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 .
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