On Amitsur's complex and restricted Lie algebras

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T h is d is s e r t a t io n h a s b e en m ic r o f ilm e d e x a c tly a s re c e iv e d

63-1267

BERK SO N , A s tr id J ., 1 9 3 6 ON A M ITSU R 'S C O M PL E X AND R E ST R IC T E D L IE A LG EBRA S. N o r th w e s te r n U n iv e rs ity , P h .D ., 1962 M a th e m a tic s

University Microfilms, Inc., Ann Arbor, Michigan

N O RTH W ESTER N UNIVERSITY

ON A M ITSU R 'S C O M PL E X AND R E ST R IC T E D L IE A LG EBRA S

A D ISSERTA TIO N SU B M IT TE D TO THE G RA DUA TE SCHOOL IN P A R T IA L F U L F IL L M E N T O F TH E REQ U IR EM EN TS f o r the d e g re e

DOCTOR O F PH ILO SO PH Y F ie ld of M a th e m a tic s

by A s tr id J . B e rk s o n

E v a n s to n . Illin o is J u n e 1962

A C K N O W LED G EM EN TS

W ith g r a titu d e an d p l e a s u r e , I w is h to a c k n o w le d g e m y d e b t to P r o f e s s o r A. R o s e n b e rg f o r h is e n c o u ra g e m e n t, g u id a n c e , a n d a s s i s t a n c e in th e w r itin g of th is d i s s e r t a t io n .

T A B L E O F C O N TE N T S

S e c tio n

I.

IL

III.

IV .

V.

V I.

Page

In tro d u c tio n .

1

N o ta tio n s .

3

T h e f i r s t c o h o m o lo g y g ro u p .

5

S p lit r e g u l a r e x te n s io n s .

7

A ll c o h o m o lo g y g ro u p s b u t on e a r e a e r o .

25

A c l a s s of F r o b e n iu s a lg e b r a s .

33

F o o tn o te s .

42

B ib lio g ra p h y .

43

1

SE C T IO N I .

IN TRO D U CTIO N

G iv en a c o m m u ta tiv e r in g C a n d a c o m m u ta tiv e C - a lg e b r a A , A m its u r in tro d u c e d a c o m p le x w h ich w ill be d e s c r ib e d in S e c tio n I I . W e w ill d e n o te th e co h o m o lo g y g ro u p s of th is c o m p le x by H ^ A ). In S e c tio n I I I w e p ro v e th a t if C i s a f ie ld , th e n f o r a n y C a lg e b r a A , H*(A) = 0. [ 1, T h e o re m

6

It w a s show n in [ 9 , T h e o re m 1, p. 3 3 0 ]

and

. 1, p. 9 9 ] t h a t if C i s a f ie ld a n d A is a fin ite n o r m a l

s e p a r a b le e x te n s io n fie ld w ith G a lo is g ro u p G , th e n H ^ A ) i s is o m o r p h ic to H ^ G , A - 1 0 j ),

th e n ** 1 co h o m o lo g y g ro u p of th e G a lo is g ro u p of

A o v e r C w ith c o e f f ic ie n ts in th e m u ltip lic a tiv e g ro u p A - i.0 J .

Sec­

tio n 111 th u s g iv e s a g e n e r a lis a tio n of H i l b e r t 's T h e o re m 90. If F i s a s e p a r a b le fie ld e x te n s io n o f C , th e g ro u p s ^ ( F ) h av e b e en d e s c r i b e d in [ 9 ] • H o w ev e r, in th e in s e p a r a b le c a s e no g e n ­ e r a l r e s u l t s on H ^ F ) w e r e know n. O u r m a in r e s u l t s d e a l w ith th a t c a s e . L e t C be a n im p e r f e c t fie ld o f c h a r a c t e r i s t i c p a n d F a fin ite p u r e ly in s e p a r a b le e x te n s io n fie ld of e x p o n e n t o n e.

H o c h sc h ild ( 7 ]

in tro d u c e d th e n o tio n of a r e g u l a r r e s t r i c t e d L ie a lg e b r a e x te n s io n o f F by T , C.

w h e re

T is th e r e s t r i c t e d L ie a lg e b r a of d e r iv a tio n s o f F o v e r

H e a ls o sh o w ed th a t th e g ro u p of r e g u l a r r e s t r i c t e d L ie a lg e b r a

e x te n s io n s of F

by

T is e q u iv a le n t to th e B r a u e r g ro u p of F

over C.

