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).
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