Finite Dimensional Extension Fields

Abstract

If Φ is a subfield of a field P, then we have seen that we can consider P as an algebra over Φ. In this chapter we shall be concerned primarily with the situation in which P is finite dimensional over the subfield Φ. We shall be concerned particularly with the general results of Galois theory that are of importance throughout algebra and especially in the theory of algebraic numbers. We shall consider the notions of normality, separability, and pure inseparability for extension fields, Galois cohomology, regular representations, traces, and norms. Also the basic results on finite fields will be derived and the notion of composites of two extension fields will be considered.