Editing Class field theory (section)

==History==

{{main|History of class field theory}}
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving [[quadratic form]]s and their '[[genus of a quadratic form|genus theory]]', work of [[Ernst Kummer]] and Leopold Kronecker/[[Kurt Hensel]] on ideals and completions, the theory of cyclotomic and [[Kummer extension]]s.

The first two class field theories were very explicit cyclotomic and complex multiplication class field theories. They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order. Much later, the theory of [[Goro Shimura|Shimura]] provided another very explicit class field theory for a class of algebraic number fields. In positive characteristic <math>p</math>, [[Yukiyosi Kawada|Kawada]] and [[Ichiro Satake|Satake]] used Witt duality to get a very easy description of the <math>p</math>-part of the reciprocity homomorphism.

However, these very explicit theories could not be extended to more general number fields. General class field theory used different concepts and constructions which work over every global field.

The famous problems of [[David Hilbert]] stimulated further development, which led to the [[reciprocity law (mathematics)|reciprocity laws]], and proofs by [[Teiji Takagi]], [[Philipp Furtwängler]], [[Emil Artin]], [[Helmut Hasse]] and many others. The crucial [[Takagi existence theorem]] was known by 1920 and all the main results by about 1930. One of the last classical conjectures to be proved was the [[principalisation property]]. The first proofs of class field theory used substantial analytic methods. In the 1930s and subsequently saw the increasing use of infinite extensions and [[Wolfgang Krull]]'s  theory of their Galois groups. This combined with [[Pontryagin duality]] to give a clearer if more abstract formulation of the central result, the [[Artin reciprocity law]]. An important step was the introduction of ideles by [[Claude Chevalley]] in the 1930s to replace ideal classes, essentially clarifying and simplifying the description of abelian extensions of global fields. Most of the central results were proved by 1940.

Later the results were reformulated in terms of [[group cohomology]], which became a standard way to learn class field theory for several generations of number theorists. One drawback of the cohomological method is its relative inexplicitness. As the result of local contributions by [[Bernard Dwork]], [[John Tate (mathematician)|John Tate]], [[Michiel Hazewinkel]] and a local and global reinterpretation by [[Jürgen Neukirch]] and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology-free presentation of class field theory was established in the 1990s. (See, for example, ''Class Field Theory'' by Neukirch.)