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 local field, and then use it to construct global class field theory.

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.

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