Class field theory

Class field theory is a major branch of algebraic number theory, including most of the central results that were proved in the period about 1900-1950.

These days the term is generally used as synonymous with the study of the abelian extensions of algebraic number fields, or more generally of global fields; an abelian extension being a Galois extension with Galois group that is an abelian group. The point in general terms is to predict or construct the extensions of this type for a general number field K, in terms of the arithmetical properties of K itself.

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. We are interested in describing G in terms of K.

The description is technical, but for example when K is the field of rational numbers the structure of G is an infinite product of the additive group of p-adic integers taken over all prime numbers p, and of a product of infinitely many finite cyclic groups. The content of this theorem, now known as the Kronecker-Weber theorem, goes back to Kronecker.

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), but is not limited to that one, classical line of generalisation.

The generalisation took place as a long-term historical project, involving quadratic forms and their 'genus theory', the reciprocity laws, work of Kummer and Kronecker/Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions, conjectures of Hilbert and proofs by numerous mathematicians (Takagi, Hasse, Artin, Furtwängler and 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.

In the 1930s and subsequently the use of infinite extensions and the theory of 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. It is also basic to Iwasawa theory.

After the results were reformulated in terms of group cohomology, the field became relatively static. The Langlands program provided a fresh impetus, in its shape as 'non-abelian class field theory', though that description should be regarded as outgrown by now if it is confined to the question of how prime ideal split in general Galois extension.

See also: local classfield theory


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