40 M. L. RACINE

§5. Maximal orders of H(& , & , *), n 2, & an associativ e division algebra.

n 0

Let ($, *) be a finite dimensional associativ e division algebra with

involution over its center $. We wish to consider subspace s $ of

U{&, *) such that 1

e

&Q and dfl d* c & V d

c

&. Since 1 e A

dd* = did *

€

fl Vd

€

&. Also d + d* = (d + 1 )(d + 1 )* - dd* - 1 e &Q$ Vd

€

A.

Elements of the form dd* are called norms, d + d* are called t r a c e s . If

char. $ t 2 then, for any d

€

M(fl, *), d = d/ 2 + d*/2 and A = H(j&, *).

PROPOSITION 6. Let ($, *) be a central finite dimensional associativ e

division algebra with involution over the field $ and let & be the subspac e

of W($, *) spanned by norms. Then $ = & ( # , * ) unles s the characteristi c of

$ is 2 and * is of symplectic type ( i . e . * induces a symplectic involution

on & ® P, where P is the algebraic closure of $ ). In that cas e

$

0

£ &(&,*) and if & c & C tt(&,*), & a subspac e such that d& d* c &

r

Vd

€

fl, then £ = M(JB, *).

PROOF. We claim that H(& ® P,*) = tt (&, *) S P. Clearly

r

JJ(&, *) S P C)f(jS® P, *). Let T a. ® A.

6

W(J&® P, *) . We may assum e

1 / w

that the A.'s are linearly independent over $ . Then I ) a. ® A . j =

/ a.* S A. and V (a, - a,*) ® A, = 0. Since the A.'s are linearly

/-s i i . ^ . i i i I

i=l i= l

independent over $ we must have a. - a.* = 0 , 1 i r, and

r

£ a. ® A,

€

H(fl, *) ® P. Also a norm of H(fl ® P, *) ha s the form

i=l L x

( / a. 8 A.]( / a. S A.) which is linear combination of terms of the form

^ i l l ' ^ i 1 l}