Editing Class field theory (section)

{{short description|Branch of algebraic number theory concerned with abelian extensions}}
In [[mathematics]], '''class field theory''' ('''CFT''') is the fundamental branch of [[algebraic number theory]] whose goal is to describe all the [[Abelian extension|abelian]] [[Galois extension]]s of [[local field|local]] and [[global field|global]] fields using objects associated to the ground field.{{sfn|Milne|2020|loc=Introduction|p=1}} 

[[David Hilbert|Hilbert]] is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to [[Leopold Kronecker|Kronecker]] and it was actually [[Eduard Ritter von Weber|Weber]] who coined the term before Hilbert's fundamental papers came out.{{sfn|Cassels|Fröhlich|1967|loc=Ch. XI by Helmut Hasse|p=266}} 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 [[Teiji Takagi|Takagi]] and [[Emil Artin|Artin]] (with the help of [[Chebotarev's density theorem|Chebotarev's theorem]]). 

One of the major results is: given a number field ''F'', and writing ''K'' for the [[Hilbert class field|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<sub>F</sub>'' for the [[idele class group]] of ''F'', and taking ''L'' to be any finite abelian extension of ''F'', this law gives a canonical isomorphism

:<math> \theta_{L/F}: C_F/{N_{L/F}(C_L)} \to \operatorname{Gal}(L/F), </math>

where <math>N_{L/F}</math> 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 <math>C_F.</math>

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 [[John Tate (mathematician)|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<ref>{{Cite journal |last=Fesenko |first=Ivan |date=2021-08-31 |title=Class field theory, its three main generalisations, and applications |url=https://ems.press/journals/emss/articles/2504062 |journal=EMS Surveys in Mathematical Sciences |language=en |volume=8 |issue=1 |pages=107–133 |doi=10.4171/emss/45 |s2cid=239667749 |issn=2308-2151|doi-access=free }}</ref> 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 <math>\Q</math>, and the theory of [[complex multiplication]] to construct abelian extensions of [[CM-field]]s. 

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