Class Field Theory

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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.

Read more about Class Field TheoryFormulation in Contemporary Language, Prime Ideals, The Role of Class Field Theory in Algebraic Number Theory, Generalizations of Class Field Theory, History

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... The origins of class field theory lie in the quadratic reciprocity law proved by Gauss ... 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 ...
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