Class field theory

These days the term is generally used as synonymous with the study of the abelian extensions of algebraic number 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 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 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 (see p-adic numbers) taken over all prime numbers p, and of a product of infinitely many finite cyclic groups. The content of this theorem goes back to Kronecker[?]. 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 ... and others). The main results were known by about 1930.

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.