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

? $ ( ) .
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