The Brauer Group

Abstract

This chapter is concerned with the classification of finite dimensional central division algebras over a given field k. In the case k = R, the Frobenius Theorem shows that R and H are the only finite dimensional central division algebras over R. This kind of classification is optimal in the sense that we have an explicit, easy-to-understand list of all finite dimensional central division algebras over R. Classifying finite dimensional central division algebras over other fields has proven much more difficult, and in fact this problem has been a focal point for research in number theory and quadratic forms. Although such an explicit list as in the case of central division algebras over R cannot always be given, there is much that can be said.KeywordsGalois GroupDivision AlgebraGalois ExtensionCentral Simple AlgebraFrobenius TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.