76
IX. TH E TRAC E FOR M I N DIMENSIO N 7
COROLLARY
31.7. For a quadratic form q of rank n to be a trace form, it is
necessary that q contains r n.
In wha t follows , thi s necessar y conditio n wil l b e calle d th e trivial one . Fo r
n 3 , ther e ar e othe r conditions , stemmin g fro m Th . 25.13 (vanishin g o f th e
higher Stiefel-Whitne y classe s of q) and fro m Th . 29. 4 (linea r relation s between th e
exterior power s o f q) .
REMARK
31.8. Th e subform r
n
i s the "larges t fixed part" o f g#, in the followin g
sense. Fi x a ground field &o as usual. Writ e g# a s rn 0 q'
E
, where q f
E
has rank n h.
Then th e "map " E i— q'
E
ha s n o fixed part , i.e. , ther e i s n o x G k$ whic h i s
represented b y ever y q'
E
(fo r ever y k/ko an d ever y E G Etn(/c)).
To prove this, on e decompose s n a s abov e int o sum s o f terms 2
mi
an d choose s
for eac h i a n z-Pfiste r for m pi. Defin e qi = (2
rni)
-pf, the for m q = 0 ^ i s a trac e
form o f rank n. (T o check this, one may assum e that n i s a power o f 2, say n
2m
,
then a tensor produc t o f m suitabl y chose n quadratic algebra s has the desire d trac e
form.) On e then use s the fac t tha t th e pur e par t o f a generic Pfiste r for m doe s no t
represent an y elemen t o f /co , as follow s fo r instanc e fro m Prop . 28.1.
31.9. TH E CASE S n = 1,2,3. I n thes e cases , th e trivia l conditio n i s sufficient .
In othe r words :
THEOREM 31.10. (1) If n = l, q is a trace form if and only if q = (1).
(2) If n = 2, q is a trace form if and only if it contains (2), i.e., if and only
ifq= (2 , t) for some t G k*.
(3) If n = 3, q is a trace form if and only if it contains (2,1), i.e., if and only
ifq= (2,1,*) for some t G k*.
PROOF.
Th e necessit y follow s fro m Corollar y 31.7. Th e sufficienc y i s clear fo r
n = l. Fo r n = 2 , if q = (2 , £), we choose for E th e quadratic algebr a k[X]/(x
2
a),
with a = 2t ; it s trac e for m i s (2,2a ) = (2,t ) = q. Fo r n = 3 , i f q = (2 , l,t), w e
choose for E th e product o f k and the quadratic algebra k[X]/(x 2 2i) as above.
31.11.
SPECIA L FORM S O F RAN K
3 . I n order t o handle the case s n = 4,5,6,7 ,
we need t o stud y som e particular quadrati c form s / o f rank 3 . Le t d = d(f) b e th e
discriminant o f / .
PROPOSITION
31.12 . The following properties are equivalent:
(1) (d) © / contains (1), i.e., (d) 0 / represents 1.
(V) (d) © / is a 2-Pfister form.
(1") d is represented by (1) 0 (d) •/.
(2) (i ) e (d) •/ s* (d) © /.
(2') A
3(/)
- A
2(/)
+ X\f) - A°(/ ) = 0 in WGr(k).
(3) There exists a G k* which is represented by f and is such that (a) -(d) = 0
inH2(k).
PROOF.
Th e equivalenc e o f (1)^ (1') follow s fro m th e fac t tha t a for m o f
rank 4 is a Pfister for m i f and onl y i f it represent s 1 and it s discriminan t i s 1. Th e
equivalence (1)^ (1") i s obvious (multiplicatio n b y (d)) .
(2) = (1) i s clear. I f (1;) i s true, the n th e Pfiste r for m (d) 0 / represent s (d) ,
hence i s isomorphi c t o (d) -((d) 0 / ) = (1) 0 (d ) •/. Henc e (2 ) i s equivalent t o (1),
(1'), an d (1").
Previous Page Next Page