Gyclotomic Division Algebras

Abstract

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).