Class field theory

In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.^[1]

Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out.^[2] The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem).

One of the major results is: given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing C_F for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism

${\displaystyle \theta _{L/F}:C_{F}/{N_{L/F}(C_{L})}\to \operatorname {Gal} (L/F),}$

where ${\displaystyle N_{L/F}}$ denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map.

The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of ${\displaystyle C_{F}.}$

A standard method for developing global class field theory since the 1930s was to construct local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory. This was first done by Emil Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. His method was explicit and algorithmic.

Inside class field theory one can distinguish^[3] special class field theory and general class field theory.

Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used to construct the abelian extensions of ${\displaystyle \mathbb {Q} }$, and the theory of complex multiplication to construct abelian extensions of CM-fields.

There are three main generalizations of class field theory: higher class field theory, the Langlands program (or 'Langlands correspondences'), and anabelian geometry.