In [ 9 ] R o s e n b e rg a n d Z e lin s k y in tro d u c e d f o r e a c h A m its u r 2 - c o c y c le t of F

a r e g u l a r r e s t r i c t e d e x te n s io n o f F

by T , d e n o te d by 0 ( t ) ,

_ 2 an d show ed th a t u n d e r th e c o r r e s p o n d e n c e in d u c e d by t - • ■ 0 ( t} , H (F )

i s is o m o rp h ic to th e g ro u p of e x te n s io n s in tro d u c e d by H o c h sc h ild . In S e c tio n IV w e d e fin e r e g u la r r e s t r i c t e d L ie a lg e b r a e x te n s io n s of M by T w h e re M i s a n a b e lia n r e s t r i c t e d L ie a lg e b r a s a tis f y in g so m e a d d i­ tio n a l h y p o th e s e s . W e a ls o d e fin e fo r e a c h n a re s trid jb d L ie a lg e b r a Kn w h ich s e r v e s as a n M .

W e th e n show th a t f o r c e r t a i n c l a s s e s of k e r n e l s ,

in c lu d in g the K ° f o r n > 2 , a n y r e g u la r r e s t r i c t e d L ie a lg e b r a e x te n s io n o f th e k e r n e ls by T m u s t s p lit. th a t of th e

0

In S e c tio n V , by a d e fin itio n s i m i l a r to

( t ) , w e d e fin e fo r e a c h n - c o c y c le t of th e A m its u r c o m ­

p le x , a r e s t r i c t e d L ie a lg e b r a e x te n s io n

0n(O° f

Kn by

T. T h is e x ­

te n s io n is r e g u la r in th e s e n s e o f S e c tio n IV an d h a s th e p r o p e r ty th a t if Q n( t ) s p lits th e n t m u s t be a c o b o u n d a ry . S e c tio n IV to p ro v e th a t f o r F

T h en w e u s e th e r e s u l t s of

a s above an d n > 2 , h V )

=

0

.

F in a lly , in S e c tio n VI, w e show th a t the r e s t r i c t e d u n iv e r s a l a s s o c ia tiv e a lg e b r a o f a fin ite r e s t r i c t e d L ie a lg e b r a is a F ro b e n iu a a lg e b r a .

3

SE C T IO N I I .

NOTATIONS

L e t A be a c o m m u ta tiv e C - a lg e b r a w ith u n it, w h e re C is a fie ld . L e t An = A m . . . C m o r p h is m s

e

W

A,

(n f a c to r s ) .

C ^ : A*1-*- An + l

We d e fin e C - a lg e b r a m o n o -

by L

£ . ( a . • . . . e a ) = a , • . . . e a . . e l e a . « . . . e a fo r i = l , . . . . n+ 1 . i n 1 i- 1 i n A s u su al we le t A a lg e b r a An .

n*

be th e m u ltip lic a tiv e g ro u p of u n its o f the

We d efin e

A : An* .n-f 1 * An A -*■ A by A J x ) = C x (x) c 2 (x " l ) . . . ^

(x * 1)

n + 1

C le a r ly A^ i s a h o m o m o rp h is m a n d i t w a s show n in [ 1. T h e o re m 5 .1 . p . 93] by A m its u r th a t th e se q u e n c e of g ro u p s an d h o m o m o rp h ism s (A

, A ) n

f o rm a c o m p le x , i. e.

A A =1. n+ 1 n

We s h a ll c a l l th is th e

A m its u r c o m p le x o v e r A a n d d e n o te i t by Q(A ). The co h o m o lo g y g ro u p K er(A 0 +1

) / 'lm ( A ) n

w ill be d e n o te d by H ^ A ). F o r th e s ig n ific a n c e o f

th e s e g ro u p s se e [ I ]

and

[ 9 ].

We s h a ll h av e u s e f o r a v a r ia n t of Q(A). L e t £ n * An— A ° + * b e g iv en by

^ n * Ll~ ZZ +

-

^n+1 *

It i s c l e a r th a t £

is a n a d d itiv e h o m o m o rp h is m . It w a s p ro v e d in [ 9 .

n

L e m m a 4 . 1, p. 347] th a t th e se q u e n c e (An , & ) of a d d itiv e g ro u p s a n d h o m o m o rp h is m s is e x a c t.

