	for	for	ADP
ejpam-4086	48	16	instance	instance	NOUN
ejpam-4086	48	17	,	,	PUNCT
ejpam-4086	48	18	take	take	VERB
ejpam-4086	48	19	f	f	NOUN
ejpam-4086	48	20	=	=	NOUN
ejpam-4086	48	21	,	,	PUNCT
ejpam-4086	48	22	k	k	PROPN
ejpam-4086	49	1	=	=	NOUN
ejpam-4086	49	2	,	,	PUNCT
ejpam-4086	49	3	v	v	NOUN
ejpam-4086	49	4	=	=	SYM
ejpam-4086	49	5	2	2	NUM
ejpam-4086	49	6	.	.	PUNCT
ejpam-4086	50	1	the	the	DET
ejpam-4086	50	2	abelian	abelian	ADJ
ejpam-4086	50	3	lie	lie	NOUN
ejpam-4086	50	4	algebra	algebra	NOUN
ejpam-4086	50	5	of	of	ADP
ejpam-4086	50	6	2×	2×	NUM
ejpam-4086	50	7	2	2	NUM
ejpam-4086	50	8	of	of	ADP
ejpam-4086	50	9	real	real	ADJ
ejpam-4086	50	10	skew	skew	ADJ
ejpam-4086	50	11	symmetric	symmetric	ADJ
ejpam-4086	50	12	matrices	matrix	NOUN
ejpam-4086	50	13	.	.	PUNCT
ejpam-4086	51	1	does	do	AUX
ejpam-4086	51	2	not	not	PART
ejpam-4086	51	3	have	have	VERB
ejpam-4086	51	4	an	an	DET
ejpam-4086	51	5	eigenvector	eigenvector	NOUN
ejpam-4086	51	6	in	in	ADP
ejpam-4086	51	7	2	2	NUM
ejpam-4086	51	8	.	.	PUNCT
ejpam-4086	51	9	example	example	NOUN
ejpam-4086	51	10	3	3	NUM
ejpam-4086	51	11	.	.	PUNCT
ejpam-4086	52	1	the	the	DET
ejpam-4086	52	2	hypothesis	hypothesis	NOUN
ejpam-4086	52	3	of	of	ADP
ejpam-4086	52	4	theorem	theorem	NOUN
ejpam-4086	52	5	1	1	NUM
ejpam-4086	52	6	can	can	AUX
ejpam-4086	52	7	not	not	PART
ejpam-4086	52	8	be	be	AUX
ejpam-4086	52	9	weakened	weaken	VERB
ejpam-4086	52	10	to	to	ADP
ejpam-4086	52	11	≫	≫	PROPN
ejpam-4086	52	12	being	be	AUX
ejpam-4086	52	13	merely	merely	ADV
ejpam-4086	52	14	solvable	solvable	ADJ
ejpam-4086	52	15	.	.	PUNCT
ejpam-4086	53	1	for	for	ADP
ejpam-4086	53	2	a	a	DET
ejpam-4086	53	3	counterexample	counterexample	NOUN
ejpam-4086	53	4	with	with	ADP
ejpam-4086	53	5	f	f	PROPN
ejpam-4086	53	6	=	=	PROPN
ejpam-4086	53	7	,	,	PUNCT
ejpam-4086	53	8	k	k	PROPN
ejpam-4086	53	9	=	=	NOUN
ejpam-4086	53	10	,	,	PUNCT
ejpam-4086	53	11	take	take	VERB
ejpam-4086	53	12	≫	≫	PROPN
ejpam-4086	53	13	to	to	PART
ejpam-4086	53	14	be	be	AUX
ejpam-4086	53	15	the	the	DET
ejpam-4086	53	16	solvable	solvable	ADJ
ejpam-4086	53	17	3	3	NUM
ejpam-4086	53	18	-	-	PUNCT
ejpam-4086	53	19	dimensional	dimensional	ADJ
ejpam-4086	53	20	real	real	ADJ
ejpam-4086	53	21	lie	lie	NOUN
ejpam-4086	53	22	algebra	algebra	NOUN
ejpam-4086	53	23	with	with	ADP
ejpam-4086	53	24	basis	basis	NOUN
ejpam-4086	53	25	(	(	PUNCT
ejpam-4086	53	26	x	x	X
ejpam-4086	53	27	,	,	PUNCT
ejpam-4086	53	28	u	u	NOUN
ejpam-4086	53	29	,	,	PUNCT
ejpam-4086	53	30	v	v	NOUN
ejpam-4086	53	31	)	)	PUNCT
ejpam-4086	53	32	such	such	ADJ
ejpam-4086	53	33	that	that	SCONJ
ejpam-4086	53	34	[	[	X
ejpam-4086	53	35	x	x	X
ejpam-4086	53	36	,	,	PUNCT
ejpam-4086	53	37	u	u	NOUN
ejpam-4086	53	38	]	]	X
ejpam-4086	53	39	=	=	SYM
ejpam-4086	53	40	−v	−v	NOUN
ejpam-4086	53	41	,	,	PUNCT
ejpam-4086	53	42	[	[	X
ejpam-4086	53	43	x	x	X
ejpam-4086	53	44	,	,	PUNCT
ejpam-4086	53	45	v	v	NOUN
ejpam-4086	53	46	]	]	X
ejpam-4086	53	47	=	=	SYM
ejpam-4086	53	48	u	u	NOUN
ejpam-4086	53	49	,	,	PUNCT
ejpam-4086	53	50	[	[	X
ejpam-4086	53	51	u	u	NOUN
ejpam-4086	53	52	,	,	PUNCT
ejpam-4086	53	53	v	v	NOUN
ejpam-4086	53	54	]	]	PUNCT
ejpam-4086	53	55	=	=	PUNCT
ejpam-4086	54	1	0	0	X
ejpam-4086	54	2	.	.	PUNCT
ejpam-4086	55	1	a	a	DET
ejpam-4086	55	2	lie	lie	NOUN
ejpam-4086	55	3	algebra	algebra	NOUN
ejpam-4086	55	4	β	β	X
ejpam-4086	55	5	over	over	ADP
ejpam-4086	55	6	f	f	PROPN
ejpam-4086	55	7	is	be	AUX
ejpam-4086	55	8	supersolvable	supersolvable	ADJ
ejpam-4086	55	9	if	if	SCONJ
ejpam-4086	55	10	the	the	DET
