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Symbol (number Theory)
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 (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue (''non-residue'') is −1. Its value at zero is 0. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol. Definition Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a and p defined as :\left(\frac\right) = \begi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Symbol
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a Jacobi symbol of −1 is a quadratic residue, and if ''k'' is a quadratic residue modulo a coprime ''n'', then , but not all entries with a Jacobi symbol of 1 (see the and rows) are quadratic residues. Notice also that when either ''n'' or ''k'' is a square, all values are nonnegative. The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography. Definition For any integer ''a'' and any positive odd integer ''n'', the Jacobi symbol is defined as the product of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Symbol
In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by . Definition Let n be a non-zero integer, with prime factorization :n=u \cdot p_1^ \cdots p_k^, where u is a unit (i.e., u=\pm1), and the p_i are primes. Let a be an integer. The Kronecker symbol \left(\frac\right) is defined by : \left(\frac\right) = \left(\frac\right) \prod_^k \left(\frac\right)^. For odd p_i, the number \left(\frac\right) is simply the usual Legendre symbol. This leaves the case when p_i=2. We define \left(\frac\right) by : \left(\frac\right) = \begin 0 & \mboxa\mbox \\ 1 & \mbox a \equiv \pm1 \pmod, \\ -1 & \mbox a \equiv \pm3 \pmod. \end Since it extends the Jacobi symbol, the quantity \left(\frac\right) is simply 1 when u=1. When u=-1, we define it by : \left(\frac\right) = \begin -1 & \mboxa 0. Table of values The following is a table of values of Kronecker symbol \left(\frac\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Residue Symbol
In algebraic number theory the ''n''-th power residue symbol (for an integer ''n'' > 2) is a generalization of the (quadratic) Legendre symbol to ''n''-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws. Background and notation Let ''k'' be an algebraic number field with ring of integers \mathcal_k that contains a primitive ''n''-th root of unity \zeta_n. Let \mathfrak \subset \mathcal_k be a prime ideal and assume that ''n'' and \mathfrak are coprime (i.e. n \not \in \mathfrak.) The norm of \mathfrak is defined as the cardinality of the residue class ring (note that since \mathfrak is prime the residue class ring is a finite field): :\mathrm \mathfrak := , \mathcal_k / \mathfrak, . An analogue of Fermat's theorem holds in \mathcal_k. If \alpha \in \mathcal_k - \mathfrak, then :\alpha^\equiv 1 \bmod. And finally, suppose \mathrm \mathfrak \equiv 1 \bmod. These facts imply that :\alpha^\equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Symbol
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields. Quadratic Hilbert symbol Over a local field ''K'' whose multiplicative group of non-zero elements is ''K''×, the quadratic Hilbert symbol is the function (–, –) from ''K''× × ''K''× to defined by :(a,b)=\begin+1,&\mboxz^2=ax^2+by^2\mbox(x,y,z)\in K^3;\\-1,&\mbox\end Equivalently, (a, b) = 1 if and only if b is equal to the norm of an element of the quadratic extension Ksqrt/math> page 110. Properties The follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Symbol
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. Statement Let L/K be a Galois extension of global fields and C_L stand for the idèle class group of L. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol mapNeukirch (1999) p.391 : \theta: C_K/ \to \operatorname(L/K)^, where \text denotes the abelianization of a group. The map \theta is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Extension
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th 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. The theory of cyclic extensions of the field ''K'' when the characteristic of ''K'' does divide ''n'' is called Artin–Schreier theory. Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Symbol
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 which 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] |
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 which 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] |
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] |
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 Matsumoto, this group is K_2 F and is part of the Milnor K-theory for a field. Properties If (⋅,⋅) 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 symbol on ''F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
