ARITHMETICS OF JORDAN ALGEBRAS 9
o-module, is spanned by the powers of x. The following proposition is an
immediate consequence of the above definitions.
PROPOSITION 1. The following are equivalent.
(i) x € P is integral.
(ii) o[x] is a finitely generated o-module.
(iii) The minimum polynomial of x is integral.
(iv) The generic minimum polynomial of x is integral.
An o-lattice M of ^ is an order if
(1) 1 € M.
(ii) x, y
M implies xU e M.
Clearly an order M of ^ is a Jordan algebra over o. If x € M then
o[x] C M. Hence o[x] is finitely generated and x is integral. M is said
to be maximal if it is maximal with respec t to inclusion. A maximal order M
is distinguished if M is a maximal lattic e of integral elements .
PROPOSITION 2. Every integral element x
^ can be embedded in an
order of ^.
PROOF. Let G be the o-algebra of transformations of % generated by
U r ... By (4) and (5) it is a commutative associativ e ring and is finitely gen -
erated as an o-module (by U ,, U . ., i, j degree of m (7\)). Let M, be
1 1 1 X 1
x x , x
an G-stabl e o -lattic e of ^ ( e . g . LG, L any o-lattice) . Since M U and
o[x]U are finitely generated o-modules there exist s a 7\
K, 7\ £ 0 such
that IV^Uj^ C A 2 M
and °[x]U C 7JM Let M
= TT1U . Then we