82 IX. TH E TRAC E FOR M I N DIMENSIO N 7
THEOREM
32.5 . Let q be a quadratic form of rank 6 such that w\(q) = 0 and
w2(q)
= 0 - For q to be a trace form, it is necessary and sufficient that:
(1) q contains (2,1);
(2) q = (1,1, c, c, c, c) for some c G fc*;
(3) g becomes isomorphic to (1,1,1,1,1,1ove ) r A:(\/2).
First w e note tw o lemmas .
LEMMA
32.6 . Every quadratic form
I/J
of rank 6 uz£/i
^i(V0 = ( 1) and W2WO ( —1)'(—1 ) + (& ) '(b) for some a, b G k*
is isotropic.
PROOF.
Thi s ca n b e foun d i n Pfiste r [Pf66 , p . 123, 3']. Pfiste r formulate s th e
hypotheses i n term s o f invariant s u d(i/j)v an d "c^)" ; thes e invariant s ar e relate d
to our s b y
dtyO = (-l ) + u;iW O an d c(^ ) = (-1) -(-1) + w
2
tyO,
so that ou r hypothese s mea n d(i/j) = 0 and c(t/j) = (a) -(b). D
LEMMA
32.7 . Le t f be a quadratic form of rank 5 suc/ i that w\(f) = 0 an d
W2(/) = 0 . T/ie n / = (l,c , c, c, c) /or som e c e k*.
PROOF.
Le t ip be th e for m (-1) 0 / . On e ha s wi(ip) = (-1) an d w 2(ip) = 0 .
This implie s tha t i/j is isotropic b y Lemm a 32. 6 applie d wit h a b = —1.
Hence / represent s 1, and w e may writ e i t a s (1) 0 g. Le t c be an y elemen t o f
k* represente d b y g. W e ma y writ e g a s (c ) -((1) 0 h) wit h rank(/i ) = 3 . On e ha s
w\(h) 0 an d w
2
(h) = 0 , henc e h i s isomorphi c t o (1,1,1) . Thi s show s tha t g i s
isomorphic t o (c , c, c, c).
PROOF O F THEORE M
32.5 . Necessity : I f q = ## , (1) i s the "trivia l condition "
of 31.7. W e may then writ e q as (1) 0 q' with rank(g
/)
= 5 . Lemm a 32. 7 applied t o
f q' show s that q has propert y (2) .
To prove property (3) , we may assume that 2 is a square in k. W e may write q =
(1,1)0(c -(1,1,1,1). B y Th. 31.26, q contains (2,1,1) ^ (1,1,1)i.e., , (c ) -(1,1,1,1)
represents 1, and henc e c is a su m o f 4 squares. Thi s give s (3) .
Sufficiency: B y Th . 31.26, w e only hav e t o chec k tha t q contains (2,1,1) ove r
fc(\/2), whic h i s obvious fro m (3) .
32.8. A
REFORMULATIO N
O F
THEOREM
32.5 . Le t us write k\ fo r the subgrou p
of k* mad e u p o f element s whic h ar e represente d b y th e 2-Pfiste r for m (1,1,1,1),
i.e., whic h ar e sum s o f four squares . W e have (c,c,c,c) = (c', c', c', c') i f and onl y if
cf lie s i n ck\.
LEMMA
32.9 . / = (1, c, c, c, c) represents 2 if and only if there exists c' G c-k%
with (2)-(c /) - 0 inH 2(k).
PROOF.
I f / represent s 2 , w e ma y writ e 2 a s x 2 0 cz , wit h eithe r z 0 o r
z G k%. I n th e firs t case , 2 is a square i n /c , and w e take d = c . I n th e secon d case ,
we take c' = cz. Sinc e (2 ) •( —1) = 0 , we hav e
(2)-(C^) = (2)-(-
C 2
) = (2)-(x 2 -2 ) = 0.
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