Let $/$ be an associative division algebra. Any involution * of
& is induced by a hermitian or skew-hermitian form h. Moreover we may
assume that h is hermitian unless & = $ a field of characteristic not 2,
the involution of $ induced by * is the identity and h is alternate
([17], Chap. 0). In that case * is the symplectic involution and
, *) = M(($ ) ), $ the split quaternion algebra with standard involution.
Therefore, at least in the case of characteristic not 2, the results of this
section, §3, I§3 and I§4 cover all special central simple Jordan algebras over
a complete discrete valuation field. In characteristic 2, outer ideals 8 con-
taining 1 also have to be considered. By Proposition 6 we have considered all
such B's whose centroid coincides with the center of $ . However as will
be seen in the next chapter this does not always hold.
In certain cases it is desirable to have , the involution of $ induced
by *, as nice as possible (e.g. the standard involution in a quaternion
algebra). Indeed if * is induced by h and if a is an involution of $ of
the same kind as , then a = c ac, Va

$, and a fixed element c e &
with c = ±c ([1], p. 154). Consider h'(x, y) = h(x, y)c; h'(x, ay) = h(x, ay)c
= h(x, y)ac = h'(x, y)a and h*(x, y) = c ch(x, y)c = ec ch(x, y)c = |i.eh'(y, x),
where e = 1 or -1 according as h is hermitian or skew-hermitian and - 1
or -1 according as c = c or -c . Therefore
is a hermitian or skew-
hermitian form which induces * on $ and a on $. But if we wish to
exercise this freedom in the choice of the involution of & we may no longer
assume that h is hermitian or alternate. Accordingly we mention a few
Previous Page Next Page