ARITHMETICS OF JORDAN ALGEBRAS 13

preceding Proposition 3 and of the above results we assum e from now on that

#/K is central simple,

THEOREM 2„ If ?/K is central simple any order of p is contained in

a maximal order. In particular p contains a maximal order.

The second statement is a consequence of the first and of Proposition 2.

If the characteristic is not 2, ^/K central simple implies g separable ([16],

p . 239). This is unfortunately not so if the characteristic is 2 ([30], p . 302).

Therefore, if the characteristic is not 2 or if $ is exceptional, Proposition 3

yields the first statement of Theorem 2 and there remains only th e cas e #

special , K of characteristic 2. We postpone the treatment of thi s c a s e until

§1 of the next chapter.

Just a s in classica l arithmetic, localization at a prime and completion

will play an important role . Let S = {p| p a prime ideal of o} . Let v be a

finite dimensional K-vector space , L an o-lattice of V. Denote by o the

r

localization of o at p, by o the completion of that localization and by

K the quotient field of o . Let v = o 0 V, L the o - module of V

P P P P ° p p

/\ /\ /\

generated by L, L the o - module of V generated by L. It is well-known

(e.g . [40], pp. 51, 62,183) that L = D L = f l L and that given {K J ,

peS * p

€

S ^ ~W

P e s * K \ an o - l a t t i c e of y such that K = L for almost all p e S,

there exist s an o-lattice K of v with IC = K

/ c

. Vp e S. A similar statement

-P

~(P)

holds for lattice s in \r . The following lemma is an immediate consequence of

t h e s e facts just a s in the associativ e c a s e (e.g . [40], p. 202; [6], VI §11).