Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic properties of such abelian extensions. A general name for such fields is global fields, or one-dimensional global fields.
The theory takes its name from the fact that it provides a one-to-one correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. For example, the Hilbert class field, which is the maximal unramified abelian extension of a number field, corresponds to a very special class of ideals. Class field theory also includes a reciprocity homomorphism which acts from the idele class group of a global field, i.e. the quotient of the ideles by the multiplicative group of the field, to the Galois group of the maximal abelian extension of the global field. Each open subgroup of the idele class group of a global field is the image with respect to the norm map from the corresponding class field extension down to the global field.
A standard method since the 1930s is to develop local class field theory which describes abelian extensions of completions of a global field, and then use it to construct global class field theory.
Formulation in contemporary language 
In modern language there is a maximal abelian extension A of K, which will be of infinite degree over K; and associated to A a Galois group G which will be a pro-finite group, so a compact topological group, and also abelian. The central aim of the theory is to describe G in terms of K. In particular to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in an appropriate object for K, such as the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields, as well as to describe those norm groups directly, e.g., such as open subgroups of finite index. The finite abelian extension corresponding to such a subgroup is called a class field, which gave the name to the theory.
The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group CK of K with respect to the natural topology on CK related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism
- Gal(L / K)ab → CK / NL/K CL
For some small fields, such as the field of rational numbers or its quadratic imaginary extensions there is a more detailed theory which provides more information. For example, the abelianized absolute Galois group G of is (naturally isomorphic to) an infinite product of the group of units of the p-adic integers taken over all prime numbers 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. Let us denote with
the group of all roots of unity, i.e. the torsion subgroup. The Artin reciprocity map is given by
when it is arithmetically normalized, or given by
if it is geometrically normalized. 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 methods to construct the reciprocity homomorphism uses class formation.
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 good for applications.
Prime ideals 
More than just the abstract description of G, it is essential for the purposes of number theory to understand how prime ideals decompose in the abelian extensions. The description is in terms of Frobenius elements, and generalises in a far-reaching way the quadratic reciprocity law that gives full information on the decomposition of prime numbers in quadratic fields. The class field theory project included the 'higher reciprocity laws' (cubic reciprocity) and so on.
The role of class field theory in algebraic number theory 
Class field theory is the key part and the heart of algebraic number theory. It has thousands of applications in number theory. Via the theory of zeta integrals initiated by Kenkichi Iwasawa and by John Tate in Tate's thesis it is related to the study of the zeta function of global fields.
Generalizations of class field theory 
One natural development in number theory is to understand and construct nonabelian class field theories which provide information about general Galois extensions of global fields. Often, the Langlands correspondence is viewed as a nonabelian class field theory and indeed when fully established it will contain a very rich 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, i.e. the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternative to the Langlands correspondence point of view.
Another natural development in arithmetic geometry is to understand and construct class field theory which describes abelian extensions of higher local and global fields. The latter come as function fields of schemes of finite type over integers and their appropriate localization and completions. Higher local and global class field theory uses algebraic K-theory and appropriate Milnor K-groups replace which is in use in one-dimensional class field theory. Higher local and global class field theory was developed by A. Parshin, Kazuya Kato, Ivan Fesenko, Spencer Bloch, Shuji Saito and other mathematicians. There are attempts to develop higher global class field theory without using algebraic K-theory (G. Wiesend), but his approach does not involve higher local class field theory and a compatibility between the local and global theories.
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalisation took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions.
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 Shimura provided another very explicit class field theory for a class of algebraic number fields. All these very explicit theories cannot be extended to work over arbitrary number field. In positive characteristic Kawada and Satake used Witt duality to get a very easy description of the -part of the reciprocity homomorphism.
However, general class field theory used different concepts and its constructions work over every global field.
The famous problems of David Hilbert stimulated further development which lead to the reciprocity laws, and proofs by Teiji Takagi, Phillip 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 the use of infinite extensions and the theory of Wolfgang Krull of their Galois groups was found increasingly useful. It combines 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 1930s. Their use replaced the classes of ideals and essentially clarified and simplified structures which describe abelian extensions of global fields. Most of the central results were proved by 1940.
After 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, 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 nineties, see e.g. the book of Neukirch.
- (Neukirch 1999, Theorems VI.5.5, VI.6.1)
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