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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 ''C''''K'' of ''K'' with respect to the natural topology on ''C''''K'' 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 → ''C''''K'' / ''N''''L''/''K'' ''C''''L'' of the maximal abelian quotient of the Galois group of the extension with the quotient of the idele class group of ''K'' by the image of the norm of the idele class group of ''L''. 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 the 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「class field theory」の詳細全文を読む スポンサード リンク
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