T h is is e a s i l y se e n :

L e t X be a C - l in e a r m ap of A o n to C s u c h th a t *f (1) = 1. T h en d e fin e

s

s

e a ) = n

(a , • . . . n ' 1

‘ An-* A n * by '

n

[ ( t.) a , t ... 1

2

• a

n

and

s. =

0

0

.

s . . f . . . = t .s f o r i ^ 1, a n d • i s th e id e n tity n tl x+i x n n+i i n+ 1 .,. m a p , th e n s . . C, + C ,s = ■ . , £ , + (-1) r n+ 1 ° n d n -l n n+ 1 1 i» 2 ' ' n+ 1 x S in ce

+

4^l+ l (-1) €m* i= 1 I n

= th e id e n tity m a p , so th a t a

n

is a c o n tr a c tin g

h o m o to p y . S ince th e o p e r a to r £ th o u g h t of a s

1• £

a n e x a c t f u n c to r ,

. n- x

= £

2

^

fo r the c o m p le x

- ... A •_ A w

i t fo llo w s th a t th e se q u e n c e

+ £ n+j

can be

a n d s in c e • _ is Wi

(An , £ ) n

i s e x a c t a ls o .

SE C T IO N I I I .

THE F IR S T COHOMOLOGY GROUP

L e t A be a c o m m u ta tiv e C * a lg e b ra w ith u n it. W e s h a ll show th a t H*(A) « 0.

To b e g in w ith w e p ro v e

L e m m a 3 .1 x * 0

L e t V a n d Wbe v e c to r s p a c e s o v e r th e fie ld C .

ia a n e le m e n t of V X

=

V. • 1

T h en if

V/, we have

w.

i

1

w h e re th e v^ a n d w. a r e l in e a r l y in d e p e n d e n t in V a n d W r e s p e c tiv e ly (c f. [ 2, $

E x e r c is e

1

6

] ).

P ro o f.

F r o m th e d e fin itio n of V W w e know th a t e v e r y e le m e n t n o f V s r W c a n be w r itte n a s -ZT f • g , w ith f. in V an d g. in W. We V i~ l 1 1 1 1 d e fin e s ® ra n k of x , a s th e s m a ll e s t p o s itiv e in te g e r s u c h th a t x is th e s u m of s

te n s o r p ro d u c ts f • g.

S uppose th e v

r

a T hen if x h a s r a n k a f x =ZZ v. e w, . i= l * i

w e re lin e a r ly d e p e n d e n t.

*

i^ r

c . v. i

i

T h e n w e c o u ld w rite

w ith c . in C. i

B u t th is w ould m a k e x = ■£— i* r

v. e (w . + c. w ) of r a n k l e s s i ' i i r

th a n

s , con*

tr a d ic tin g th e d e fin itio n of s.

S im ila r ly , th e w ^ 's m u s t be lin e a r ly

in d ep e n d en t. T h e o re m 1 w ith u n it.

L e t C be a fie ld a n d le t A b e any c o m m u ta tiv e C * a lg e b ra T h en

P ro o f. x * ZT v^ •

H*(A) » 0.

L et

x be a c o c y c le of r a n k s in A

2*

. We m a y th e n w r ite

w ith th e v^, w^ e le m e n ts of A a n d th e s e ta f v^ j

l in e a r ly in d e p e n d e n t o v e r C .

and £

S ince x i s a c o c y c le w e h av e ^ ( x ) ■ 1 , o r

* (

8

1

*

8

• v. • w .) ( £ v. e w. e X J J

1

) b ZZ v, • 1

1

• w,

w h ic h c a n be w r itte n a s (3 . 1)

a {f2> • e 2 - ^ ‘f t e ^ f f j ) • If D * 'f ( e j) *

+

'f ( « i e 1 ) P '

1

(«,) • e ,

+ f xf 2 [ e ^ e j

a p p ly in g (4 .8 ) w ith f 2 » 1 and e 2 a m in M w e fin d

(f^D) m « f j . D (m ). A lso (4. 8 ) w ith f j = 1 an d f 2 « f „ e 2 = m in M y ie ld s O (fm ) = D(f) • m + f . D (m ). T h u s M is a r e g u la r T -m o d u le u n d e r th e n a tu r a l a c tio n of T on M. It is w o rth notin g th a t (4. 6 ) m u s t be a s s u m e d if (4. 7) is to be v a lid fo r e^ in M.

