Editing Class field theory (section)

==Generalizations of class field theory==

There are three main generalizations, each of great interest. They are: the [[Langlands program]], [[anabelian geometry]], and higher class field theory.

Often, the Langlands correspondence is viewed as a [[nonabelian class field theory]]. If or when it is fully established, it would contain a certain theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory: the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.

Another generalization of class field theory is [[anabelian geometry]], which studies algorithms to restore the original object (e.g. a number field or a hyperbolic curve over it) from the knowledge of its full absolute Galois group or [[algebraic fundamental group]].<ref>{{Citation|first=Ivan|last=Fesenko|year=2015|title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015|url=https://ivanfesenko.org/wp-content/uploads/2021/10/notesoniut.pdf}}</ref><ref>{{Citation|first=Ivan|last=Fesenko|year=2021 |title=Class field theory, its three main generalisations, and applications, May 2021, EMS Surveys 8(2021) 107-133 |url=https://ivanfesenko.org/wp-content/uploads/2021/11/232.pdf}}</ref>

Another natural generalization is higher class field theory, divided into ''higher local class field theory'' and ''higher global class field theory''. It describes abelian extensions of [[higher local field]]s and higher global fields. The latter come as function fields of [[scheme (mathematics)|scheme]]s of [[Scheme of finite type|finite type]] over integers and their appropriate localizations and completions. It uses [[algebraic K-theory]], and appropriate [[Milnor K-group]]s generalize the <math>K_1</math> used in one-dimensional class field theory.