ejpam-4086	55	11	spectrum	spectrum	NOUN
ejpam-4086	55	12	of	of	ADP
ejpam-4086	55	13	the	the	DET
ejpam-4086	55	14	linear	linear	ADJ
ejpam-4086	55	15	map	map	NOUN
ejpam-4086	55	16	ad	ad	NOUN
ejpam-4086	55	17	a	a	X
ejpam-4086	55	18	:	:	PUNCT
ejpam-4086	55	19	β	β	X
ejpam-4086	55	20	→	→	SYM
ejpam-4086	55	21	β	β	X
ejpam-4086	55	22	lies	lie	VERB
ejpam-4086	55	23	in	in	ADP
ejpam-4086	55	24	f	f	PROPN
ejpam-4086	55	25	for	for	ADP
ejpam-4086	55	26	all	all	DET
ejpam-4086	55	27	a	a	DET
ejpam-4086	55	28	∈	∈	NOUN
ejpam-4086	55	29	β	β	X
ejpam-4086	55	30	.	.	PUNCT
ejpam-4086	56	1	if	if	SCONJ
ejpam-4086	56	2	β	β	X
ejpam-4086	56	3	is	be	AUX
ejpam-4086	56	4	not	not	PART
ejpam-4086	56	5	supersolvable	supersolvable	ADJ
ejpam-4086	56	6	it	it	PRON
ejpam-4086	56	7	need	need	AUX
ejpam-4086	56	8	not	not	PART
ejpam-4086	56	9	have	have	VERB
ejpam-4086	56	10	an	an	DET
ejpam-4086	56	11	eigenvector	eigenvector	NOUN
ejpam-4086	56	12	,	,	PUNCT
ejpam-4086	56	13	as	as	SCONJ
ejpam-4086	56	14	is	be	AUX
ejpam-4086	56	15	shown	show	VERB
ejpam-4086	56	16	by	by	ADP
ejpam-4086	56	17	example	example	NOUN
ejpam-4086	56	18	3	3	X
ejpam-4086	56	19	.	.	PUNCT
ejpam-4086	57	1	we	we	PRON
ejpam-4086	57	2	do	do	AUX
ejpam-4086	57	3	n’t	not	PART
ejpam-4086	57	4	know	know	VERB
ejpam-4086	57	5	if	if	SCONJ
ejpam-4086	57	6	theorem	theorem	ADJ
ejpam-4086	57	7	1	1	NUM
ejpam-4086	57	8	extends	extend	VERB
ejpam-4086	57	9	to	to	ADP
ejpam-4086	57	10	supersolvable	supersolvable	ADJ
ejpam-4086	57	11	lie	lie	NOUN
ejpam-4086	57	12	algebras	algebra	NOUN
ejpam-4086	57	13	,	,	PUNCT
ejpam-4086	57	14	except	except	SCONJ
ejpam-4086	57	15	for	for	ADP
ejpam-4086	57	16	the	the	DET
ejpam-4086	57	17	following	follow	VERB
ejpam-4086	57	18	special	special	ADJ
ejpam-4086	57	19	case	case	NOUN
ejpam-4086	57	20	:	:	PUNCT
ejpam-4086	57	21	m.	m.	PROPN
ejpam-4086	57	22	w.	w.	PROPN
ejpam-4086	57	23	hirsch	hirsch	PROPN
ejpam-4086	57	24	,	,	PUNCT
ejpam-4086	57	25	j.	j.	PROPN
ejpam-4086	57	26	w.	w.	PROPN
ejpam-4086	57	27	robbin	robbin	PROPN
ejpam-4086	57	28	/	/	SYM
ejpam-4086	57	29	eur	eur	PROPN
ejpam-4086	57	30	.	.	PUNCT
ejpam-4086	58	1	j.	j.	PROPN
ejpam-4086	58	2	pure	pure	PROPN
ejpam-4086	58	3	appl	appl	PROPN
ejpam-4086	58	4	.	.	PROPN
ejpam-4086	58	5	math	math	PROPN
ejpam-4086	58	6	,	,	PUNCT
ejpam-4086	58	7	14	14	NUM
ejpam-4086	58	8	(	(	PUNCT
ejpam-4086	58	9	4	4	NUM
ejpam-4086	58	10	)	)	PUNCT
ejpam-4086	58	11	(	(	PUNCT
ejpam-4086	58	12	2021	2021	NUM
ejpam-4086	58	13	)	)	PUNCT
ejpam-4086	58	14	,	,	PUNCT
ejpam-4086	58	15	1108	1108	NUM
ejpam-4086	58	16	-	-	SYM
ejpam-4086	58	17	1111	1111	NUM
ejpam-4086	58	18	1110	1110	NUM
ejpam-4086	58	19	theorem	theorem	NOUN
ejpam-4086	58	20	2	2	NUM
ejpam-4086	58	21	.	.	PUNCT
ejpam-4086	59	1	a	a	DET
ejpam-4086	59	2	supersolvable	supersolvable	ADJ
ejpam-4086	59	3	lie	lie	NOUN
ejpam-4086	59	4	algebra	algebra	NOUN
ejpam-4086	59	5	β	β	X
ejpam-4086	59	6	of	of	ADP
ejpam-4086	59	7	linear	linear	ADJ
ejpam-4086	59	8	transformations	transformation	NOUN
ejpam-4086	59	9	of	of	ADP
ejpam-4086	59	10	3	3	NUM
ejpam-4086	59	11	has	have	VERB
ejpam-4086	59	12	an	an	DET
ejpam-4086	59	13	eigenvector	eigenvector	NOUN
ejpam-4086	59	14	.	.	PUNCT
ejpam-4086	59	15	proof	proof	NOUN
ejpam-4086	59	16	.	.	PUNCT
ejpam-4086	60	1	lacking	lack	VERB
ejpam-4086	60	2	an	an	DET
ejpam-4086	60	3	algebraic	algebraic	ADJ
ejpam-4086	60	4	proof	proof	NOUN
ejpam-4086	60	5	,	,	PUNCT
ejpam-4086	60	6	we	we	PRON
ejpam-4086	60	7	use	use	VERB
ejpam-4086	60	8	a	a	DET
