Editing Class field theory (section)

==Formulation in contemporary language==

In modern mathematical language, class field theory (CFT) can be formulated as follows. Consider the ''maximal'' abelian extension ''A'' of a local or [[global field]] ''K''. It is of infinite degree over ''K''; the Galois group ''G'' of ''A'' over ''K'' is an infinite [[profinite group]], so a [[compact topological group]], and it is abelian. The central aims of class field theory are: to describe ''G'' in terms of certain appropriate topological objects associated to ''K'', to describe finite abelian extensions of ''K'' in terms of open subgroups of finite index in the topological object associated to ''K''. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of ''K'' and their norm groups in this topological object for ''K''. This topological object is the [[multiplicative group]] in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding to an open subgroup of finite index is called the class field for that subgroup, which gave the name to the theory.

The fundamental result of general class field theory states that the group ''G'' is naturally isomorphic to the [[profinite group|profinite completion]] of ''C<sub>K</sub>'', the multiplicative group of a local field or the idele class group of the global field, with respect to the natural topology on ''C<sub>K</sub>'' related to the specific structure of the field ''K''. Equivalently, for any finite Galois extension ''L'' of ''K'', there is an isomorphism (the [[Artin reciprocity law|Artin reciprocity map]])

:<math>\operatorname{Gal}(L/K)^{\operatorname{ab}} \to C_K/N_{L/K} (C_L)</math>

of the [[abelianization]] of the Galois group of the extension with the quotient of the idele class group of ''K'' by the image of the [[field norm|norm]] of the idele class group of ''L''.

For some small fields, such as the field of rational numbers <math>\Q</math> or its [[quadratic extension|quadratic imaginary extension]]s there is a more detailed ''very explicit but too specific'' theory which provides more information. For example, the abelianized [[absolute Galois group]] ''G'' of <math>\Q</math> is (naturally isomorphic to) an infinite product of the group of units of the [[p-adic integer]]s taken over all [[prime number]]s ''p'', and the corresponding maximal abelian extension of the rationals is the field generated by all [[roots of unity]]. This is known as the [[Kronecker–Weber theorem]], originally conjectured by [[Leopold Kronecker]]. In this case the reciprocity isomorphism of class field theory (or Artin reciprocity map) also admits an explicit description due to the Kronecker–Weber theorem. However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory.

The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when defined on the [[idele]] group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the ''global reciprocity law'' and is a far reaching generalization of the Gauss [[quadratic reciprocity law]].

One of the methods to construct the reciprocity homomorphism uses [[class formation]] which derives class field theory from axioms of class field theory. This derivation is purely topological group theoretical, while to establish the axioms one has to use the ring structure of the ground field.<ref>[https://ivanfesenko.org/wp-content/uploads/2021/10/jl.pdf Reciprocity and IUT, talk at RIMS workshop on IUT Summit, July 2016, Ivan Fesenko]</ref>

There are methods which use cohomology groups, in particular the [[Brauer group]], and there are methods which do not use cohomology groups and are very explicit and fruitful for applications.