Steinitz classes of central simple algebras

Abstract

1. Introduction. The Steinitz class of an extension of number fields is a familiar object of study. Numerous algebraic results concerning it are known, and recently ([2], [4]) density and distribution results have been obtained for suitable families of extensions. Our purpose here is to present analogous results for the case where the extension field is replaced by a central simple algebra over the base field. Since central simple algebras generally have several conjugacy classes of maximal orders, it is not obvious at first that a Steinitz class can be defined in this setting. However, it emerges (see Corollary 3.2) that all maximal orders are isomorphic as modules over the integers of the base field, and so we may define the Steinitz class of a central simple algebra to be the common Steinitz invariant of these modules. Next we determine (see Proposition 3.5) precisely which classes may appear as Steinitz classes of central simple algebras. In Section 4, we find the asymptotic frequency with which each of these possible classes occurs as the algebra is varied in the family of central division algebras with fixed dimension and fixed splitting type at infinity, ordered by the norm of the discriminant. The distribution of Steinitz classes is not generally uniform, but is so in some cases, including that of algebras of prime degree. The possible non-uniformity is demonstrated in the last section by giving a specific example.

Characterizing quaternion rings over an arbitrary base

“…(Compare this with Kable et al [8], who considers the Steinitz class of a central simple algebra over a number field, and Peters [17] who works over a Dedekind domain.) Cognizant of (3.12), we make the following definition.…”