mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, non-abelian class field theory is a catchphrase, meaning the extension of the results of

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

, the relatively complete and classical set of results on

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...

s of any

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

''K'', to the general

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

''L''/''K''. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense.

History

A presentation of class field theory in terms of

group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...

was carried out by

Claude Chevalley Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a foundin ...

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...

and others, mainly in the 1940s. This resulted in a formulation of the central results by means of the group cohomology of the

idele class group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...

. The theorems of the cohomological approach are independent of whether or not the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

''G'' of ''L''/''K'' is abelian. This theory has never been regarded as the sought-after ''non-abelian'' theory. The first reason that can be cited for that is that it did not provide fresh information on the splitting of prime ideals in a Galois extension; a common way to explain the objective of a non-abelian class field theory is that it should provide a more explicit way to express such patterns of splitting. The cohomological approach therefore was of limited use in even formulating non-abelian class field theory. Behind the history was the wish of Chevalley to write proofs for class field theory without using

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analyti ...

: in other words to eliminate

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...

s. The first wave of proofs of the central theorems of class field theory was structured as consisting of two 'inequalities' (the same structure as in the proofs now given of the

fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basi ...

, though much more complex). One of the two inequalities involved an argument with L-functions. In a later reversal of this development, it was realised that to generalize

Artin reciprocity Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun Harutyun ( hy, Հարություն and in Western Armenian Յարութիւն) also spelled Haroutioun, Harut ...

to the non-abelian case, it was essential in fact to seek a new way of expressing

Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. T ...

s. The contemporary formulation of this ambition is by means of the

Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

: in which grounds are given for believing Artin L-functions are also L-functions of

automorphic representation In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

s. As of the early twenty-first century, this is the formulation of the notion of ''non-abelian class field theory'' that has widest expert acceptance.''The matter of reciprocity laws and symbols for non-Abelian field extensions more properly fits into non-Abelian class field theory and the Langlands program'': from {{SpringerEOM, title=Hilbert problems , id=Hilbert_problems , oldid=18549 , first=M. , last=Hazewinkel

History

See also

Notes