ejpam-4086	60	9	dynamical	dynamical	ADJ
ejpam-4086	60	10	argument	argument	NOUN
ejpam-4086	60	11	.	.	PUNCT
ejpam-4086	61	1	let	let	VERB
ejpam-4086	61	2	g	g	PROPN
ejpam-4086	61	3	⊂	⊂	PROPN
ejpam-4086	61	4	gl(3	gl(3	PROPN
ejpam-4086	61	5	,	,	PUNCT
ejpam-4086	61	6	)	)	PUNCT
ejpam-4086	61	7	be	be	VERB
ejpam-4086	61	8	the	the	DET
ejpam-4086	61	9	connected	connected	ADJ
ejpam-4086	61	10	lie	lie	NOUN
ejpam-4086	61	11	subgroup	subgroup	NOUN
ejpam-4086	61	12	having	having	AUX
ejpam-4086	61	13	lie	lie	VERB
ejpam-4086	61	14	algebra	algebra	NOUN
ejpam-4086	61	15	β	β	NOUN
ejpam-4086	61	16	.	.	PUNCT
ejpam-4086	62	1	the	the	DET
ejpam-4086	62	2	natural	natural	ADJ
ejpam-4086	62	3	action	action	NOUN
ejpam-4086	62	4	of	of	ADP
ejpam-4086	62	5	g	g	NOUN
ejpam-4086	62	6	on	on	ADP
ejpam-4086	62	7	the	the	DET
ejpam-4086	62	8	projective	projective	ADJ
ejpam-4086	62	9	plane	plane	NOUN
ejpam-4086	62	10	¶2	¶2	NOUN
ejpam-4086	62	11	of	of	ADP
ejpam-4086	62	12	lines	line	NOUN
ejpam-4086	62	13	in	in	ADP
ejpam-4086	62	14	3	3	NUM
ejpam-4086	62	15	through	through	ADP
ejpam-4086	62	16	the	the	DET
ejpam-4086	62	17	origin	origin	NOUN
ejpam-4086	62	18	fixes	fix	VERB
ejpam-4086	62	19	some	some	DET
ejpam-4086	62	20	l	l	NOUN
ejpam-4086	62	21	∈	∈	PROPN
ejpam-4086	62	22	¶2	¶2	NOUN
ejpam-4086	62	23	.	.	PUNCT
ejpam-4086	63	1	this	this	PRON
ejpam-4086	63	2	follows	follow	VERB
ejpam-4086	63	3	from	from	ADP
ejpam-4086	63	4	supersolvability	supersolvability	NOUN
ejpam-4086	63	5	because	because	SCONJ
ejpam-4086	63	6	dim(¶2	dim(¶2	X
ejpam-4086	63	7	)	)	PUNCT
ejpam-4086	63	8	=	=	SYM
ejpam-4086	64	1	2	2	NUM
ejpam-4086	64	2	,	,	PUNCT
ejpam-4086	64	3	the	the	DET
ejpam-4086	64	4	action	action	NOUN
ejpam-4086	64	5	on	on	ADP
ejpam-4086	64	6	¶2	¶2	NOUN
ejpam-4086	64	7	is	be	AUX
ejpam-4086	64	8	effective	effective	ADJ
ejpam-4086	64	9	and	and	CCONJ
ejpam-4086	64	10	analytic	analytic	ADJ
ejpam-4086	64	11	,	,	PUNCT
ejpam-4086	64	12	and	and	CCONJ
ejpam-4086	64	13	the	the	DET
ejpam-4086	64	14	euler	euler	NOUN
ejpam-4086	64	15	characteristic	characteristic	NOUN
ejpam-4086	64	16	of	of	ADP
ejpam-4086	64	17	¶2	¶2	NOUN
ejpam-4086	64	18	is	be	AUX
ejpam-4086	64	19	nonzero	nonzero	NOUN
ejpam-4086	64	20	(	(	PUNCT
ejpam-4086	64	21	hirsch	hirsch	PROPN
ejpam-4086	64	22	&	&	CCONJ
ejpam-4086	64	23	weinstein	weinstein	PROPN
ejpam-4086	65	1	[	[	X
ejpam-4086	65	2	1	1	NUM
ejpam-4086	65	3	]	]	PUNCT
ejpam-4086	65	4	)	)	PUNCT
ejpam-4086	65	5	.	.	PUNCT
ejpam-4086	66	1	the	the	DET
ejpam-4086	66	2	nonzero	nonzero	PROPN
ejpam-4086	66	3	points	point	NOUN
ejpam-4086	66	4	of	of	ADP
ejpam-4086	66	5	l	l	NOUN
ejpam-4086	66	6	are	be	AUX
ejpam-4086	66	7	eigenvectors	eigenvector	NOUN
ejpam-4086	66	8	for	for	ADP
ejpam-4086	66	9	β	β	X
ejpam-4086	66	10	.	.	PROPN
ejpam-4086	66	11	2.1	2.1	NUM
ejpam-4086	66	12	.	.	PUNCT
ejpam-4086	67	1	proof	proof	NOUN
ejpam-4086	67	2	of	of	ADP
ejpam-4086	67	3	theorem	theorem	NOUN
ejpam-4086	67	4	1	1	NUM
ejpam-4086	67	5	we	we	PRON
ejpam-4086	67	6	rely	rely	VERB
ejpam-4086	67	7	on	on	SCONJ
ejpam-4086	67	8	jacobson	jacobson	PROPN
ejpam-4086	67	9	’s	’s	PART
ejpam-4086	67	10	primary	primary	ADJ
ejpam-4086	67	11	decomposition	decomposition	NOUN
ejpam-4086	67	12	theorem	theorem	VERB
ejpam-4086	67	13	[	[	PUNCT
ejpam-4086	67	14	2	2	NUM
ejpam-4086	67	15	,	,	PUNCT
ejpam-4086	67	16	ii.4	ii.4	PROPN
ejpam-4086	67	17	,	,	PUNCT
ejpam-4086	67	18	theorem	theorem	VERB
ejpam-4086	67	19	5	5	NUM
ejpam-4086	67	20	]	]	PUNCT
ejpam-4086	67	21	.	.	PUNCT
ejpam-4086	68	1	this	this	PRON
