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}
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