10

C o n v e rs e ly , if M is a r e g u l a r T -m o d u le w h ic h is a n a b e lia n r e s t r i c t e d L ie a lg e b r a su c h th a t (4 .6 ) is v a lid , we s h a ll c o n s id e r r e g u l a r r e s t r i c t e d e x te n s io n s of M by T w h ic h in d u c e th e g iv en r e g u la r T -m o d u le s t r u c tu r e a n d p -m a p on M. A s u s u a l w e s h a ll sa y th a t a r e g u l a r e x te n s io n E s p lits if t h e r e is a n F - l i n e a r r e s t r i c t e d L ie a lg e b ra h o m o m o rp h is m Y" o f T to E su c h th a t

th e id e n tity on T.

It is th e m a in p u rp o s e of th is

s e c tio n to show th a t f o r a la r g e c l a s s of r e g u la r T -m o d u le s , a ll r e g u la r r e s t r i c t e d e x te n s io n s s p lit. T h e o re m 2 .

L e t F be a fin ite p u re ly in s e p a r a b le e x te n s io n fie ld of C

o f ex p o n en t 1 an d l e t T be th e r e s t r i c t e d L ie a lg e b r a of d e riv a tio n s of F over C.

L e t M be a r e g u la r T -m o d u le w h ich is a s tro n g ly a b e lia n r e ­

s t r i c t e d L ie a lg e b ra i. e. 0 -* M ^

s 0 » [ M, Mj .

L et

E —T —0

be a r e g u la r r e s t r i c t e d e x te n s io n of M by T w h ich in d u c e s th e giv en T > m odule s t r u c tu r e on M .

T hen th e e x te n sio n s p lits a s a r e g u la r r e s tr ic te d

e x ie n si on. B e fo re p ro v in g T h e o re m 2, we r e c a l l th a t the g e n e r a to r s a o f F ij in A m a y be c h o s e n in su c h a w ay th a t w ith 0 - i - p - 1 a r e a C -b a s is of F (cf. e .g . [ 3, P r o p o s itio n 1, p. 190] ).

L e t TQ be th e r e ­

s t r i c t e d L ie a lg e b r a sp a n n e d o v e r C by th e d e r iv a tio n s p

(a ) ■ S

i 1 y

g iv en by

ij

It is e a s ily v e r if ie d th a t [ D ., D^] > 0

an d

D.P = 0

[ 3 , p. 192, (6 )] so

th a t Tq is i t s e l f a s tro n g ly a b e lia n r e s t r i c t e d L ie a lg e b r a .

T hen th e f i r s t p a r t of the p ro o f of T h e o re m 2 c o n s is ts of L em m a 4. j

G iven

T f a n d the e x te n s io n 0 —*-M -*■ E

?

• T —0

a s in T h e o re m 2, th e r e is a C - lin e a r r e s t r i c t e d L ie a lg e b r a is o m o rp h is m o f Tq in to E w h ich i s in v e r s e to V \ P ro o f.

If

[ Tq ! C ] « 1 th en we c o n s tr u c t a n is o m o rp h is m of

T Q in to E q a Vf ’ 1 ) ] ‘ Cp 0 ( D 2 , , [ e 2 * ^ 0 ( D l >1 ] s

is a h om om orphism and

[«£» p^(D^) ] * 0.

°

“lttC®

P 0

H e n c e [ p ^ ( D ^ ) , e 3] a 0.

14 R e p e a tin g th is p r o c e d u r e w ill f in a lly y ie ld a n e le m e n t e w ith Y (e^) = D ^,

e^ =

T h en d e fin in g Y q = p ^

0

,

an d

[ p ^ D ^ ) , e^] =

on V Q a n d

f o r i= l,

0

2

n

of E n 0

............. n - 1 .

Y q (Dft) = ®n g iv e s th e d e s i r e d

m a p , p ro v in g L e m m a 4. 1. P r o o f o f T h e o r e m 2,

G iven

0 -»

M -► E

y -*

T -♦ 0 , l e t

^

b e th e r e s t r i c t e d L ie a lg e b r a m a p f r o m T ^ to E q a s g iv e n by L e m m a 4. 1. S in ce the

c o n s titu te a le ft F - b a s i s o f T [ 3 , P r o p o s itio n 2, p, 191] •

w e m a y d e fin e

T -* E

Dj in Tp an d a d d itiv ity .

by

'f'{i D^) = f * Y q (D^)

T h is Y

f o r f in F

an d

c l e a r l y F - l i n e a r , an d if w e c a n show

th a t i t is a r e s t r i c t e d L ie a lg e b r a h o m o m o rp h is m , th e n i t i s th e r e q u ir e d s p littin g m ap .

