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,