ejpam-4086	68	2	states	state	VERB
ejpam-4086	68	3	that	that	SCONJ
ejpam-4086	68	4	v	v	NOUN
ejpam-4086	68	5	has	have	VERB
ejpam-4086	68	6	a	a	DET
ejpam-4086	68	7	≫-invariant	≫-invariant	ADJ
ejpam-4086	68	8	direct	direct	ADJ
ejpam-4086	68	9	sum	sum	NOUN
ejpam-4086	68	10	decomposition	decomposition	NOUN
ejpam-4086	68	11	⊕vi	⊕vi	ADP
ejpam-4086	68	12	where	where	SCONJ
ejpam-4086	68	13	each	each	DET
ejpam-4086	68	14	primary	primary	ADJ
ejpam-4086	68	15	component	component	NOUN
ejpam-4086	68	16	vi	vi	PROPN
ejpam-4086	68	17	has	have	VERB
ejpam-4086	68	18	the	the	DET
ejpam-4086	68	19	following	follow	VERB
ejpam-4086	68	20	property	property	NOUN
ejpam-4086	68	21	:	:	PUNCT
ejpam-4086	68	22	for	for	ADP
ejpam-4086	68	23	each	each	DET
ejpam-4086	68	24	a	a	DET
ejpam-4086	68	25	∈≫	∈≫	NOUN
ejpam-4086	68	26	the	the	DET
ejpam-4086	68	27	minimal	minimal	ADJ
ejpam-4086	68	28	polynomial	polynomial	NOUN
ejpam-4086	68	29	of	of	ADP
ejpam-4086	68	30	a|vi	a|vi	PROPN
ejpam-4086	68	31	is	be	AUX
ejpam-4086	68	32	a	a	DET
ejpam-4086	68	33	prime	prime	ADJ
ejpam-4086	68	34	power	power	NOUN
ejpam-4086	68	35	in	in	ADP
ejpam-4086	68	36	f	f	PROPN
ejpam-4086	68	37	[	[	X
ejpam-4086	68	38	t	t	X
ejpam-4086	68	39	]	]	PUNCT
ejpam-4086	68	40	.	.	PUNCT
ejpam-4086	68	41	condition(c2	condition(c2	NOUN
ejpam-4086	68	42	)	)	PUNCT
ejpam-4086	68	43	implies	imply	VERB
ejpam-4086	68	44	the	the	DET
ejpam-4086	68	45	dimension	dimension	NOUN
ejpam-4086	68	46	of	of	ADP
ejpam-4086	68	47	some	some	DET
ejpam-4086	68	48	primary	primary	ADJ
ejpam-4086	68	49	component	component	NOUN
ejpam-4086	68	50	is	be	AUX
ejpam-4086	68	51	/∈	/∈	NOUN
ejpam-4086	68	52	m.	m.	NOUN
ejpam-4086	68	53	to	to	PART
ejpam-4086	68	54	prove	prove	VERB
ejpam-4086	68	55	theorem	theorem	VERB
ejpam-4086	68	56	1	1	NUM
ejpam-4086	68	57	it	it	PRON
ejpam-4086	68	58	therefore	therefore	ADV
ejpam-4086	68	59	suffices	suffice	VERB
ejpam-4086	68	60	to	to	PART
ejpam-4086	68	61	apply	apply	VERB
ejpam-4086	68	62	the	the	DET
ejpam-4086	68	63	following	following	ADJ
ejpam-4086	68	64	result	result	NOUN
ejpam-4086	68	65	to	to	ADP
ejpam-4086	68	66	such	such	DET
ejpam-4086	68	67	a	a	DET
ejpam-4086	68	68	primary	primary	ADJ
ejpam-4086	68	69	component	component	NOUN
ejpam-4086	68	70	:	:	PUNCT
ejpam-4086	68	71	theorem	theorem	NOUN
ejpam-4086	68	72	3	3	NUM
ejpam-4086	68	73	.	.	NUM
ejpam-4086	68	74	assume(c1	assume(c1	NOUN
ejpam-4086	68	75	)	)	PUNCT
ejpam-4086	68	76	and(c2	and(c2	PROPN
ejpam-4086	68	77	)	)	PUNCT
ejpam-4086	68	78	.	.	PUNCT
ejpam-4086	69	1	if	if	SCONJ
ejpam-4086	69	2	πa	πa	PRON
ejpam-4086	69	3	is	be	AUX
ejpam-4086	69	4	a	a	DET
ejpam-4086	69	5	prime	prime	ADJ
ejpam-4086	69	6	power	power	NOUN
ejpam-4086	69	7	in	in	ADP
ejpam-4086	69	8	f	f	PROPN
ejpam-4086	69	9	[	[	X
ejpam-4086	69	10	t	t	X
ejpam-4086	69	11	]	]	PUNCT
ejpam-4086	69	12	for	for	ADP
ejpam-4086	69	13	each	each	DET
ejpam-4086	69	14	a	a	DET
ejpam-4086	69	15	∈≫	∈≫	NOUN
ejpam-4086	69	16	then	then	ADV
ejpam-4086	69	17	the	the	DET
ejpam-4086	69	18	following	follow	VERB
ejpam-4086	69	19	hold	hold	NOUN
ejpam-4086	69	20	:	:	PUNCT
ejpam-4086	69	21	(	(	PUNCT
ejpam-4086	69	22	a	a	NOUN
ejpam-4086	69	23	)	)	PUNCT
ejpam-4086	69	24	πa(t	πa(t	PUNCT
ejpam-4086	69	25	)	)	PUNCT
ejpam-4086	69	26	=	=	SYM
ejpam-4086	69	27	(	(	PUNCT
ejpam-4086	69	28	t−	t−	PROPN
ejpam-4086	69	29	ra	ra	PROPN
ejpam-4086	69	30	)	)	PUNCT
ejpam-4086	70	1	n	n	CCONJ
ejpam-4086	70	2	,	,	PUNCT
ejpam-4086	70	3	ra	ra	PROPN
ejpam-4086	70	4	∈	∈	PROPN