B u t by r e p e a te d a p p lic a tio n s o f (4. 1), ( 4 .2 ) , ( 4 .7 ) an d

( 4 .8 ) w e have f o r

I t ( t Di ).y ,(g D j )]-

S in c e i i . a d d itiv e . (4. 9 ) e x te n d , to s u m s a n d w e s e e th a t L ie a lg e b r a h o m o m o rp h is m .

Y la

a

15

(4. 10) % i D .)P) n f i (f D i)p ‘ l C 2 t x - ' | . . .

n + 1 0

n+1D £ ♦------------

£ 2 (x

( x ' 1) \ ---

)

* C n + l> ~ i r ~

H en ce in th e n o ta tio n of S ectio n I I ,

"+ l D £ n . i ( x t l ) + • • • + ---------- i s i t . —

^ n+ l (x

*

b* ,4 - l l »

23 C o m bin ing the la u t two fo r m u la e .

t 2n+3 ,2 n . ^ n + i'* * 1 » _ D“ * ‘ J . .. , 2n - l . " . , 2n

2n+3_.A

2

n+ 2

*

>

1

**

2n +2 t

x 2n - l

S ince w (D ) is in F , fo r

3

^ i + 1 *W

=

3 —

e l

by

'

r

/ "

°

c

Dt

2n + 2\

/

P (D) =

^

2

n+ 2

3

( - f i r )

-

a K2n+2.

(

) • I

^ 2 n - 1

2 ” * 1

We s h a ll show th a t

-

w (D )

( j '+ P ) •

.

2 n , {w (D )» 1 n )

i > 2,

^

2

■ • * - ')

We saw in S e c tio n I I th a t the c o m p le x ( F n , • i

.

*

T h e n $ 2n+Z *W

* l2U *‘

D* \ » '

1

t 2n » l

/

' O 2n +2 V.

3_ ^

I

_ "

•

n

* l2 n H ence

■ ) is e x a c t.

T h ere -

m w

m1

_ q ( we

2“

is in

I m ( £ 2 n + l)

1

T —E

is a n F - U n e a r r e s t r i c t e d

L ie a lg e b r a h o m o m o rp h ism . F i r s t l e t u s no te th a t by [9» L e m m a 4 . 4 , p. 349] , fo r and

D, D* in

T,

t in F n *

24

( 4 . 14 ) " s !i_

.

( J £ i;p

+

v

*

(

^ e ± .j

C o m b in in g (4. 12) an d (4. 14) y ie ld s

(4 .1 5 )

p([D ,D '])P (DP) - p (D)P -

- > W

2

n+ 2 D p (D ') 2 n + 2

2

and

D P’ *p (D) o - w(DP) + w ( o f 4

1

w (D)

- ^ ( D P> .

T h en ( f + p) ( [ D . D*] ) “ [W D ), y 'd f t ] by (4 .1 5 ).

n+ 2 D '(3(D )

B ut

+ 2n+2Dp (D*) -

2

n+ 2 D 'p (D)

[ ( f + P ) D , (Y#+ P ) D , J = [ ? ( D ) .Y ( D ') ] + [ f ( D ) ,P (D #) ]

+ [ P ( D ) . ^ ( D ') ] + [ p ( D ) , P (D ‘) ] « [ r ( D ) i y '( D ,) ] + 2 n+ 2 D p (D )- 2 ll+ 2 t f p P ) s in c e P (D) is in So

K2 n + 2 and [ e , k ] *

Y *(e)(k) .

(V^+ P) i s a L ie a lg e b r a h o m o m o rp h is m .

(Y '+P) (DP> b ^ (D P) + P (D P) =f { D ) P + P(D )P + B ut

2

n+ 2 D P

[ (Y'+PHD)] P = y ( D ) P + p (D )P + 2n+2DPml ( p(D) ) T h e re fo re

Y' and

p

r e g u l a r e x te n s io n .

P(D) by (4 . 15). by (4. 5)

(Y ^p) i s a r e s t r i c t e d L ie a lg e b r a h o m o m o rp h ism . S in ce

a r e b o th F - l i n e a r , so is t h e ir s u m .

so th a t Y ( f + P) =

" 1

Y/= id e n tity on T .

