ARITHMETICS OF JORDAN ALGEBRAS 31
Since M p and M are maximal orders, as was noted above (^nE(L.A)) 1 =
E(LA) and ( ^ n E ( L J ) ' = E(L
2
). Therefore E(L A) = E(L2) which implies
L A = QL (by Lemma 4) for a an o-ideal, so a L A = L . We have
-1 -1 -2 1 -2 -2
a
2
= g(L
2
,L
2
) = g(a LjA, a LjA) = a gfL^LjAA ) = o g(L
] [
,L
1
7) = 7 a a}
- 1 2
Hence 0 = 7 0 a and
\ Q
1
0 /
\ Q
2
0 /
q . e . d .
REMARK 2. If Q is trivial there is only one isomorphism clas s of
maximal orders of # = U(C ), C split.
COROLLARY 1. An order M of ? = M ( C J ) n 3 , C a split
n 7
associativ e composition algebra over K, is maximal if and only if M' is a
maximal order of C (a fortiori a maximal J -stabl e order of C ).
n 7 n
§4. Maximal ^-stabl e orders in associativ e algebras with involution over a
complete discrete valuation field.
Let ($, -) be a finite dimensional central associativ e division
algebra with involution over the field § (i. e. $ = W(C(&), - ) , C($) the
center of $). A hermitian form is a mapping h : \s x V -* & such that
(i) h(ax, y) = ah(x, y) a

fl, x, y

U.
(ii) h(y, x) = h(x, y).
Accordingly we say that A = (a..)
e
& is hermitian if a.. = a... A
U n i; 31
skew-hermitian form h is a mapping h : V x 1/ -* $ such that
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