92
IX. TH E TRAC E FOR M I N DIMENSIO N 7
We hav e
w2(qx) = (-l)-(A) + (2HB).
Since W2(q
x
) 0, thi s implie s tha t (-l)-(A) = (2 ) -(B). Th e tw o quadrati c form s
(1,2,5) an d (A,A,2B) hav e th e sam e w\ an d W2, hence ar e isomorphic . Thi s
allows u s to rewrit e q
x
a s
qx = (1,2,B) ©{1,2,5} = (1,1,2,2,5,5).
Since 2 and B ar e sum s o f two squares, w e hav e
(2,2) = (1,1 ) = (5,5) ,
hence q
x
= (1,1,1,1,1,1 ) an d e(q x) = 0 .
34.5. No w le t G b e a finite grou p wit h a 2-Sylo w subgrou p H isomorphi c t o
QIQ. Le t e G Inv(i7, Z/2Z) b e th e invarian t define d i n (34.3) .
PROPOSITION
34.6 . Let b e Inv(G , Z/2Z) be the corestriction of e as defined
in 14.3. Then:
(i) The restriction of b to H is equal to e.
(ii) The invariant b is unramified.
(iii) We have b ^ 0 if the ground field ko is Q .
PROOF. Le t N b e th e normalize r o f H i n G . Th e grou p N/H i s of odd order .
On th e othe r hand , th e grou p o f automorphism s o f H i s easily see n t o b e o f orde r
16. Thi s show s tha t th e natura l homomorphis m N/H Out(H) i s trivial . I n
other words , an y automorphis m o f H o f the for m x i— gxg~
x
wit h g G N i s inner .
By Prop . 13.1 , this implie s tha t suc h a n automorphis m fixes e . Henc e e ha s th e
first propert y o f Prop. 15.9. B y Prop . 34.4 , it als o has th e secon d propert y o f 15.9.
This implie s (i) .
Property (ii ) follows fro m th e analogou s property o f e, combined wit h the com -
putation o f th e residu e o f a corestrictio n (se e 8.6) . A s fo r (iii) , i t i s obviou s sinc e
e = Res(6 ) i s nonzero whe n ko = Q , se e 33.26 an d 32.31.
THEOREM
34.7 . If G has a 2-Sylow subgroup isomorphic to the quaternion
group Qi6, then Rat(G/Q ) is false.
This follow s fro m Prop . 34. 6 combined wit h Cor . 33.11.
COROLLARY
34.8 . Noether's problem has a negative solution for G over Q .
Examples o f such group s G are SL2(F
g
) wit h q = 7 or 9 (mo d 16). (Not e tha t
the case s q = 7 and q = 9 have alread y bee n treate d i n 33.27 . Not e als o tha t th e
results abov e contai n a s special case s thos e o f Th. 33.2 6 an d 33.27. )
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