F in a lly p (T ) e K e r ( Y )

T h en th e e x te n s io n s p lits a s a

25

SEC TIO N V

A L L COHOMOLOGY GRO UPS BU T ONE A R E Z E R O

In th is s e c tio n w e s h a ll show th a t a ll co h o m o lo g y g ro u p s b u t th e se c o n d of A m its u r ’s c o m p le x f o r F , a fin ite p u r e ly in s e p a r a b le ex ten * s io n fie ld of ex p o n en t o n e , a r e z e r o . “ + lDA x

_ ,"Dx . £ n \ ------- ~ J to r an y

We have by (4 .1 3 ) th a t n x

in F

n*

,

Thus fo r

x

«d x S n(" 1*x"~"^ “

a c o c y c le ,

n by th e e x a c tn e s s of th e c o m p le x (F , £ *• D in T , w e have

and each a coset

V (D j t)

m o d u lo

a

.

K®

L et

" 1

n ^ —- — t

is in

Im ( £

.) , n» l

We s h a ll d e fin e fo r e a c h c o c y c le t in of

F

n*

tor

n*

T hus th e r e is

F n "* su c h th a t

( F n " 2) B K e r < / n - 1 )

a lg e b r a e x te n s io n

T h 6 n ' afialn

) , fo r e a c h c o c y c le t in F

K e r ^ ^ j ) in

S £

°*

j(V (D ,t) ) n > 2.

a r e s t r i c t e d L ie

Kn * by T,

We se t 0 w h e re

n - 1 ( t ) * ^ n " l D + L (q ) | q in

f “ ’ 1J

,

II* 1 ~ D and L(q) a r e d e fin e d a s in S e ctio n IV. W ith th is d e fin itio n

i t is c l e a r th a t

L e m m a 5. 1

0

V ( B , t)

. (t)c w n- 1 If

t

in

Horn

F ,

_ , j # p n~c

n > 2 ,

( F *1 \

Fn V

is a n A m its u r c o c y c le , th e n

, ( t ) is a r e g u la r r e s t r i c t e d L ie a lg e b r a e x te n s io n of K n- 1

n

1

by T .

3

26

P ro o f. *

[ n “ *D + L ( q ),

a * 1 D , + L ( q ') J

[ n ‘ l D, n ’ l D '] + L < n ‘ 1 D (q ') - n " l D ' ) .

But S n -1

£ a £,< *> • • •

Th*n £1 V‘>* C, €»•*D+i C,"'1**» s

Jn ^ n j th e n a a • • •a

is c o m p le te ly s e ttle d .

1

le a s t in d e x su c h th a t j Deg

and

1

A s su m e now th a t th e le m m a is t r u e fo r a Deg a

a^n

so

n

i

a ,• i

^JEL —:

a p = 1

= It

a

j'

n

n

anda

= aJ

=* k - 1,

b y th e f i r s t in d u c tio n

i s th e s a m e a s i ts e x p o n e n t in

is p - 1

th en

D eg a^ a

J*

—

k -1

36 by th e f i r s t in d u c tio n h y p o th e s is . th e n

j' a tz k

Deg

le t

a. a

D eg

a

J'

J

— I -2-— c_ a . X

3

if

k

in d u c tio n h y p o th e s is . Deg a ^ a ^

k -1 + 1

Deg a^ a ^ ^ k + 1 .

L et

T h en if th e

P ro o f.

x

3

a^

• • • a^ I

D eg * } £ s - 1

)

6

c_ a*) I I

+1 in a

J

by th e

se c o n d

i s p -1 ,

a ^ a^£- k+1 if th e e x p o n en t o f

if

a^ in a ^

th e e x p o n en t of a^ in a ^ i s p - 1 , c o m ­

. 1. r 7T

a , , th e p ro d u c t of ‘j

r f a c t o r s a* in a n *J

T hen D eg x - r ,

T h is i s obv io u s if r

a n d c o n s id e r

1

th e e x p o n e n t of a ^ in a ^ is n o t p - 1,

H ence Deg k

( ^ 7

e x p o n e n t of

k , an d if

1

a rb itra ry o rd e r.

