ARITHMETICS OF JORDAN ALGEBRAS 57
results on involutions induced by a skew-hermitian form.
Let K be a complete discret e valuation field of characteristic not 2,
(&, -) a finite dimensional central associativ e division algebra with involu-
tion over K, S($, -) = {d c $| d = - d } , the subspac e of skew-hermitian elements
of &. Note that | | and v do not depend in any way on the involution.
Consider v(S(£,-)). We may assum e that S(fl,-) £ {0}. Let c e S(&,-).
Then a* = c " a c defines an involution * of & and S(#, -) = ctf($, *) .
Since v(S(fl, -)) = v(c) + v(H(&, *)), v(S(&, -)) = Z , 2Z or 2Z + 1
( i . e . {2m + l | m
c
Z } ) according a s v(M(fl,*)) = Z or 2Z with v(c) even
or odd.
We show by an example that all three possibilitie s can occur. Let
C be the unique quaternion division algebra over a local field K of
characteristi c not 2, the standard involution. Then with respect to the
norm form, C =* ( l ) J - ( - A ) _ L (TT) X ( -ATT) where T T is a prime of o and A
is a non-squar e unit of o ([37], p. 169). Let l , x x x be a K-basis of
C corresponding to this decomposition; S(C, -) - Kx + Kx + Kx , v(x ) = 0,
v(x ) = 1 and v(S(C,-)) = Z . Also H ( C , - ) = K , so v(H(C,-)) = 2Z. Hence if
o is defined by a = x ax , v(S(C, o)) = 2Z, and if T is defined by
a = x ax v(S(C, T ) ) = 2Z + 1. This algebra als o provides an example
showing that even when H(&, *) generates & associatively , the associativ e
o -algebra generated by Jt(C, *) need not be £. Indeed if we assum e that 2
is a unit of K then © = ol + ox + ox + ox &(©, a) = ol + ox + ox for, a
a s above and since x x = TTX &(©, cr)1 = ol + px + ox + ox C ©. There -
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