ejpam-4086	70	5	f	f	X
ejpam-4086	70	6	(	(	PUNCT
ejpam-4086	70	7	b	b	NOUN
ejpam-4086	70	8	)	)	PUNCT
ejpam-4086	70	9	there	there	PRON
ejpam-4086	70	10	is	be	VERB
ejpam-4086	70	11	a	a	DET
ejpam-4086	70	12	basis	basis	NOUN
ejpam-4086	70	13	putting	put	VERB
ejpam-4086	70	14	≫	≫	PROPN
ejpam-4086	70	15	in	in	ADP
ejpam-4086	70	16	triangular	triangular	NOUN
ejpam-4086	70	17	form	form	NOUN
ejpam-4086	70	18	assertion	assertion	NOUN
ejpam-4086	70	19	(	(	PUNCT
ejpam-4086	70	20	a	a	X
ejpam-4086	70	21	)	)	PUNCT
ejpam-4086	70	22	is	be	AUX
ejpam-4086	70	23	equivalent	equivalent	ADJ
ejpam-4086	70	24	to	to	ADP
ejpam-4086	70	25	πa	πa	ADP
ejpam-4086	70	26	having	have	VERB
ejpam-4086	70	27	a	a	DET
ejpam-4086	70	28	root	root	NOUN
ejpam-4086	71	1	ra	ra	PROPN
ejpam-4086	71	2	∈	∈	PROPN
ejpam-4086	72	1	f	f	X
ejpam-4086	72	2	.	.	PUNCT
ejpam-4086	73	1	therefore	therefore	ADV
ejpam-4086	73	2	(	(	PUNCT
ejpam-4086	73	3	a	a	PRON
ejpam-4086	73	4	)	)	PUNCT
ejpam-4086	73	5	follows	follow	VERB
ejpam-4086	73	6	from	from	ADP
ejpam-4086	73	7	:	:	PUNCT
ejpam-4086	73	8	lemma	lemma	PROPN
ejpam-4086	73	9	1	1	X
ejpam-4086	73	10	.	.	PUNCT
ejpam-4086	74	1	let	let	VERB
ejpam-4086	74	2	α	α	PRON
ejpam-4086	74	3	∈	∈	PROPN
ejpam-4086	74	4	f	f	PROPN
ejpam-4086	75	1	[	[	X
ejpam-4086	75	2	t	t	X
ejpam-4086	75	3	]	]	PUNCT
ejpam-4086	75	4	be	be	AUX
ejpam-4086	75	5	a	a	DET
ejpam-4086	75	6	polynomial	polynomial	NOUN
ejpam-4086	75	7	of	of	ADP
ejpam-4086	75	8	degree	degree	NOUN
ejpam-4086	76	1	n	n	NOUN
ejpam-4086	76	2	that	that	PRON
ejpam-4086	76	3	splits	split	VERB
ejpam-4086	76	4	in	in	ADP
ejpam-4086	76	5	k[t	k[t	PROPN
ejpam-4086	76	6	]	]	PUNCT
ejpam-4086	76	7	.	.	PUNCT
ejpam-4086	77	1	if	if	SCONJ
ejpam-4086	77	2	n	n	ADV
ejpam-4086	77	3	/∈	/∈	PUNCT
ejpam-4086	78	1	m	m	AUX
ejpam-4086	78	2	then	then	ADV
ejpam-4086	78	3	α	α	PRON
ejpam-4086	78	4	has	have	VERB
ejpam-4086	78	5	a	a	DET
ejpam-4086	78	6	root	root	NOUN
ejpam-4086	78	7	in	in	ADP
ejpam-4086	78	8	f	f	PROPN
ejpam-4086	78	9	,	,	PUNCT
ejpam-4086	78	10	and	and	CCONJ
ejpam-4086	78	11	the	the	DET
ejpam-4086	78	12	sum	sum	NOUN
ejpam-4086	78	13	of	of	ADP
ejpam-4086	78	14	the	the	DET
ejpam-4086	78	15	multiplicities	multiplicity	NOUN
ejpam-4086	78	16	of	of	ADP
ejpam-4086	78	17	such	such	ADJ
ejpam-4086	78	18	roots	root	NOUN
ejpam-4086	78	19	is	be	AUX
ejpam-4086	78	20	/∈	/∈	NOUN
ejpam-4086	78	21	m.	m.	NOUN
ejpam-4086	78	22	proof	proof	NOUN
ejpam-4086	78	23	.	.	PUNCT
ejpam-4086	79	1	let	let	VERB
ejpam-4086	79	2	r	r	PRON
ejpam-4086	79	3	⊂	⊂	PROPN
ejpam-4086	79	4	k	k	PROPN
ejpam-4086	79	5	denote	denote	VERB
ejpam-4086	79	6	the	the	DET
ejpam-4086	79	7	set	set	NOUN
ejpam-4086	79	8	of	of	ADP
ejpam-4086	79	9	roots	root	NOUN
ejpam-4086	79	10	of	of	ADP
ejpam-4086	79	11	π	π	PROPN
ejpam-4086	79	12	,	,	PUNCT
ejpam-4086	79	13	and	and	CCONJ
ejpam-4086	79	14	rj	rj	PROPN
ejpam-4086	79	15	⊂	⊂	PROPN
ejpam-4086	79	16	r	r	VERB
ejpam-4086	79	17	the	the	DET
ejpam-4086	79	18	set	set	NOUN
ejpam-4086	79	19	of	of	ADP
ejpam-4086	79	20	roots	root	NOUN
ejpam-4086	79	21	of	of	ADP
ejpam-4086	79	22	multiplicity	multiplicity	NOUN
ejpam-4086	79	23	j.	j.	PROPN
ejpam-4086	79	24	the	the	DET
ejpam-4086	79	25	galois	galois	PROPN
ejpam-4086	79	26	group	group	NOUN
ejpam-4086	79	27	γ	γ	PROPN
ejpam-4086	79	28	has	have	VERB
ejpam-4086	79	29	order	order	NOUN
ejpam-4086	80	1	[	[	X
ejpam-4086	80	2	k	k	X
ejpam-4086	80	3	:	:	PUNCT
ejpam-4086	80	4	f	f	X
ejpam-4086	80	5	]	]	PUNCT
ejpam-4086	80	6	and	and	CCONJ
ejpam-4086	80	7	acts	act	VERB