Deg

1

p le tin g th e r o o f of L e m m a .2

J

ex p o n en t of a ^ i n a ^ is n o t p - 1 .

c . a 1) ^

J

3

a

e x p o n en t of a ^ in a ^ is p - 1 .

th e

i s < p -1 a n d Deg a ^ a ^

6

I a ), D eg £

th e

if

6

Now

a 1, k)

w h e re D eg a} i— / k - 1

Deg a

i s < p -1

We h av e th a t

n m a x (D eg a # ( X

S in ce a < $ t

j*

a g a in by th e f i r s t in d u c tio n h y p o th e s is .

m a x (D eg 2 Z c . a l i e

C o r o lla r y

If th e ex p o n en t of a ^ in a

3

1. Now

s

A s s u m e th e le m m a is tr u e f o r r < a a^

• • • a^ 2

by th e in d u c tio n h y p o th e s is .'

c^ a *

3

s

i

T h en

w h e re

a,

. & 1

. . . a . = a. le ll

2

L em m a

.3

6

(£Z c . a *) h a s d e g re e ^ s by L e m m a i

L e t S be th e s u b s p a c e of L sp a n n e d by the a*

D eg a*< n ( p - l ) .

L e t x = a.

a.

X1

n lp -1 ) o f the a 's . r i

...

a. ln < p -l)

l 2

D~ 1 a, ” ... I n

T hen x -

n« J

a

6

. 1.

w ith

be a p ro d u c t of

m od S if e a c h a. i

o c c u r s e x a c tly p - 1 t im e s in x , w h ile x » 0 m o d S o th e r w is e . P ro o f.

We u s e in d u c tio n on th e n u m b e r of tr a n s p o s itio n s r e q u i r e d to

t r a n s f o r m th e o r d e r e d s e t (a. , • •• , a . ) in to th e o r d e r e d s e t ll Xn ( p - 1 ) ................................ ) » «ch th a t 31 n (p -1 )

j

^ j *

. . k+i

If no tr a n s p o s itio n s a r e r e q u ir e d an d e a c h a ^ a p p e a r s p - 1 t im e s th e n by d e fin itio n a. . . . a , = a, ll n (p -1 ) I

p-

1

pa r

... n

1

_ p- 1 = a. ...a i n

p-

1

. , (m od S) .

If no tr a n s p o s itio n s a r e r e q u i r e d , an d if q is th e g r e a t e s t in d e x s u c h th a t a

q

a p p e a r s m o r e th a n p - 1 t im e s , a n d a

q

a p p e a rs

p + t tim e s ,

th en x = a. • • * a . = a. • • • a ‘l ‘M p -O ‘l q

t

. a

• a q

p- 1

. . . a.

q

• n (p -1)

t

L e t th e n u m b e r of f a c t o r s up

to an d in c lu d in g a ^ be u a n d th e re m a in in g

n u m b er fro m a

be v.

to a .

q L em m a

6

. 1 we

T h u s u + v = n ( p - l)

.

Now

ln(p-l) h a v e , s in c e a ^ * * • • a. is an q ln ( p - l )

a*,

by

D eg a

• a p * • . . a. - D en a p " * . . . a . “ q ln (p -D q V

27 c .

T hus a q

. a p“ * . . . a. = q l n ( p - l) I

a* w ith Deg a * v

= v - 1 . o - 1.

H en ce

1

x Is a l i n e a r c o m b in a tio n of t e r m s e a c h of w h ic h h a s u + v - l a n ( p - l ) - l f a c to r s a . .

T hen x l i e 3 In

S by C o r o lla r y

6

.2 .

J Now a s s u m e th a t L e m m a tio n s

6

. 3 is t r u e if l e s s th a n k t r a n s p o s i ­

a r e r e q u i r e d to t r a n s f o r m the o r d e r e d s e t

(a , ll

a.

a. ) to (a l n (p - l)

L e t sr. =■ (a . , a . ). ls l o+l

) su c h th a t j. *= J. . w(p” l)

T hen

x s a . . . . a. a, a, a , . . . a, I Xs - l a+1 s s+ 2 n (p -l) + a. • • • a. [ a . * a. ] a, ... II s-1 a ls+ l a+ 2 Since

a, n (p - l)

k -1 tr a n s p o s itio n s a r e r e q u ir e d f o r th e s e t

(a, • • • • . a ,

’ *{ s-1

1

'

*••*» a .

s+1

s

s+2

) * a (p -l)

th e n by th e in d u c tio n a s s u m p tio n ,

r p- 1 ... )a l V

1

,,ai

. - 1

i a i . +a1. * 1 . &i . «a /

S ince by C o r o lla r y