More on Embeddings of Local Fields in Simple Algebras

Abstract

Let A | F be a central simple algebra over a p-field F of arbitrary characteristic. Then concretely A may be represented as a complete m_m matrix algebra A=M m (D d ), where D d =D denotes a central division algebra of index d over F. Thus the reduced degree of A over F is N=dm. 1 We write o F , respectively O D , for the ring of integers of F, respectively D, and P F =? F o F , respectively P D =? D O D , for the maximal ideals of o F , respectively O D . We write k F , respectively k D , for the residual fields of F and D.An o F -order of A is any subring of A containing the identity element of A which is also a finitely generated o F submodule of A containing an F basis for A. Let A denote an o F -order of A. We call A hereditary [R, p. 27] if every left ideal of A is a projective left A module. The order A has a Jacobson radical P A [R, p. 77ff ]; it is the minimal (two-sided) ideal of A such that the quotient ring AÂP A is semi-simple. If A is hereditary, then AÂP A is a direct product of complete matrix algebras with entries in k D , and ? F A=P rd A with a positive integer r, called the period of A. Following Benz [B], Bushnell and Fro hlich [BF], and Fro hlich [F] we callThe period of a principal order A determines A up to conjugacy. If A is principal with period r, then A is conjugate to the standard principal order A r /M r (M s (O D )) such that the r_r matrix g=( g ij ) belongs to A r if and only if g ij # M s (P D ) for i> j. Thus the set of standard principal orders A r of A, and hence the set of conjugacy classes of principal orders of A, corresponds bijectively to the set of factors r of m.

Continuation of Hereditary Orders in Local Central Simple Algebras

“…Clearly, h # K B 1 /K C 1 /K A 1 , and from [Z,Theorem 2(iii) and Corollary 2(ii)(and its proof)] we obtain…”

“…By [Z,Theorem 2], C 1 :=A 1 & C is a principal order in C which continues B 1 and is continued by A 1 . Clearly, h # K B 1 /K C 1 /K A 1 , and from [Z,Theorem 2(iii) and Corollary 2(ii)(and its proof)] we obtain…”

“…The proof requires the handling of several special cases and uses the results of [Be2,F,Z] about principal orders. So first, in Section 1, we consider the relationship between principal and arbitrary hereditary orders.…”

Simple characters for principal orders in Mm(D)

“…Moreover, taking subfields of fields embedded sound with maximal possible invariant preserves maximality of the invariant. This enables us to use this property in a similar manner as a type of embedding in the sense of [BG] or the notion of (e, f )-purity in the sense of [Z1,Definition 4] (see the following Definition 2.4).…”

“…By [BG,3.2] and [Z1,Proposition 6] two A-pure elements fulfilling the same notion of purity and with the same minimal polynomial over F are conjugated over K A . It is not hard to see that the results of [BG,4,5] are valid for all notions of purity.…”