33. APPLICATION OF TRACE FORMS TO NOETHER'S PROBLEM 89

In particular :

PROPOSITION 33.22 . The image of the map

tf^An) -+ H\k,An) - H\k,S n) = Et n(fc)

is equal to the classes of rank n etale algebras E with Wi(qE) = 0 and

W2{C[E)

— 0.

33.23. A COHOMOLOGICA L INVARIAN T FO R AQ AND A

7

. Se t U = 6 Or 7, le t

t b e a n elemen t o f H

l(k,

A n), an d le t E(t) i n Et n(fc) b e th e correspondin g etal e

algebra. B y 33.22, we may appl y the result s of §32 to E(t). I n particular, th e trac e

form o f E(t) i s of the for m

(1,1, c , c, c, c) (fo r n — 6) o r (1,1,1c, , c, c, c) (fo r n = 7 )

for a suitabl e c G k*. Moreover , th e elemen t (c ) •(—1) •(—1) G H 3(k) doe s no t

depend o n the choic e of c (as explained i n 32.13); let u s denote i t b y e(i). W e ma y

view e as a cohomological invariant

ei&ikM^H^k).

PROPOSITION

33.24 . The invariant e defined above is normalized and unrami-

fied; if the ground field ko is Q, this invariant is not 0 .

PROOF. Tha t e is normalized i s clear; tha t i t i s unramified follow s fro m The -

orem 32.10 an d Lemm a 32.18. T o prov e tha t e ^ 0 i f ko = Q , on e need s onl y

to construc t a n extensio n k/Q an d a n elemen t t o f H

l(k,

A n) wit h e(t) ^ 0 ; thi s

has bee n don e i n 32.25 , by choosin g k t o b e th e functio n fiel d o f the affln e quadri c

X\ + • • • + Xl + 7 = 0 . •

THEOREM

33.25 . Noether's problem has a negative solution for

AQ

and Aj

over Q .

This follow s fro m Remar k 33.12.1.

The metho d w e have use d give s a bit more , namely :

THEOREM

33.26 . Let G be a subgroup of odd index of Af. Then Noether's

problem has a negative solution for G over Q .

(In fact , w e shall prov e th e stronge r resul t tha t Rat(G/Q ) i s false. )

PROOF. B y composition , e defines a cohomologica l invarian t

eG: H l(k,G) - • H^k.Aj) U H 3{k).

It i s clear that e

G

i s unramified an d normalized . Le t u s show that i t is not 0 . Le t H

be a 2-Sylow subgroup of A-j containe d i n G. (I t i s easy to see that H i s isomorphi c

to th e quaternio n grou p QIQ of order 16, but thi s wil l no t pla y an y rol e here. ) B y

Remark 32.31, there exist s a n etal e algebr a E o f rank 7 over a suitable field k suc h

that e(E) i s not zer o and E i s split b y a Galoi s extension o f k o f degree a power of

2. Suc h a n algebr a come s fro m a homomorphism 1^— • Aj — A? whos e imag e i s a

2-group. Hence , i t come s fro m a homomorphis m r & — • iJ , henc e fro m a n elemen t

of H x{k, G). Thi s show s that e

G

i s not 0 . •