68
VIII. WIT T INVARIANT S
The proo f i s the sam e a s that o f Theore m 21.6.
REMARK
27.23 . Th e theore m above , a s well as Theorem 21.6 giving the coho -
mological invariant s o f Herm n, ca n b e state d a s
InvnOTmHerm„)fel/feo=ker [lnvnorm
Quad
n/fc0
- + Inv
norm
Quad
n/fcl
where Inv
norm
denote s th e normalize d invariants .
28. Quadrati c for m invariant s o f Pfister
n
an d Quad
n
In thi s section , w e conside r morphism s A Quad^ , whic h w e cal l quadratic
form invariants; w e write Inv(A , Quadjv) fo r th e se t o f al l suc h morphisms . (Not e
that thes e ar e not strictl y invariant s i n the sens e of §1because Quad ^ take s value s
in th e categor y Set s instea d o f th e categor y Abelia n Groups. ) Firs t a n auxiliar y
result:
PROPOSITION
28.1. Let x\, ..., x
n
be indeterminates, and k = fco(#i xn).
For each Id [l,n ] let xj denote Yl iejXi, and let pi be an anisotropic quadratic
form over ko . Then the quadratic form
^2^i i
xi)
i
is anisotropic over k.
PROOF.
Th e case n 1 is standard. Th e general case follows by induction.
COROLLARY
28.2 . Let po and p\ be anisotropic quadratic forms over k$. For
each n 1, there is an n-Pfister form q defined over an extension k of ko with
^o + V? i '{Q ~ 1) anisotropic over k.
Recall fro m 27.2 0 that , i f q is a quadrati c for m o f ran k n whic h represent s 1,
we write q 1 for th e quadrati c for m q' of rank n 1 such tha t g = (l)®g' .
PROOF. Se t k = ko(xi,... ,x
n
) r xi,...,x
n
indeterminates , an d se t q
(1, x\) 0 (g) (1, x
n
). I n th e notatio n o f Proposition 28.1,
A) + V? i '(Q ~ 1) = ^2 fr(xi), wit h pi =
I
Proposition 28. 1 gives the result . D
COROLLARY
28.3 . Letpo, .. ., p
n
be anisotropic quadratic forms overko- There
exists an n-dimensional quadratic form q defined over an extension k of ko with
Yl7=o ^ '^(Q) anisotropic over k.
PROOF. Se t k = ko(xi,... ,x n) fo r x\,...,x
n
indeterminates , an d se t q
(xi,... ,x
n
). Th e clai m i s no w reduce d t o Propositio n 28.1, just a s th e proo f o f
Corollary 28. 2 did .
THEOREM
28.4 . Every invariant in Inv(Pfistern, Quad^ ) can be written uniquely
as
q^ PoBpiiq- 1)
with po, pi G Quad(A:o), with p\ anisotropic.
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