Class field theory

The class field theory is a major branch of algebraic number theory, which deals with the study of abelian extensions of algebraic number fields, or more generally of global bodies. Roughly speaking it comes to describe such extensions of a number field of arithmetic properties of or construct.

There are a maximum abelian extension of infinite degree, and the profinite Galois group to be described by starting out.

If, for example, so is isomorphic to an infinite product of the additive group of p- adic integers over all primes and a product of an infinite number of finite cyclic groups. This set, the set of Kronecker -Weber, goes back to Leopold Kronecker.

The description of the decomposition of prime ideals in abelian extensions of is very important for the theory of numbers. This is done using the Frobeniuselements, and provides a far-reaching generalization of the quadratic reciprocity law is describing the decomposition of primes in quadratic fields.

This generalization has a long history, beginning with Carl Friedrich Gauss, quadratic forms and gender theory, work by Ernst Eduard Kummer, Kronecker and Kurt Hensel about ideals and completions, the theory of cyclotomic extensions and Kummer extensions, conjectures by David Hilbert and evidence of many mathematicians such as Teiji Takagi, Helmut Hasse, Emil Artin, Phillip Furtwängler and others. The decisive existence theorem of Takagi was known since 1920 and all main results since about 1930. A classical conjecture, which was recently proved the principal ideal theorem was.

In the 1930s and thereafter, with the theory of infinite Galois extensions of Wolfgang Krull and the Pontryagin duality for a clearer, albeit more abstract formulation of the main theorem, the Artin reciprocity law given. Infinite extensions are also the subject of Iwasawa theory.

After Claude Chevalley (1909-1984) had built the global class field theory by using Idelen and their characters at the local, rather than require, as before analytical methods, it remained fairly constant. The Langlands program as a "non- abelian class field theory ," even if this goes much further than the question of how prime ideals are broken down into general Galois extensions, brought new impetus.