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.