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 :
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