18 M . L. RACINE
z ^ 0 and ^ is a Jordan division algebra. By Lemma 2, since
(x ) -1 (x )
Nv '(y) = N(x) N(y) e o, y is integral in $ '. But by Proposition 2
(x ) (x ) (x )
1 + y is integral in £ . Recall that 1 - x. Applying Lemma 2,
N
( x )
( 1
(x )
+ y
j
=
N(x)" 1 N(x + y) o. Finally we must show that M is
finitely generated. If the generic trace form T is non-degenerate pick a
bas e x _ , . . . , x of J in M (any x can be scale d into M by multiplica-
1 m
tion by a non-zero 7\ K). Let L = ox, + . . . + ox . The submodule M is
1 m
contained in L, the dual of L, and is therefore finitely generated. If ^ is
of degree 2, N is a quadratic form and # = #(N, 1). By Proposition 5, any
lattice L C M such that 1 c L is an order. By Corollary 2, L is contained
in a maximal order, say M'. But if x e M, N is integral on M' + ox which
is therefore still an order by Proposition 5. Hence x
e
M1 and M' = M is
finitely generated. We are therefore left with # special of degree 3, K of
characteristic 2. From structure theory ^ c & , & a central K division
algebra, or ^ c &($, j) = {x
e
&| x = x } , (&, j) a division algebra with involu-
tion such that K is the j-fixed subfield of the center. In either cas e
M C {x e $| reduced norm of x = o} which is the unique maximal order of &
([16], p. 99; [40], p. 220) and hence finitely generated.
q. e. d.
REMARK. There exist exceptional Jordan division algebras over com-
plete discrete valuation fields. To show thi s it suffices to give an example of
a separable associativ e division algebra Q of degree 3 over K such that the
reduced norm of G does not represent all of K ([29], Theorem 6). Let Q,
Previous Page Next Page