Class field theory
In mathematics, class field theory is a major branch of algebraic number theory.
Most of the central results in this area were proved in the period between 1900 and 1950. The theory takes its name from some of the early ideas, conjectures and results such as those on the Hilbert class field, which took a generation to settle up to 1930. The ideal class group (which is a basic object of study inside a single field of numbers K, such as a quadratic field), is also seen as a Galois group of a field extension L/K: a structure built on top of K and possibly involving irrational numbers going beyond square roots.
These days the term is generally used synonymously with the study of all 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.
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. We are interested in describing G in terms of K.
The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group of K. For example when K is the field of rational numbers the Galois group G 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 stated by Kronecker.
For a description of the general case see class formation.
Prime ideals
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.
History
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 ideals split in general Galois extensions.