[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/power-residue-symbol-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/power-residue-symbol-wikipedia\/","headline":"Power residue symbol – Wikipedia","name":"Power residue symbol – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In algebraic number theory the n-th power residue symbol (for an integer n","datePublished":"2017-09-02","dateModified":"2017-09-02","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/78758c163c0f13c6c4ebabc6e1e35d64f1688395","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/78758c163c0f13c6c4ebabc6e1e35d64f1688395","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/power-residue-symbol-wikipedia\/","wordCount":8643,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In 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[1]reciprocity laws.[2]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsBackground and notation[edit]Definition[edit]Properties[edit]Relation to the Hilbert symbol[edit]Generalizations[edit]Power reciprocity law[edit]See also[edit]References[edit]Background and notation[edit]Let k be an algebraic number field with ring of integers Ok{displaystyle {mathcal {O}}_{k}} that contains a primitive n-th root of unity \u03b6n.{displaystyle zeta _{n}.}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Let p\u2282Ok{displaystyle {mathfrak {p}}subset {mathcal {O}}_{k}} be a prime ideal and assume that n and p{displaystyle {mathfrak {p}}} are coprime (i.e. n\u2209p{displaystyle nnot in {mathfrak {p}}}.)The norm of p{displaystyle {mathfrak {p}}} is defined as the cardinality of the residue class ring (note that since p{displaystyle {mathfrak {p}}} is prime the residue class ring is a finite field):Np:=|Ok\/p|.{displaystyle mathrm {N} {mathfrak {p}}:=|{mathcal {O}}_{k}\/{mathfrak {p}}|.}An analogue of Fermat’s theorem holds in Ok.{displaystyle {mathcal {O}}_{k}.} If \u03b1\u2208Ok\u2212p,{displaystyle alpha in {mathcal {O}}_{k}-{mathfrak {p}},} then\u03b1Np\u22121\u22611modp.{displaystyle alpha ^{mathrm {N} {mathfrak {p}}-1}equiv 1{bmod {mathfrak {p}}}.}And finally, suppose Np\u22611modn.{displaystyle mathrm {N} {mathfrak {p}}equiv 1{bmod {n}}.}  These facts imply that\u03b1Np\u22121n\u2261\u03b6nsmodp{displaystyle alpha ^{frac {mathrm {N} {mathfrak {p}}-1}{n}}equiv zeta _{n}^{s}{bmod {mathfrak {p}}}}is well-defined and congruent to a unique n{displaystyle n}-th root of unity \u03b6ns.{displaystyle zeta _{n}^{s}.}Definition[edit]This root of unity is called  the n-th power residue symbol for Ok,{displaystyle {mathcal {O}}_{k},} and is denoted by(\u03b1p)n=\u03b6ns\u2261\u03b1Np\u22121nmodp.{displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}=zeta _{n}^{s}equiv alpha ^{frac {mathrm {N} {mathfrak {p}}-1}{n}}{bmod {mathfrak {p}}}.}Properties[edit]The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol (\u03b6{displaystyle zeta } is a fixed primitive n{displaystyle n}-th root of unity):(\u03b1p)n={0\u03b1\u2208p1\u03b1\u2209p\u00a0and\u00a0\u2203\u03b7\u2208Ok:\u03b1\u2261\u03b7nmodp\u03b6\u03b1\u2209p\u00a0and there is no such\u00a0\u03b7{displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}={begin{cases}0&alpha in {mathfrak {p}}\\1&alpha not in {mathfrak {p}}{text{ and }}exists eta in {mathcal {O}}_{k}:alpha equiv eta ^{n}{bmod {mathfrak {p}}}\\zeta &alpha not in {mathfrak {p}}{text{ and there is no such }}eta end{cases}}}In all cases (zero and nonzero)(\u03b1p)n\u2261\u03b1Np\u22121nmodp.{displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}equiv alpha ^{frac {mathrm {N} {mathfrak {p}}-1}{n}}{bmod {mathfrak {p}}}.