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Norm Residue Symbol
In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality. * Legendre symbol \left(\frac\right) defined for ''p'' a prime, ''a'' an integer, and takes values 0, 1, or −1. * Jacobi symbol \left(\frac\right) defined for ''b'' a positive odd integer, ''a'' an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of ''b''. * Kronecker symbol \left(\frac\right) defined for ''b'' any integer, ''a'' an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of ''b''. * Power residue symbol \left(\frac\right)=\left(\frac\right)_m is defined for ''a'' in some global field containing the ''m''th roots of 1 ( for some ''m''), ''b'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable group, solvable, i.e., if the group can be decomposed into a series of normal Group extension, extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Description Class field theory provides detailed information about the abelian extensions of number fields, function field of an algebraic variety, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contou-Carrère Symbol
In mathematics, the Contou-Carrère symbol 〈''a'',''b''〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series over an Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ... ''k'', taking values in the group of units of ''k''. It was introduced by . Definition If ''k'' is an Artinian local ring, then any invertible formal Laurent series ''a'' with coefficients in ''k'' can be written uniquely as :a=a_0t^\prod_(1-a_it^i) where ''w''(''a'') is an integer, the elements ''a''''i'' are in ''k'', and are in ''m'' if ''i'' is negative, and is a unit if ''i'' = 0. The Contou-Carrère symbol 〈''a'',''b''〉 of ''a'' and ''b'' is defined to be :\langle a,b\rangle=(-1)^\frac References * Number theory {{numthe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Symbol
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime \ell and any natural number n. John MilnorMilnor (1970) speculated that this theorem might be true for \ell=2 and all n, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of ''L''-functions.Bloch, Spencer and Kato, Kazuya, "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, 333–400, Progr. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinberg Symbol
In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg. For a field ''F'' we define a ''Steinberg symbol'' (or simply a ''symbol'') to be a function ( \cdot , \cdot ) : F^* \times F^* \rightarrow G, where ''G'' is an abelian group, written multiplicatively, such that * ( \cdot , \cdot ) is bimultiplicative; * if a+b = 1 then (a,b) = 1. The symbols on ''F'' derive from a "universal" symbol, which may be regarded as taking values in F^* \otimes F^* / \langle a \otimes 1-a \rangle. By a theorem of Hideya Matsumoto, this group is K_2 F and is part of the Milnor K-theory for a field. Properties If (â‹…,â‹…) is a symbol then (assuming all terms are defined) * (a, -a) = 1 ; * (b, a) = (a, b)^ ; * (a, a) = (a, -1) is an element of order 1 or 2; * (a, b) = (a+b, -b/a) . Examples * The trivial symbol which is identically 1. * The Hilbert symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm Residue Symbol
In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality. * Legendre symbol \left(\frac\right) defined for ''p'' a prime, ''a'' an integer, and takes values 0, 1, or −1. * Jacobi symbol \left(\frac\right) defined for ''b'' a positive odd integer, ''a'' an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of ''b''. * Kronecker symbol \left(\frac\right) defined for ''b'' any integer, ''a'' an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of ''b''. * Power residue symbol \left(\frac\right)=\left(\frac\right)_m is defined for ''a'' in some global field containing the ''m''th roots of 1 ( for some ''m''), ''b'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class that includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients :\frac, if . Ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Extension
Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Christopher Kummer (born 1975), German economist * Corby Kummer (born 1957), American journalist * Dirk Kummer (born 1966), German actor, director, and screenwriter * Eberhard Kummer (1940–2019), Austrian concert singer, lawyer, and medieval music expert * Eduard Kummer, also known as the following Ernst Kummer * Eloise Kummer (1916–2008), American actress *Ernst Kummer (1810–1893), German mathematician ** Kummer configuration, a mathematical structure discovered by Ernst Kummer ** Kummer surface, a related geometrical structure discovered by Ernst Kummer * Ferdinand von Kummer (1816–1900), German general * Frederic Arnold Kummer (1873–1943), American author, playwright, and screenwriter * Friedrich August Kummer (1797–1879), German ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idele 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 definition of the appropriate topology is straightforward only in case ''G'' is a linear algebraic group. In the case of ''G'' being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms. In case ''G'' is a linear algebraic group, it is an affine algebraic variety in affine ''N''-space. The topology on the adelic algebraic group G(A) is taken to be the subspace topology in ''A''''N'', the Cartesian product of ''N'' copies of the adele ring. In this case, G(A) is a topological group. Histo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
