Fields and Galois Theory

Abstract

The first principal theme of this chapter is the structure theory of fields. We shall study a field F in terms of a specified subfield K (F is said to be an extension field of K). The basic facts about field extensions are developed in Section 1, in particular, the distinction between algebraic and transcendental extensions. For the most part we deal only with algebraic extensions in this chapter. Arbitrary field extensions are considered in Chapter VI. The structure of certain fields and field extensions is thoroughly analyzed: simple extensions (Section 1); splitting fields (normal exten-sions) and algebraic closures (Section 3); finite fields (Section 5); and separable algebraic extensions (Sections 3 and 6).