ejpam-4086	80	8	on	on	ADP
ejpam-4086	80	9	r	r	NOUN
ejpam-4086	80	10	by	by	ADP
ejpam-4086	80	11	permutations	permutation	NOUN
ejpam-4086	80	12	.	.	PUNCT
ejpam-4086	81	1	the	the	DET
ejpam-4086	81	2	cardinality	cardinality	NOUN
ejpam-4086	81	3	of	of	ADP
ejpam-4086	81	4	each	each	DET
ejpam-4086	81	5	orbit	orbit	NOUN
ejpam-4086	81	6	divides	divide	VERB
ejpam-4086	81	7	[	[	X
ejpam-4086	81	8	k	k	X
ejpam-4086	81	9	:	:	PUNCT
ejpam-4086	81	10	f	f	X
ejpam-4086	81	11	]	]	X
ejpam-4086	81	12	,	,	PUNCT
ejpam-4086	81	13	and	and	CCONJ
ejpam-4086	81	14	r	r	NOUN
ejpam-4086	81	15	∩	∩	NOUN
ejpam-4086	81	16	f	f	PROPN
ejpam-4086	81	17	is	be	AUX
ejpam-4086	81	18	the	the	DET
ejpam-4086	81	19	set	set	NOUN
ejpam-4086	81	20	of	of	ADP
ejpam-4086	81	21	fixed	fix	VERB
ejpam-4086	81	22	points	point	NOUN
ejpam-4086	81	23	of	of	ADP
ejpam-4086	81	24	this	this	DET
ejpam-4086	81	25	action	action	NOUN
ejpam-4086	81	26	.	.	PUNCT
ejpam-4086	82	1	each	each	DET
ejpam-4086	82	2	rj	rj	PROPN
ejpam-4086	82	3	is	be	AUX
ejpam-4086	82	4	a	a	DET
ejpam-4086	82	5	union	union	NOUN
ejpam-4086	82	6	of	of	ADP
ejpam-4086	82	7	orbits	orbit	NOUN
ejpam-4086	82	8	,	,	PUNCT
ejpam-4086	82	9	as	as	SCONJ
ejpam-4086	82	10	is	be	AUX
ejpam-4086	82	11	rj\f	rj\f	PUNCT
ejpam-4086	82	12	.	.	PUNCT
ejpam-4086	83	1	it	it	PRON
ejpam-4086	83	2	follows	follow	VERB
ejpam-4086	83	3	that	that	SCONJ
ejpam-4086	83	4	#	#	SYM
ejpam-4086	83	5	(	(	PUNCT
ejpam-4086	83	6	rj	rj	PROPN
ejpam-4086	83	7	\	\	PROPN
ejpam-4086	83	8	f	f	X
ejpam-4086	83	9	)	)	PUNCT
ejpam-4086	83	10	∈	∈	PROPN
ejpam-4086	83	11	m.	m.	NOUN
ejpam-4086	83	12	let	let	VERB
ejpam-4086	83	13	k	k	PROPN
ejpam-4086	83	14	≤	≤	PROPN
ejpam-4086	83	15	n	n	CCONJ
ejpam-4086	83	16	denote	denote	VERB
ejpam-4086	83	17	the	the	DET
ejpam-4086	83	18	sum	sum	NOUN
ejpam-4086	83	19	of	of	ADP
ejpam-4086	83	20	the	the	DET
ejpam-4086	83	21	multiplicities	multiplicity	NOUN
ejpam-4086	83	22	of	of	ADP
ejpam-4086	83	23	the	the	DET
ejpam-4086	83	24	roots	root	NOUN
ejpam-4086	83	25	that	that	PRON
ejpam-4086	83	26	are	be	AUX
ejpam-4086	83	27	not	not	PART
ejpam-4086	83	28	in	in	ADP
ejpam-4086	83	29	f	f	PROPN
ejpam-4086	83	30	.	.	PUNCT
ejpam-4086	84	1	then	then	ADV
ejpam-4086	84	2	k	k	PROPN
ejpam-4086	84	3	=	=	PUNCT
ejpam-4086	84	4	n∑	n∑	PROPN
ejpam-4086	84	5	j=2	j=2	PROPN
ejpam-4086	85	1	j	j	PROPN
ejpam-4086	85	2	·	·	PUNCT
ejpam-4086	85	3	#	#	PROPN
ejpam-4086	85	4	(	(	PUNCT
ejpam-4086	85	5	rj	rj	PROPN
ejpam-4086	85	6	\	\	PROPN
ejpam-4086	85	7	f	f	X
ejpam-4086	85	8	)	)	PUNCT
ejpam-4086	85	9	therefore	therefore	ADV
ejpam-4086	85	10	k	k	PROPN
ejpam-4086	85	11	∈	∈	PROPN
ejpam-4086	85	12	m	m	VERB
ejpam-4086	85	13	because	because	SCONJ
ejpam-4086	85	14	m	m	NOUN
ejpam-4086	85	15	is	be	AUX
ejpam-4086	85	16	closed	close	VERB
ejpam-4086	85	17	under	under	ADP
ejpam-4086	85	18	addition	addition	NOUN
ejpam-4086	85	19	.	.	PUNCT
ejpam-4086	86	1	by	by	ADP
ejpam-4086	86	2	hypothesis	hypothesis	NOUN
ejpam-4086	86	3	n	n	NOUN
ejpam-4086	86	4	/∈	/∈	PUNCT
ejpam-4086	86	5	m	m	ADP
ejpam-4086	86	6	,	,	PUNCT
ejpam-4086	86	7	hence	hence	ADV
ejpam-4086	86	8	n	n	ADV
ejpam-4086	86	9	−	−	PROPN
ejpam-4086	86	10	k	k	INTJ
ejpam-4086	86	11	/∈	/∈	PUNCT
ejpam-4086	87	1	m	m	VERB
ejpam-4086	87	2	and	and	CCONJ
ejpam-4086	87	3	n	n	CCONJ
ejpam-4086	87	4	−	−	PROPN
ejpam-4086	87	5	k	k	X
ejpam-4086	88	1	>	>	X
ejpam-4086	88	2	0	0	X
ejpam-4086	88	3	.	.	PUNCT
ejpam-4086	89	1	as	as	ADP
ejpam-4086	89	2	n	n	PROPN
ejpam-4086	89	3	−	−	PROPN
ejpam-4086	89	4	k	k	PROPN
