Separable Extensions of Finite Fields and Finite Rings

Abstract

In this chapter we want to deal with separable extension theory, since it will be a fundamental tool to describe the Galois extensions of local rings and to construct Galois rings in the subsequent chapters. We start by recalling the main ideas of the abstract theory of this kind of extensions in the case of fields (see, for example, ([44] or [64]), with a particular interest in finite fields. After that, we shall consider the separable extensions of finite, local rings ([56]); we will give the crucial definition of unramified extension of a local ring and then show the equivalence of these two notions. This will lead us to the characterization theorem of separable extensions of finite, local rings and provide some particularly interesting examples.