31. CHARACTERIZATIO N O F TRAC E FORM S I N DIMENSIO N 7 7 7
We have A
3(/)
= (d) , A
2(/)
= (d) •/, A^/ ) = / , an d A°(/ ) = (1). Henc e (2' )
is equivalent t o (2) .
We now sho w (1)^ (3), which wil l conclud e th e proof .
(1) = (3): B y (1), we have a n equatio n 1 = dx 2 + a, wher e a i s represented b y
/ . I f a i s not 0 , we hav e
(a) \d) = (a) -(dx2) = (a ) -(1 - a) = 0 ,
hence (3) . I f a = 0 , then d is a squar e an d (3 ) i s obviously true .
(3) = (1): I f a i s a s i n (3) , th e fac t tha t (a) -(d) = 0 mean s tha t a ma y b e
written a s u 2 d-v 2 wit h u, v i n k. Thi s show s tha t u 2 = d-v 2 0 a i s represente d
by (d) 0 / . I f u ^ 0 , thi s implie s tha t 1 is represente d b y (d) 0 / ; i f w = 0 , the n
(d) 0 / i s isotropic, henc e represent s 1.
A quadrati c for m / o f ran k 3 wit h propertie s (1) throug h (3 ) abov e wil l b e
called special Not e that , ove r a finite field, ever y / i s special : use , fo r instance ,
(1) o r (3) . (Th e sam e i s true ove r an y field k suc h that cd 2(k) 2, see Prop. 31.14
below.)
REMARK
31.13I . f / become s specia l ove r a n odd-degre e extensio n o f k, the n
/ i s special. Thi s follows , fo r instance , fro m conditio n (1).
Here i s a less elementary characterizatio n o f special forms :
PROPOSITION
31.14 . Properties (1), ... , (3 ) are equivalent to:
(4) w 3(f) = 0mH
3(k).
PROOF.
Le t u s write / a s (a , 6, c) wit h d = abc. Defin e
q2 = 1 0 (d) -f = (1, ab, ac, be) an d q
3
= (1, —d) -q2.
These form s ar e Pfiste r forms . Th e Elman-La m invarian t e
2
(q2) a s i n §18 is give n
by:
e2te) = {-ab)'(-ac)
= (a)-(6 ) + (6)-(c ) + (c)-(a ) + (-l)-(-a6c )
= w2(f) + (-l)i-d).
The invarian t o f q% i s thus:
e3(/3) = (d ) -62(92) = (d ) -w2(f) (sinc e (d ) -(-d) = 0) .
= w 1(f)-w2(f)=w3(f), cf . (19.3).
By a well-know n theore m o f Merkurje v (se e e.g. [Ar75 , Prop. 2]) , q
3
i s hyperboli c
if an d onl y i f it s invarian t i s 0 , i.e. , i f an d onl y i f (4 ) i s true . O n th e othe r hand ,
the ver y definitio n o f ^ 3 shows tha t i t i s hyperboli c i f an d onl y i f d i s represente d
by q
2
, whic h i s condition (1). Henc e (1) an d (4 ) ar e equivalent .
In wha t follows , w e shall nee d anothe r characterizatio n o f special forms .
PROPOSITION
31.15 . Assume \k\ 3. Properties (1), ... , (4 ) are equivalent to:
(5) There exist A, B in k* with A 2 - B ^ 0 such that
f ^ (A, A 2 - B,AB(A 2 - B)).
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