ARITHMETICS OF JORDAN ALGEBRAS
27
6 -p
•7
a)
(the split quaternion case) .
In both c a s e s M(C . J ) = U(C ) = W(C , J, ). All symmetric elements of
n* 7 n n 1
(C
3
J ) have diagonal entries in K except when C is a split quaternion algebra
n 7
and K has characteristic 2. In that cas e Jf(C, -) has dimension 3. The
algebra obtained by considering all symmetric elements of (C , J ) is an
isotope of M($ , T ), T the transpos e involution. In that sens e it is not
really a split algebra. Moreover it does not seem to yield to our methods
unles s the bas e field is assumed to be complete discrete . That cas e will be
treated in §5.
If C =K © K, $ =M(C ) s K . By Proposition 1, maximal orders of
n n
K coincide with maximal orders of K . But thes e are of the form E(L), L
n n
an o-lattice of an n-dimensional K-vector spac e If. An isomorphism of
maximal orders induces either an automorphism K or an antiautomorphism
of K . Any antiautomorphism can be obtained by composing the transpos e
antiautomorphism with an automorphism. By Lemma 3
L = ox © . . . 0 ox 0 ax , E(L) is isomorphic to
a \
and the transpos e sends thi s order
-1
Q 0
onto an order isomorphic to E(L') where L1 = ox ox . 0 0 x .
n- 1 n
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