CHAPTE R I X
The trac e for m i n dimensio n 7
In thi s chapte r (excep t i n 33. 1 throug h 33.13) th e characteristi c i s assume d t o
be + 2.
As i n Chapte r VI , w e pu t E\k) = iT(/c , Z/2Z) an d H(k) = ©iT(/c) .
3 1 . C h a r a c t e r i z a t i o n o f t r a c e form s i n d i m e n s i o n 7
In thi s section , q i s a quadrati c for m ove r k o f ran k n. Fo r n 7 , w e giv e a
criterion fo r q t o b e isomorphi c t o a trac e for m qg wher e E i s a n etal e algebr a o f
rank n ove r k.
31.1. A NECESSAR Y CONDITION . (Cf . [Se84 , App . I] ) Le t E b e a n etal e algebr a
over k o f ran k n. Writ e n i n dyadi c for m a s n = Yli=i 2 m* fo r 0 m i m^ .
P R O P O S I T I ON 31.2. There exists a finite extension k'/k of odd degree such that
E' = k' (g ) E splits as a product of etale algebras E[ with ranki^ ' = 2 m i .
P R O O F . B y usin g Sylow' s theore m fo r th e prim e 2 , on e show s tha t k'/k ca n
be chose n suc h tha t th e "Galoi s group " o f E' k'' ® E (i.e. , th e imag e o f pE f) is
a 2-group . Th e resul t follow s b y decomposin g [l,n ] int o th e orbit s o f thi s group ,
cf. [Se84 , Lemm a 3] .
Let u s no w defin e a quadrati c for m r
n
o f ran k h (wher e h i s a s above ) b y th e
formula
(31.3) r
n
= ( 2 m i , . . . , 2 m ^ ) .
Note tha t (2 m ) i s isomorphi c t o (1) i f m i s eve n an d t o (2 ) i f m i s odd ; moreove r
(2,2) i s isomorphi c t o (1,1) . Usin g thes e facts , on e see s tha t r
n
ca n b e writte n
more simpl y a s
f ( l , l , . . . , l ) i f £ ^ i i s eve n
I ( 2 , 1 , . . . , 1) i f 2 ^ mi i s odd .
E X A M P L ES 31.5. Fo r n = 1 , . . . , 8 , w e have :
n = (l) , r
2
= (2) , r
3
= (2,l) , r
4
= (1),
r
5
= (1,1) , r
6
= (2,1), r
7
= (2,1,1), r
8
= (2) .
Recall tha t a quadrati c for m q i s sai d t o contain a quadrati c for m q' i f q f i s
isomorphic t o a subfor m o f q.
P R O P O S I T I ON 31.6. [Se84 , Prop . 4 ] The trace form q
E
contains r
n
.
P R O O F . B y a well-know n theore m o f Springe r [La m 73, VII.2.3], i t i s sufficien t
to prov e thi s afte r makin g a n odd-degre e extensio n o f k. W e the n appl y Propositio n
31.2, togethe r wit h th e obviou s fac t tha t th e trac e for m o f E[ contain s (2 rrii).
75
http://dx.doi.org/10.1090/ulect/028/11
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