Abelian extensions

Introduction

In the field of algebraic number theory, an Abelian extension is a type of Galois extension where the Galois group is an Abelian group. This concept is named after the Norwegian mathematician Niels Henrik Abel, who made significant contributions to the understanding of these extensions. Abelian extensions play a crucial role in many areas of mathematics, including number theory, algebra, and geometry.

A representation of an Abelian extension, showing the relationship between the base field and the extended field.

Definition

Formally, let F be a field, and let K be a field extension of F. The extension K/F is said to be Abelian if the Galois group Gal(K/F) is an Abelian group. This means that for any two automorphisms σ and τ in Gal(K/F), the commutativity στ = τσ holds.

Properties

Abelian extensions have several important properties that distinguish them from other types of extensions.

Closed under Compositum

One of the key properties of Abelian extensions is that they are closed under compositum. This means that if K1/F and K2/F are both Abelian extensions, then the compositum K1K2/F is also an Abelian extension.

Kronecker's Theorem

Kronecker's theorem states that every Abelian extension of the field of rational numbers Q is contained in a Cyclotomic field. This theorem is a cornerstone in the study of Abelian extensions and has profound implications in the theory of algebraic number fields.

Abelian Extensions and Class Field Theory

Class field theory, a major branch of algebraic number theory, studies Abelian extensions of number fields. The main goal of class field theory is to describe the Abelian extensions of a given number field K. The theory provides a deep connection between the Abelian extensions of K and the ideal class groups of K.

Examples

There are several notable examples of Abelian extensions.

Quadratic Extensions

A Quadratic extension is an example of an Abelian extension. In a quadratic extension Q(√d)/Q, where d is a square-free integer, the Galois group is either trivial or isomorphic to Z/2Z, both of which are Abelian groups.

Cyclotomic Extensions

A Cyclotomic extension is another example of an Abelian extension. For any positive integer n, the extension Q(ζn)/Q, where ζn is a primitive nth root of unity, is an Abelian extension. The Galois group Gal(Q(ζn)/Q) is isomorphic to (Z/nZ)×, which is an Abelian group.