32. TRAC E FORM S WIT H VANISHIN G STIEFEL-WHITNE Y CLASSE S 8 5
P R O P O S I T I ON 32.25 . The quadratic form q = ( 1 , 1 , - 1 , - 1 , - 1 , - 1is ) a trace
form over k with trivial Stiefel-Whitney classes; it is not isomorphic to the unit
form (1,1,1,1,1,1).
COROLLARY 32.26 . The field k does not have property OQ.
P R O O F O F P R O P. 32.25 . T o se e tha t q i s a trac e form , on e applie s Th . 32. 5
with c = —1. Conditio n 32.5. 1 i s (2)*(—1) = 0 , whic h i s clear ; conditio n 32.5. 2
is obvious . T o chec k 32.5.3 , observ e tha t 7 ca n b e writte n a s u 2 + v 2 + w 2 ove r
k\ k(y/2) b y choosing , e.g. , u = 1, v = \/2 , an d w = 2 . I f xi denote s th e imag e
in k o f th e indeterminat e Xi, w e hav e
x\-\ h x\ + u 2 + v 2 + w 2 = 0 .
Hence th e ran k 8 for m I/J (1,1,...,1 ) i s isotropi c ove r k\, whic h i s equivalen t t o
saying tha t 1 i s a su m o f 4 square s ove r k\ .
It remain s t o prov e tha t 1 i s no t a su m o f 4 square s ove r k. Le t fi b e th e
quadratic for m (1,1,1,1,1,7 ) an d le t Q(0 ) b e th e functio n field o f th e affin e con e
0 = 0 , i.e. , th e field o f fraction s o f Q[Y
U
. . . , Y
6
]/(Y2 + ---+7
5
2 - f 7Y 2). Th e field k
defined abov e embed s int o Q(0 ) b y Xi \-^ YI/YQ, fo r i = 1 , . . . , 5 . I f 1 is a su m o f
4 square s ove r fc, then ip splits ove r Q(0) . B y [Scha85 , p . 155, Th . 5.4(h)] , applie d
with K = Q an d a 1, thi s implie s tha t ^ contain s 0 , henc e (1,1,1 ) contain s (7) .
That is , 7 i s representabl e b y (1,1,1 ) ove r Q , whic h i s no t true . Henc e 1 is no t a
sum o f 4 square s ove r k.
(The onl y propert y o f 7 tha t w e hav e use d i s tha t i t i s no t a su m o f 3 square s
over Q , bu t become s on e ove r Q(v / 2).)
Q U E S T I ON 32.27 . Th e field k fro m 32.2 4 i s a n extensio n o f Q o f transcendenc e
degree 4 . D o ther e exis t extension s o f Q o f smalle r transcendenc e degre e (e.g. ,
equal t o 1) whic h fai l t o hav e propert y OQ?
32.28. T H E CASE n = 7 . Thi s cas e reduce s t o th e cas e n = 6 , a s explaine d i n
§31. I n particular :
T H E O R E M 32.29 . Let q be a quadratic form of rank 7 over k, with w\(q) 0
and W2{q) = 0 . For q to be a trace form, it is necessary and sufficient that:
(a) q contains (2,1,1);
(b) q = (1,1, l,c, c , c, c) for some c G &*;
(c) q ^ (1,1,1,1,1,1,1 ) over k(V2)-
This follow s fro m Theore m 32. 5 b y usin g th e fac t (Cor . 31.24) tha t q is a trac e
form i f an d onl y i f i t ma y b e writte n a s (1) ®g ; wher e q' i s a ran k 6 trac e form .
Similarly:
T H E O R E M 32.30 . The following are equivalent:
(1) k has property OQ.
(2) The only rank 7 quadratic form having properties (a) , (b) , and (c ) of
Theorem 32.2 9 is the unit form (1,1,1,1,1,1,1).
In othe r words , i f on e define s "propert y O7 " i n a n obviou s way , the n O 7 i s
equivalent t o OQ.
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