In mathematics, class field theory is a major branch of algebraic number theory. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

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. In algebraic number theory, the Hilbert class field E of a number field is the maximal abelian unramified extension of Note that in this context, unramified is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. ... In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or...

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. In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... The term global field refers to either of the following: a number field, i. ... In mathematics, a Galois extension is a field extension that has a Galois group. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

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. In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ...

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 taken over all prime numbers p, and of a product of infinitely many finite cyclic groups. The content of this theorem, now known as the Kronecker-Weber theorem, goes back to Kronecker. The title given to this article is incorrect due to technical limitations. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... In algebraic number theory, the Kronecker-Weber theorem states that every finite abelian extension of the field of rational numbers , or in other words every algebraic number field whose Galois group over is abelian, is a subfield of a cyclotomic field, i. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ...

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. In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ... In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ...

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 mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, reciprocity is applied to a number of theorems, and at times certain relationships. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Abigail and Brittany Hensel (Hensel twins) (-) Kurt Hensel (1861-1941), mathematician Walther Hensel (1887-1956) Hensel Family Wilhelm Hensel (1794-1861), painter = (married) Fanny Mendelssohn Hensel (-), sister of Felix Mendelssohn Paul Hensel (1860-1930) Luise Hensel (1798-1876), poet Hensels lemma See also Hensell Hänsel(Haensel), Hänsell... In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots of... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Teiji Takagi (高木 貞治 Takagi Teiji, April 21, 1875 - February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. ... Helmut Hasse (pronounced HAHS uh) (25 August 1898- 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry (Hasse principle), and to local zeta functions. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... Phillip Furtwängler was a 20th century German number theorist. ... In class field theory, the Takagi existence theorem states in part that if K is a number field with class group G, there exists a unique abelian extension L/K with Galois group G, such that every ideal in K becomes principal in L, and that L is characterized by...

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. Krull is a 1983 heroic fantasy film directed by Peter Yates and produced by Ron Silverman. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, Artin reciprocity refers to various results connecting Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to Heckes grossencharacters of that number field. ... In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields. ...

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. In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ...

See also: local classfield theory In mathematics, local classfield theory is the study in number theory of the abelian extensions of local fields. ...