Concept
Class field theory
Related concepts (25)
Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
Global field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: Algebraic number field: A finite extension of Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of , the field of rational functions in one variable over the finite field with elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.
Artin reciprocity law
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
Helmut Hasse
Helmut Hasse (ˈhasə; 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 class field theory and diophantine geometry (Hasse principle), and to local zeta functions. Hasse was born in Kassel, Province of Hesse-Nassau, the son of Judge Paul Reinhard Hasse, also written Haße (12 April 1868 – 1 June 1940, son of Friedrich Ernst Hasse and his wife Anna Von Reinhard) and his wife Margarethe Louise Adolphine Quentin (born 5 July 1872 in Milwaukee, daughter of retail toy merchant Adolph Quentin (b.
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Then: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form for an odd prime ; that is, to determine the "perfect squares" modulo .
Local field
In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below.
Reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial splits into linear terms when reduced mod . That is, it determines for which prime numbers the relationholds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes the polynomial splits into linear factors, denoted .
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Kummer was born in Sorau, Brandenburg (then part of Prussia). He was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay (De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis), which was published a year later.
Complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points.
Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer n – and therefore belong to abstract algebra.