Abelian Extensions

Abstract

In this chapter we shall investigate several types of abelian extension fields. First, we shall consider cyclotomic fields over the field of rational numbers and we shall determine their dimensionalities and Galois groups. Next we shall consider Kummer extensions, which are obtained by adjoining the roots of a finite number of pure equations x m = α to a field containing m distinct m-th roots of 1. Finally, we shall study the so-called abelian p-extensions, which are defined to be abelian extensions of p f dimensions of a field of characteristic p ≠ 0. The theory of characters of finite commutative groups is a basic tool for the investigation of Kummer extensions and abelian p-extensions. Besides this, our study of abelian p-extensions will be based on a certain type of ring, a ring of Witt vectors which can be constructed from any commutative algebra U over a field of characteristic p ≠ 0. For any such U and integer m = 1, 2, ···, we have a ring of Witt vectors M m(U) of characteristic p m . In the theory of valuations it is useful to pass to the limit as m → ∞ and to consider also rings M(U) of infinite Witt vectors. This will be considered in Chapter V.