78 IX. TH E TRAC E FOR M I N DIMENSIO N 7
PROOF.
(3 ) = (5) : I f k i s finite, pu t B = d, an d choos e A e k* suc h tha t
i
2
- 5 / 0 . (Thi s i s possibl e sinc e |fc | 3. ) Th e for m (A,
A2
- B,AB{A
2
- B))
has the sam e discriminan t a s /, henc e is isomorphic t o / . I f k i s infinite, writ e / a s
(a, 6, c) wit h d = abc and (a ) -(d) = 0 , henc e a
u2

d-v2
wit h u, v e k. Sinc e /c
is infinite, on e ma y choos e u, v bot h nonzero . (Otherwise , th e correspondin g coni c
in th e (-u , v)-plane woul d hav e a t mos t 4 points.) Defin e A, B E k* b y
A = 6 , B = d-(bv/u) 2.
We hav e A 2 - 5 = b 2 -(1 - d^ 2/?/2) = b 2-a/u2, henc e (,4 2 - £ ) ^ (a ) an d
(A,
A2
- B, AB(A
2
- 5) ) 9 * (6, a, frda) ^ (6 , a, c) ^ / .
(5)= . (1): I f ^
7
B ar e a s i n (5) , w e hav e (d) = (B), an d th e for m (d) 0 /
contains (B,A 2 B), whic h represent s 1.
REMARK
31.16I . f |A: | = 3 , th e for m (1,1,1i ) s special , bu t doe s no t hav e
property (5) .
31.17. TH E CAS E n = 4 . W e no w retur n t o th e questio n "When is q a trace
form?" i n th e cas e wher e q has ran k n = 4 . I n tha t case , th e trivia l conditio n i s
that q represents 1, i.e., q can b e writte n a s (1) 0 / , wit h rank / = 3 .
THEOREM 31.1 8 ([Se94 ] an d [EK94]) . For q = ( 1 ) 0 / to be a trace form, it is
necessary and sufficient that f be special (in the sens e define d i n §31.1 2 above) .
PROOF.
Necessity : I f q = (1) 0 / i s a trac e form , w e kno w b y Th . 29. 4 tha t
A3(/
(1)) = 0 . Bu t i t i s easy t o se e tha t
A3(/
- (1)) = A
3(/)
- A
2(/)
+ \\f) - A°(/) .
Hence / ha s propert y (2
;)
o f Prop. 31.12, which mean s tha t / i s special.
[Alternately, Th . 25.13 shows that ws(q) = 0 , hence ws(f) = 0, and on e applie s
Prop. 31.14.]
Sufficiency: I f k i s finite, ever y quadrati c for m o f ran k 1 i s a trac e for m
(just choos e E = k x x k or E = k x k x fc
2
wher e & 2 i s
a
quadrati c
extension o f k). Whe n k i s infinite , w e ma y appl y Prop . 31.1 5 an d writ e / a s
(A, A
2
- B, AB(A
2
- B)). Le t P b e th e polynome bicarre
P(X) = X
A
- 2AX
2
+ B.
Let E(A,B) b e th e ran k 4 algebr a k[X]/(P); th e followin g lemm a show s tha t i t
has th e require d trac e form .
LEMMA
31.19 . The algebra E(A,B) is etale, and its trace form is isomorphic
to (1,A,A2-B,AB(A2 -B)).
PROOF.
Th e discriminan t o f P i s 6 4 B-(A
2
- B)
2,
whic h i s no t 0 . Henc e
E(A,B) i s etale . I f x denote s th e imag e o f X i n E(A,B), on e check s tha t th e
elements
_L, X^ X ./i , X s\ IX
make u p a n orthogona l basi s o f E(A,B), an d tha t th e trace s o f thei r square s ar e
respectively
4, 4,4 , 4(A
2-B),
4A(A
2-B)/B.
Hence th e trac e for m o f E(A, B) i s
(4,4A,
4(A2
- B) ,
4A(A2
- B)/B) = (1, A,
A2
- B, AB(A
2
- B)).
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