ejpam-4086	89	5	is	be	AUX
ejpam-4086	89	6	the	the	DET
ejpam-4086	89	7	sum	sum	NOUN
ejpam-4086	89	8	of	of	ADP
ejpam-4086	89	9	the	the	DET
ejpam-4086	89	10	multiplicities	multiplicity	NOUN
ejpam-4086	89	11	of	of	ADP
ejpam-4086	89	12	the	the	DET
ejpam-4086	89	13	roots	root	NOUN
ejpam-4086	89	14	in	in	ADP
ejpam-4086	89	15	f	f	PROPN
ejpam-4086	89	16	,	,	PUNCT
ejpam-4086	89	17	the	the	DET
ejpam-4086	89	18	conclusion	conclusion	NOUN
ejpam-4086	89	19	follows	follow	VERB
ejpam-4086	89	20	.	.	PUNCT
ejpam-4086	90	1	now	now	ADV
ejpam-4086	90	2	that	that	PRON
ejpam-4086	90	3	(	(	PUNCT
ejpam-4086	90	4	a	a	X
ejpam-4086	90	5	)	)	PUNCT
ejpam-4086	90	6	of	of	ADP
ejpam-4086	90	7	theorem	theorem	NOUN
ejpam-4086	90	8	3	3	NUM
ejpam-4086	90	9	is	be	AUX
ejpam-4086	90	10	proved	prove	VERB
ejpam-4086	90	11	,	,	PUNCT
ejpam-4086	90	12	assertion	assertion	NOUN
ejpam-4086	90	13	(	(	PUNCT
ejpam-4086	90	14	b	b	NOUN
ejpam-4086	90	15	)	)	PUNCT
ejpam-4086	90	16	is	be	AUX
ejpam-4086	90	17	a	a	DET
ejpam-4086	90	18	consequence	consequence	NOUN
ejpam-4086	90	19	of	of	ADP
ejpam-4086	90	20	the	the	DET
ejpam-4086	90	21	following	following	ADJ
ejpam-4086	90	22	result	result	NOUN
ejpam-4086	90	23	:	:	PUNCT
ejpam-4086	90	24	references	reference	NOUN
ejpam-4086	90	25	1111	1111	PROPN
ejpam-4086	90	26	lemma	lemma	PROPN
ejpam-4086	90	27	2	2	X
ejpam-4086	90	28	.	.	PUNCT
ejpam-4086	90	29	let	let	AUX
ejpam-4086	90	30	be	be	AUX
ejpam-4086	90	31	a	a	DET
ejpam-4086	90	32	nilpotent	nilpotent	ADJ
ejpam-4086	90	33	lie	lie	NOUN
ejpam-4086	90	34	algebra	algebra	NOUN
ejpam-4086	90	35	of	of	ADP
ejpam-4086	90	36	linear	linear	PROPN
ejpam-4086	90	37	operators	operator	NOUN
ejpam-4086	90	38	on	on	ADP
ejpam-4086	90	39	v	v	NUM
ejpam-4086	90	40	.	.	PUNCT
ejpam-4086	91	1	assume	assume	VERB
ejpam-4086	91	2	that	that	SCONJ
ejpam-4086	91	3	for	for	ADP
ejpam-4086	91	4	all	all	DET
ejpam-4086	91	5	a	a	DET
ejpam-4086	91	6	∈	∈	NOUN
ejpam-4086	91	7	there	there	PRON
ejpam-4086	91	8	exists	exist	VERB
ejpam-4086	91	9	ra	ra	PROPN
ejpam-4086	91	10	∈	∈	PROPN
ejpam-4086	92	1	f	f	PROPN
ejpam-4086	92	2	such	such	ADJ
ejpam-4086	92	3	that	that	PRON
ejpam-4086	92	4	πa(t	πa(t	PUNCT
ejpam-4086	92	5	)	)	PUNCT
ejpam-4086	92	6	=	=	SYM
ejpam-4086	92	7	(	(	PUNCT
ejpam-4086	92	8	t−	t−	PROPN
ejpam-4086	92	9	ra	ra	PROPN
ejpam-4086	92	10	)	)	PUNCT
ejpam-4086	92	11	n.	n.	NOUN
ejpam-4086	93	1	then	then	ADV
ejpam-4086	93	2	v	v	NOUN
ejpam-4086	93	3	has	have	VERB
ejpam-4086	93	4	a	a	DET
ejpam-4086	93	5	basis	basis	NOUN
ejpam-4086	93	6	putting	put	VERB
ejpam-4086	93	7	in	in	ADP
ejpam-4086	93	8	triangular	triangular	NOUN
ejpam-4086	93	9	form	form	NOUN
ejpam-4086	93	10	.	.	PUNCT
ejpam-4086	94	1	proof	proof	NOUN
ejpam-4086	94	2	.	.	PUNCT
ejpam-4086	95	1	every	every	DET
ejpam-4086	95	2	a	a	DET
ejpam-4086	95	3	∈	∈	PROPN
ejpam-4086	95	4	can	can	AUX
ejpam-4086	95	5	be	be	AUX
ejpam-4086	95	6	written	write	VERB
ejpam-4086	95	7	uniquely	uniquely	ADV
ejpam-4086	95	8	as	as	ADP
ejpam-4086	95	9	rai	rai	X
ejpam-4086	95	10	+	+	CCONJ
ejpam-4086	95	11	na	na	NOUN
ejpam-4086	95	12	with	with	ADP
ejpam-4086	95	13	na	na	PART
ejpam-4086	95	14	nilpotent	nilpotent	ADJ
ejpam-4086	95	15	and	and	CCONJ
ejpam-4086	95	16	i	i	PRON
ejpam-4086	95	17	the	the	DET
ejpam-4086	95	18	identity	identity	NOUN
ejpam-4086	95	19	map	map	NOUN
ejpam-4086	95	20	of	of	ADP
ejpam-4086	95	21	v	v	NOUN
ejpam-4086	95	22	.	.	PUNCT
ejpam-4086	96	1	it	it	PRON
ejpam-4086	96	2	is	be	AUX
ejpam-4086	96	3	easy	easy	ADJ
ejpam-4086	96	4	to	to	PART
