Division Algebras over Transcendental Fields

Abstract

In this chapter we consider the central simple algebras over fields that are transcendental extensions of their prime fields. In contrast with the abundant information about algebras over local and global fields that was presented in the last two chapters, our knowledge about division algebras over transcendental fields is sparse. The most important result on this subject is Tsen’s Theorem. It will be proved in Section 4. Tsen’s Theorem is a generalization of Wedderburn’s Theorem on finite division algebras. Using properties of the reduced norm we will prove a result that includes Tsen’s Theorem and Wedderburn’s Theorem as special cases. Tsen’s Theorem is the basis of most work on the Brauer groups of transcendental extensions. In Section 5 we will use it to prove a relativized version of the Auslander—Brumer—Faddeev Theorem that describes B(F(x)). The study of B(F(x)) leads to a construction of division algebras in Section 6 that clarifies the relation between the Schur index and the exponent of a central simple algebra. The last three sections of the chapter examine algebras over Laurent series fields.KeywordsPrime IdealGalois GroupDivision AlgebraLaurent SeriesGalois ExtensionThese 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.