}(\u03b1p)n(\u03b2p)n=(\u03b1\u03b2p)n{displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}left({frac {beta }{mathfrak {p}}}right)_{n}=left({frac {alpha beta }{mathfrak {p}}}right)_{n}}\u03b1\u2261\u03b2modp\u21d2(\u03b1p)n=(\u03b2p)n{displaystyle alpha equiv beta {bmod {mathfrak {p}}}quad Rightarrow quad left({frac {alpha }{mathfrak {p}}}right)_{n}=left({frac {beta }{mathfrak {p}}}right)_{n}}Relation to the Hilbert symbol[edit]The n-th power residue symbol is related to the Hilbert symbol (\u22c5,\u22c5)p{displaystyle (cdot ,cdot )_{mathfrak {p}}} for the prime p{displaystyle {mathfrak {p}}} by(\u03b1p)n=(\u03c0,\u03b1)p{displaystyle left({frac {alpha }{mathfrak {p}}}right)_{n}=(pi ,alpha )_{mathfrak {p}}}in the case p{displaystyle {mathfrak {p}}} coprime to n, where \u03c0{displaystyle pi } is any uniformising element for the local field Kp{displaystyle K_{mathfrak {p}}}.[3]Generalizations[edit]The n{displaystyle n}-th power symbol may be extended to take non-prime ideals or non-zero elements as its “denominator”, in the same way that the Jacobi symbol extends the Legendre symbol.Any ideal a\u2282Ok{displaystyle {mathfrak {a}}subset {mathcal {O}}_{k}} is the product of prime ideals, and in one way only:a=p1\u22efpg.{displaystyle {mathfrak {a}}={mathfrak {p}}_{1}cdots {mathfrak {p}}_{g}.}The n{displaystyle n}-th power symbol is extended multiplicatively:(\u03b1a)n=(\u03b1p1)n\u22ef(\u03b1pg)n.{displaystyle left({frac {alpha }{mathfrak {a}}}right)_{n}=left({frac {alpha }{{mathfrak {p}}_{1}}}right)_{n}cdots left({frac {alpha }{{mathfrak {p}}_{g}}}right)_{n}.}For 0\u2260\u03b2\u2208Ok{displaystyle 0neq beta in {mathcal {O}}_{k}} then we define(\u03b1\u03b2)n:=(\u03b1(\u03b2))n,{displaystyle left({frac {alpha }{beta }}right)_{n}:=left({frac {alpha }{(beta )}}right)_{n},}where (\u03b2){displaystyle (beta )} is the principal ideal generated by \u03b2.{displaystyle beta .}Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.If \u03b1\u2261\u03b2moda{displaystyle alpha equiv beta {bmod {mathfrak {a}}}} then (\u03b1a)n=(\u03b2a)n.{displaystyle left({tfrac {alpha }{mathfrak {a}}}right)_{n}=left({tfrac {beta }{mathfrak {a}}}right)_{n}.}(\u03b1a)n(\u03b2a)n=(\u03b1\u03b2a)n.{displaystyle left({tfrac {alpha }{mathfrak {a}}}right)_{n}left({tfrac {beta }{mathfrak {a}}}right)_{n}=left({tfrac {alpha beta }{mathfrak {a}}}right)_{n}.}(\u03b1a)n(\u03b1b)n=(\u03b1ab)n.{displaystyle left({tfrac {alpha }{mathfrak {a}}}right)_{n}left({tfrac {alpha }{mathfrak {b}}}right)_{n}=left({tfrac {alpha }{mathfrak {ab}}}right)_{n}.}Since the symbol is always an n{displaystyle n}-th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an n{displaystyle n}-th power; the converse is not true.Power reciprocity law[edit]The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as[4](\u03b1\u03b2)n(\u03b2\u03b1)n\u22121=\u220fp|n\u221e(\u03b1,\u03b2)p,{displaystyle left({frac {alpha }{beta }}right)_{n}left({frac {beta }{alpha }}right)_{n}^{-1}=prod _{{mathfrak {p}}|ninfty }(alpha ,beta )_{mathfrak {p}},}whenever \u03b1{displaystyle alpha } and \u03b2{displaystyle beta } are coprime.See also[edit]^ Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.^ All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2^ Neukirch (1999) p. 336^ Neukirch (1999) p. 415References[edit]Gras, Georges (2003), Class field theory. From theory to practice, Springer Monographs in Mathematics, Berlin: Springer-Verlag, pp.\u00a0204\u2013207, ISBN\u00a03-540-44133-6, Zbl\u00a01019.11032Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN\u00a00-387-97329-XLemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer Science+Business Media, doi:10.1007\/978-3-662-12893-0, ISBN\u00a03-540-66957-4, MR\u00a01761696, Zbl\u00a00949.11002Neukirch, J\u00fcrgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol.\u00a0322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN\u00a03-540-65399-6, Zbl\u00a00956.11021       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/power-residue-symbol-wikipedia\/#breadcrumbitem","name":"Power residue symbol – Wikipedia"}}]}]