72 VIII. W I T T INVARIANT S
By combinin g (30.2) , (30.3) , an d (30.4) , on e obtain s th e require d formula :
P R O P O S I T I O N 30.5 . q
E
= 5 - 2q
E
+ (2)X 2qE
Note tha t th e formul a hold s i n WGr(k), no t jus t i n W(fc) , sinc e bot h side s
have ran k 3 .
R E M A R K 30.6 . I f k contain s th e 8t h root s o f unity , w e hav e (2 ) = (—1) = 1
and 2 = 0 i n W(k), s o tha t 30. 5 ca n b e rewritte n i n a simplifie d way :
(30.50 IE* = 1 + A 2 ^ i n W(k).
30.7. T H E SEXTI C RESOLVEN T O F A SEXTI C EQUATION . Le t E b e a n etal e
algebra o f ran k 6 , an d le t E' b e th e etal e algebr a obtaine d fro m E vi a a n oute r
automorphism o f SQ ("sexti c resolvent") . Le t q an d q f denot e th e respectiv e trac e
forms.
By Theore m 29.1, i n orde r t o describ e q f i n term s o f q, i t i s sufficien t t o d o
so whe n E i s multiquadratic , i.e. , whe n E split s a s a produc t o f thre e quadrati c
algebras:
E = E
a
xEbx E
c
,
where E
a
= k[x]/(x 2 - a) , ... , E
c
= k[x]/(x 2 - c) wit h a , 6 , c G k*. The n
(30.8) q = (2)(1,1, l,a,6,c - 3(2 ) 0 (2(a,6,c .
Let C = Z/2 Z x Z/2 Z x Z/2 Z an d identif y C wit h it s imag e i n 5
6
unde r th e
embedding correspondin g t o th e partitio n o f { 1 , . . . , 6 } a s {1, 2} U {3,4} U {5, 6}.
To determin e th e structur e o f E', w e hav e t o loo k a t th e imag e C o f C unde r a n
outer automorphis m o f 56 -
Such a n automorphis m transform s a transpositio n int o a produc t o f thre e dis -
joint transposition s an d conversely . B y construction , th e grou p C contain s onl y
one produc t o f thre e disjoin t transpositions , represente d b y th e diagona l subgrou p
D. Thu s th e grou p C contain s onl y on e transposition , whic h generate s th e imag e
D' o f D. Thi s give s a decompositio n o f { 1 , . . . , 6 } int o tw o orbit s (say ) {1,2} an d
{ 3 , . . . , 6} , where the actio n o n { 3 , . . . , 6 } is the fre e actio n o f C'/D' = Z/2 Z x Z/2Z .
This correspond s t o a decompositio n o f E' a s a direc t su m o f th e quadrati c algebr a
Eabc an d th e biquadrati c Galoi s algebr a E
a
b®Eac (whic h i s isomorphic t o E^^E^
and E
ac
0 E
hc
). Tha t is ,
E = E
abc
X (E
a
b 0 E
ac
)-
Hence:
q' = (2 , 2abc) 0 (2 , 2ab) 0 (2 , 2ac) = (2 , 2abc) 0 (1, aft, ac, be).
Setting
7i = (a , b, c), 7 2 = (ab , ac, 6c, 7 3 = (a6c) ,
one has :
(7=2}(3 + 7i )
A2g - 3 + 37 1 + 7 2
A 3 g = ( 2 ) ( l + 3
7
i + 3
7
2 + 7 3 )
^ = (1,2) + 7 2 + (2)
7 3
.
This implies :
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