32. TRAC E FORM S WIT H VANISHIN G STIEFEL-WHITNE Y CLASSE S 8 3

Conversely, if c' G c-k\ an d (2 ) •(c/) = 0, the form (2 , c') is isomorphic to (1, 2d).

Hence

(1) 0 (1, c', c', c', d) 9 * (2, 2, c', c', c', c') ^ (2,1, 2c', c', c', c',

i.e., (l,d ,d ,d ,d) = (2,2d,d

1

dJd)1whic h show s tha t (1, c^c', c',c') represent s 2 ;

since thi s for m i s isomorphic t o / , w e see that / represent s 2 . •

This lemm a allow s u s to rewrit e Theore m 32. 5 in the followin g way :

THEOREM

32.10. Let q be a quadratic form of rank 6 with Wi(q) = 0 and

w2(q)

= 0 . For q to be a trace form, it is necessary and sufficient that there exists

c e k* with the following properties:

(1) (2)-(c ) = 0 inH

2(k);

(2) ^ ( l , l , c , c , c , c ) ;

(3) c is a sum of 4 squares in k(y/2).

Indeed w e have

32.5.1 and 32.5. 2 = 32.10.1 an d 32.10.2

by Lemm a 32.9 , and 32.10.3 is equivalent t o 32.5.3 .

32.11.

PROPERT Y OQ.

Not e tha t q = (1, l,c, c, c, c) i s isomorphi c t o th e uni t

form (1,1,1,1,1,1 ) i f an d onl y i f c i s a su m o f 4 squares . Ou r origina l questio n

about trac e form s ca n the n b e translate d int o th e following :

/"}o 1 o\ I s it true that 32.10.1 and 32.10.3 imply that c is a sum of 4 squares

{61Al)

ink?

If i t i s s o (fo r a give n field fc), we sa y tha t k has property OQ. I n othe r words , k

has propert y OQ if an d onl y i f the onl y trac e for m o f ran k 6 over k wit h vanishin g

Stiefel-Whitney classe s i s the uni t for m (1,1,1,1,1,1).

REMARK

32.13. Fo r c e k*, th e followin g ar e equivalent :

(i) c is a su m o f 4 squares.

(ii) Th e 3-Pfiste r for m (1, -c) -(1,1,1,1 ) i s hyperbolic .

(hi) On e ha s (c ) -(-1) -(-1) = 0 in H 3(k).

((i)44 (ii) i s standard ; (ii ) ^ (iii ) follow s fro m th e theore m o f Merkurje v quote d

in 31.14.)

Moreover, i f w e multipl y c by a su m o f 4 squares , th e elemen t (c ) •(—1) •( —1)

does no t change . I t i s thu s a n invariant o f th e for m q = (1,1 , c, c, c, c), whic h w e

shall denot e b y e(q); this invarian t wil l play a n essentia l rol e i n §33.

32.14.

EXAMPLE S O F FIELD S WHIC H HAV E PROPERT Y

0

6

.

32.15. An y field for whic h k\ = k* has property OQ. Thi s applie s if cd2(&) 2

or i f (—1)-( —1) = 0 i n H

2(k)

(sinc e i n tha t case , th e Pfiste r for m (1,1,1,1 ) i s

hyperbolic). I n particular , a field of nonzero characteristi c ha s propert y OQ.

32.16. A p-adi c field (wit h finite residu e field) ha s propert y 06 . Thi s follow s

from 32.15.

32.17. A numbe r field ha s propert y OQ. Indeed , i f c has propert y 32.10.3, i t

is totally positiv e i n fc(\/2), henc e als o i n & , and i t i s well-known tha t thi s implie s

that c is a su m o f 4 squares.

In orde r t o giv e further examples , w e need th e followin g result :