In mathematics, class field theory is a major branch of algebraic number theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers
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. In Algebraic number theory, the Hilbert class field E of a Number field K is the Maximal abelian Unramified 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 Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain In Mathematics, a quadratic field is an Algebraic number field K of degree two over Q. In Mathematics, a Galois group is a group associated with a certain type of Field extension. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose
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. In Abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, the term global field refers to either of the following a number field, i In Mathematics, a Galois extension is an algebraic field extension E / F satisfying certain conditions (described below one also says that the In Mathematics, a Galois group is a group associated with a certain type of Field extension. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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.
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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. In Mathematics, profinite groups are Topological groups that are in a certain sense assembled from Finite groups they share many properties with their finite In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. 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. In Mathematics, profinite groups are Topological groups that are in a certain sense assembled from Finite groups they share many properties with their finite In Mathematics, an adelic algebraic group is a Topological group defined by an Algebraic group G over a Number field 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. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 This is known as the Kronecker-Weber theorem, originally stated by Kronecker. In Algebraic number theory, the Kronecker–Weber theorem states that every finite Abelian extension of the field of Rational numbers Q, or in Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued
For a description of the general case see class formation. In mathematics a class formation is a structure used to organize the various Galois groups and modules that appear in Class field theory.
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. 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 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. In Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability In Mathematics, a quadratic field is an Algebraic number field K of degree two over Q. 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, cubic reciprocity is any of various results connecting the solvability of two related Cubic equations in Modular arithmetic.
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). In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In mathematics reciprocity may mean Quadratic reciprocity, a fundamental result in number theory Hensel is a surname and may refer to Abigail and Brittany Hensel ( Hensel twins) (1990- Albert Hensel Bruce In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers In Mathematics, Kummer theory provides a description of certain types of Field extensions involving the adjunction of n th roots of elements of David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Teiji Takagi (高木 貞治 Takagi Teiji, April 21, 1875 - February 28, 1960) was a Japanese Mathematician, best Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Philipp Furtwängler ( April 21, 1869, Elze, Germany – May 19, 1940, Vienna, Austria) was a The crucial Takagi existence theorem was known by 1920 and all the main results by about 1930. In Class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite 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. In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier It is also basic to Iwasawa theory. In Number theory, Iwasawa theory is a Galois module theory of Ideal class groups initiated by Kenkichi Iwasawa, in the 1950s as part of the
After the results were reformulated in terms of group cohomology, the field became relatively static. In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper 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. The Langlands program is a web of far-reaching and influential Conjectures that connect Number theory and the representation theory of certain groups In Mathematics, non-abelian class field theory is a catchphrase meaning the extension of the results of Class field theory, the relatively complete and classical
See also
References
- E. Artin, J. Tate, Class field theory ISBN 0-201-51011-1
- J. John Torrence Tate Jr born March 13, 1925 in Minneapolis Minnesota, is an American Mathematician, distinguished for many fundamental W. S. Cassels, A. Frohlich, Algebraic Number Theory ISBN 0-12-163251-2
- J. Neukirch, Class Field Theory, Berlin, Heidelberg [et al. ] : Springer, 1986
- J. Neukirch, Algebraische Zahlentheorie, 1992, ISBN 3-540-54273-6 (German)
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