4 M. L. RACINE

(4) (U f = U , V =V ^ ([17], p. 1.26)

v ' x n n m m+n J

x x , x x

(5) x m o x n = 2 x m + n ([17], p. 1.23)

2

An element e e ^ is an idempotent if e = e; two idempotents e and

f are orthogonal if e © f = 0 and eUf = 0 = fU (if P is special,

0 = e © f = ef + ef, therefore e(ef + fe) = ef + efe = ef = 0 so fe = 0 and e

and f are orthogonal in the usual sense). An idempotent e is called

completely primitive if («?U , U, e) is a Jordan division algebra. $ is said to

have a capacity if it contains a supplementary set of orthogonal completely

primitive idempotents. The minimum number of elements in such a set is

called the capacity of ^. A non-zero idempotent e is called absolutely

primitive if every element of JU is of the form ae + z where a e $ and z

n

is nil potent. If 1 = ) e, where the e, are absolutely primitive idempotents

1=1

l

'

then 9 is said to be reduced.

If $ is a field and 9 is finite dimensional over $, Jacobson and

Katz [18] have extended the definition of generic minimum polynomial ([16],

p. 223) to the characteristic 2 case. So for x e $, let m (\) -

A - (J, (x) 7\ + (y2(x) ^ + . . . + (-1) a (x) denote the generic minimum

polynomial of x; call T(x) = a. (x) the generic trace of x, N(x) = cr (x) the

1 n

generic norm of x, n the (generic) degree of ^. We mention a few results

of [18].

(6) m U) =N(M - x), where M - x

€

? $ ( ) .