Class field theory | Wikipedia audio article
Channel: wikipedia tts
Date: 27/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Class_field_theory


00:03:18 1 Formulation in contemporary language
00:07:56 2 Prime ideals
00:08:39 3 Applications
00:09:28 4 Generalizations of class field theory
00:11:51 5 History



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SUMMARY
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In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups  associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.
The abelian topological group CK associated to such a field K is the multiplicative group of a local field or the idele class group of a global field.
One of the fundamental results of class field theory is a construction of a nontrivial reciprocity homomorphism, which acts from CK to the Galois group of the maximal abelian extension of the field K. The existence theorem of class field theory states that each open subgroup of finite index of CK is the image with respect to the norm map from the corresponding class field extension down to K.
The theory takes its name from the fact that it includes a one-to-one correspondence between finite abelian extensions of a fixed local or global field and appropriate open subgroup of finite index in CK. For example, in the case of number fields, the latter are classes of ideals of the field or open subgroups of the idele class group of the field; the  Hilbert class field, which is the maximal unramified abelian extension of a number field, corresponds to a very special class of ideals.
A standard method since the 1930s is to develop local class field theory,  which describes abelian extensions of completions of a local field, and then use it to construct global class field theory.
Origins of class field theory (reciprocity laws) can be traced to Gauss. Class field theory is the top achievement of algebraic number theory of the 20th century. There is a variety of presentations of class field theory, ranging from using Brauer groups or not using them, from using Galois cohomology or not using it, using features of characteristic zero or of positive characteristic.
There are two types of class field theories for number fields: (1) very explicit but restricted class field theory such as cyclotomic and CM which work over very special number fields using additional structures (roots of unity, torsion points of elliptic curves with CM), (2) general class field theory which works over any global field (any number field) and which follows different conceptual vision and which, remarkably, is simpler than the very explicit class field theory.
There are three main generalizations of class field theory: the Langlands program, higher class field theory, anabelian geometry, each leading to its own insights into key aspects of number theory.