31. CHARACTERIZATIO N O F TRAC E FORM S I N DIMENSIO N 7 7 9

EXERCISE 31.20. Sho w tha t th e trac e for m o f E(A,B) i s isomorphic t o
(1,AB,A2 -B,A(A 2 -B)).
31.21. T H E CASE n = 5 . Her e th e trivia l conditio n o n q i s tha t i t contain s
{1,1), i.e. , tha t q = (1,1) 0 / , wit h r a n k / = 3 .
T H E O R E M 31.22. For q to be a trace form, it is necessary and sufficient that f
be special
PROOF. Necessity : I f q i s isomorphi c t o a trac e for m g# , ther e i s (Prop . 31.2)
an odd-degre e extensio n k'jk ove r whic h E split s a s a produc t o f k' an d o f a ran k 4
etale algebra . B y 31.18, this implie s tha t / become s specia l ove r /c' , hence i s specia l
over k (cf . Remar k 31.13).
Sufficiency: B y Theore m 31.1 8 ther e i s a n etal e algebr a E' o f ran k 4 whos e
trace for m i s isomorphi c t o (1) 0 / . Th e algebr a E = k x E' ha s trac e for m q.
We coul d als o hav e deduce d Theore m 31.22 directl y fro m Theore m 31.1 8 an d
from th e followin g genera l result :
T H E O R E M 31.23. If E is an etale algebra of odd rankn, there is an etale algebra
E' of rank n 1 such that E and k x E' have the same trace form.
In characteristi c 0 , thi s ca n b e deduce d fro m [Mes90 , Prop . 1 an d Prop . 3] ,
cf. [EK94 , p . 425] . Th e resul t als o hold s i n characteristi c p 2 (Mestre , unpub -
lished); w e omi t th e proof .
COROLLARY 31.24. Let q be a quadratic form of odd rank n. Then q is a trace
form if and only if q is isomorphic to (1) (&q', where q' is a trace form of rank n 1.
31.25. T H E CASE n = 6 . Le t q b e a quadrati c for m o f ran k 6 ; le t d = d(q) b e
its discriminant .
T H E O R E M 31.26. For q to be a trace form, it is necessary and sufficient that
the following two conditions be fulfilled:
(a) q contains (2,1);
(b) q contains (2,1,1over ) the field k{y/2d).
P R O O F . Necessity : Conditio n (a ) i s th e trivia l conditio n i n ran k 6 . W e ma y
then writ e q a s (2,1) 0 Q , wher e rank(Q ) = 4 , an d d(Q) = 2d. T o prov e (b) , w e
may replac e k b y k(\/2d), i.e. , assum e tha t 2d i s a square , i.e. , d(Q) = 1. W e ma y
also replac e k b y an y odd-degre e extension . B y Propositio n 31.2, thi s allow s u s t o
assume tha t q is isomorphi c t o qE , wher e E split s a s E' x E" wit h ran k E' 2 an d
rank E" = 4 . W e thu s hav e
q g *
qE
, 0 q
E
n = * (2, 2x) 0 (1) 0 / , wit h rank(/ ) = 3 .
Hence Q ^ (2x) 0 / . Sinc e d(Q) = 1, we hav e (2x) = (d(f)), henc e Q ^ (d(/) ) 0 / .
By Theore m 31.18 , / i s special , henc e ha s propert y (1) o f Propositio n 31.12 ; thi s
means tha t Q contain s (1), henc e q contain s (2,1,1).
Sufficiency: W e us e th e followin g lemma :
L E M M A 31.27. Let Q be a quadratic form of rank 1, and let a be an element
of k* . Suppose that Q represents 1 over the field k(y/a). Then Q contains a binary
form (x,y) with (x) -(ay) 0 in H 2(k).
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