The Structure of Fields

Abstract

In this chapter we shall analyze arbitrary extension fields of a given field. Since algebraic extensions were studied in some detail in Chapter V, the emphasis here will be on transcendental extensions. As the first step in this analysis, we shall show that every field extension K ⊂ F is in fact a two-step extension K ⊂ E ⊂ F, with F algebraic over E and E purely transcendental over K (Section 1). The basic concept used here is that of a transcendence base, whose cardinality (called the transcendence degree) turns out to be an invariant of the extension of K by F (Section 1). The notion of separability is extended to (possibly) nonalgebraic extensions in Section 2 and separable extensions are characterized in several ways.