ejpam-4086	96	5	see	see	VERB
ejpam-4086	96	6	that	that	SCONJ
ejpam-4086	96	7	the	the	DET
ejpam-4086	96	8	set	set	NOUN
ejpam-4086	96	9	comprising	comprise	VERB
ejpam-4086	96	10	the	the	DET
ejpam-4086	96	11	na	na	NOUN
ejpam-4086	96	12	is	be	AUX
ejpam-4086	96	13	closed	close	VERB
ejpam-4086	96	14	under	under	ADP
ejpam-4086	96	15	commutator	commutator	NOUN
ejpam-4086	96	16	brackets	bracket	NOUN
ejpam-4086	96	17	.	.	PUNCT
ejpam-4086	97	1	therefore	therefore	ADV
ejpam-4086	97	2	v	v	X
ejpam-4086	97	3	has	have	VERB
ejpam-4086	97	4	a	a	DET
ejpam-4086	97	5	basis	basis	NOUN
ejpam-4086	97	6	triangularizing	triangularize	VERB
ejpam-4086	97	7	all	all	DET
ejpam-4086	97	8	the	the	DET
ejpam-4086	97	9	na	na	NOUN
ejpam-4086	97	10	(	(	PUNCT
ejpam-4086	97	11	jacobson	jacobson	PROPN
ejpam-4086	97	12	[	[	X
ejpam-4086	97	13	2	2	NUM
ejpam-4086	97	14	,	,	PUNCT
ejpam-4086	97	15	ii.2	ii.2	PROPN
ejpam-4086	97	16	,	,	PUNCT
ejpam-4086	97	17	theorem	theorem	VERB
ejpam-4086	97	18	1′	1′	NUM
ejpam-4086	97	19	]	]	PUNCT
ejpam-4086	97	20	)	)	PUNCT
ejpam-4086	97	21	,	,	PUNCT
ejpam-4086	97	22	and	and	CCONJ
ejpam-4086	97	23	such	such	DET
ejpam-4086	97	24	a	a	DET
ejpam-4086	97	25	basis	basis	NOUN
ejpam-4086	97	26	triangularizes	triangularizes	NOUN
ejpam-4086	97	27	.	.	PUNCT
ejpam-4086	98	1	this	this	PRON
ejpam-4086	98	2	completes	complete	VERB
ejpam-4086	98	3	the	the	DET
ejpam-4086	98	4	proof	proof	NOUN
ejpam-4086	98	5	of	of	ADP
ejpam-4086	98	6	theorem	theorem	NOUN
ejpam-4086	98	7	1	1	NUM
ejpam-4086	98	8	.	.	PUNCT
ejpam-4086	98	9	references	reference	NOUN
ejpam-4086	98	10	[	[	X
ejpam-4086	98	11	1	1	NUM
ejpam-4086	98	12	]	]	X
ejpam-4086	98	13	m	m	VERB
ejpam-4086	98	14	hirsch	hirsch	PROPN
ejpam-4086	98	15	and	and	CCONJ
ejpam-4086	98	16	a	a	DET
ejpam-4086	98	17	weinstein	weinstein	NOUN
ejpam-4086	98	18	.	.	PUNCT
ejpam-4086	99	1	fixed	fix	VERB
ejpam-4086	99	2	points	point	NOUN
ejpam-4086	99	3	of	of	ADP
ejpam-4086	99	4	analytic	analytic	ADJ
ejpam-4086	99	5	actions	action	NOUN
ejpam-4086	99	6	of	of	ADP
ejpam-4086	99	7	supersoluble	supersoluble	ADJ
ejpam-4086	99	8	lie	lie	NOUN
ejpam-4086	99	9	groups	group	NOUN
ejpam-4086	99	10	on	on	ADP
ejpam-4086	99	11	compact	compact	ADJ
ejpam-4086	99	12	surfaces	surface	NOUN
ejpam-4086	99	13	.	.	PUNCT
ejpam-4086	100	1	ergod	ergod	NOUN
ejpam-4086	100	2	.	.	PUNCT
ejpam-4086	101	1	th	th	X
ejpam-4086	101	2	.	.	PUNCT
ejpam-4086	101	3	dyn	dyn	PROPN
ejpam-4086	101	4	.	.	PUNCT
ejpam-4086	102	1	sys	sys	PROPN
ejpam-4086	102	2	.	.	PROPN
ejpam-4086	102	3	,	,	PUNCT
ejpam-4086	102	4	21(6):1783–1787	21(6):1783–1787	NUM
ejpam-4086	102	5	,	,	PUNCT
ejpam-4086	102	6	2001	2001	NUM
ejpam-4086	102	7	.	.	PUNCT
ejpam-4086	103	1	[	[	X
ejpam-4086	103	2	2	2	NUM
ejpam-4086	103	3	]	]	PUNCT
ejpam-4086	103	4	n	n	PRON
ejpam-4086	103	5	jacobson	jacobson	PROPN
ejpam-4086	103	6	.	.	PROPN
ejpam-4086	104	1	lie	lie	PROPN
ejpam-4086	104	2	algebras	algebras	PROPN
ejpam-4086	104	3	.	.	PUNCT
ejpam-4086	105	1	interscience	interscience	NOUN
ejpam-4086	105	2	tracts	tract	NOUN
ejpam-4086	105	3	in	in	ADP
ejpam-4086	105	4	pure	pure	ADJ
ejpam-4086	105	5	mathematics	mathematic	NOUN
ejpam-4086	105	6	no	no	INTJ
ejpam-4086	105	7	.	.	PROPN
ejpam-4086	105	8	10	10	NUM
ejpam-4086	105	9	,	,	PUNCT
ejpam-4086	105	10	john	john	PROPN
ejpam-4086	105	11	wiley	wiley	PROPN
ejpam-4086	105	12	,	,	PUNCT
ejpam-4086	105	13	new	new	PROPN
ejpam-4086	105	14	york	york	PROPN
ejpam-4086	105	15	,	,	PUNCT
ejpam-4086	105	16	1962	1962	NUM
ejpam-4086	105	17	.	.	PUNCT