Class field theory

The class field theory is a large branch of algebraic number theory , which deals with the investigation of Abelian extensions of algebraic number fields , or more generally of global fields . Roughly speaking, it is about describing or constructing such extensions of a number field from the arithmetic properties of . ${\ displaystyle K}$ ${\ displaystyle K}$

There is a maximal Abelian extension of from infinite degree over , and the pro- finite Galois group shall be described starting from. ${\ displaystyle A}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle G = G (A | K)}$ ${\ displaystyle K}$

For example , is isomorphic to an infinite product of the additive groups of the p -adic integers over all prime numbers and a product of an infinite number of finite cyclic groups. This sentence, the sentence of Kronecker-Weber , goes back to Leopold Kronecker . ${\ displaystyle K = \ mathbb {Q}}$ ${\ displaystyle G}$ ${\ displaystyle p}$

For number theory, the description of the decomposition of prime ideals from into Abelian extensions is very important. This is done with the help of the Frobenius element , and represents a far-reaching generalization of the quadratic reciprocity law , which describes the decomposition of prime numbers into quadratic number fields . ${\ displaystyle K}$

This generalization has a long history, starting with Carl Friedrich Gauß , quadratic forms and their gender theory , works by Ernst Eduard Kummer , Kronecker and Kurt Hensel on ideals and completions, the theory of circle division extensions and Kummer extensions , conjectures by David Hilbert and evidence from many mathematicians such as Teiji Takagi , Helmut Hasse , Emil Artin , Phillip Furtwängler and others. Takagi's crucial existential theorem had been known since 1920, and all of the main results since about 1930. One of the classic conjectures that was last proven was the main ideal theorem .

In the 1930s and thereafter, Wolfgang Krull's theory of the infinite Galois extensions and the Pontryagin duality gave a clearer, if more abstract, formulation of the main clause, Artin's law of reciprocity . Infinite extensions are also the subject of Iwasawa theory .

After Claude Chevalley (1909–1984) had built the global class field theory with the help of ideals and their characters on the local one, instead of requiring analytical methods as before, it remained fairly constant. The Langlands program as a “ non-Abelian class field theory ”, even if it goes much further than the question of how primitive ideals are broken down into general Galois extensions, brought new impulses.

literature

James Milne, Class Field Theory .
Jürgen Neukirch: Algebraic number theory. Springer, Berlin et al. 1992, ISBN 3-540-54273-6 .