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Quadratic reciprocity
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<p>In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the law of quadratic reciprocity is a theorem about <a href="/facts/Modular_arithmetic/0wtRa5dU">modular arithmetic</a> that gives conditions for the solvability of <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equations</a> modulo <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>. Due to its subtlety, it has many formulations, but the most standard statement is: </p> <p>Law of quadratic reciprocity—Let <i>p</i> and <i>q</i> be distinct odd prime numbers, and define the <a href="/facts/Legendre_symbol/MwHrJ7as">Legendre symbol</a> as </p> ( q p ) = { 1 if n 2 ≡ q mod p for some integer n − 1 otherwise . {\displaystyle \left({\frac {q}{p}}\right)={\begin{cases}1&{\text{if }}n^{2}\equiv q{\bmod {p}}{\text{ for some integer }}n\\-1&{\text{otherwise}}.\end{cases}}} <p>Then </p> ( p q ) ( q p ) = ( − 1 ) p − 1 2 q − 1 2 . {\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.} <p>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 x 2 ≡ a mod p {\displaystyle x^{2}\equiv a{\bmod {p}}} for an odd prime p {\displaystyle p} ; that is, to determine the "perfect squares" modulo p {\displaystyle p} . However, this is a <a href="/facts/Constructivism_(mathematics)/1VtNuLwT">non-constructive</a> result: it gives no help at all for finding a <i>specific</i> solution; for this, other methods are required. For example, in the case p ≡ 3 mod 4 {\displaystyle p\equiv 3{\bmod {4}}} using <a href="/facts/Euler%2527s_criterion/JkbpqqB1">Euler's criterion</a> one can give an explicit formula for the "square roots" modulo p {\displaystyle p} of a quadratic residue a {\displaystyle a} , namely, </p> ± a p + 1 4 {\displaystyle \pm a^{\frac {p+1}{4}}} <p>indeed, </p> ( ± a p + 1 4 ) 2 = a p + 1 2 = a ⋅ a p − 1 2 ≡ a ( a p ) = a mod p . {\displaystyle \left(\pm a^{\frac {p+1}{4}}\right)^{2}=a^{\frac {p+1}{2}}=a\cdot a^{\frac {p-1}{2}}\equiv a\left({\frac {a}{p}}\right)=a{\bmod {p}}.} <p>This formula only works if it is known in advance that a {\displaystyle a} is a <a href="/facts/Quadratic_residue/rHD2uQdQ">quadratic residue</a>, which can be checked using the law of quadratic reciprocity. </p><p>The quadratic reciprocity theorem was conjectured by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> and <a href="/facts/Adrien-Marie_Legendre/2a7CAolM">Adrien-Marie Legendre</a> and first proved by <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>, who referred to it as the "fundamental theorem" in his <i><a href="/facts/Disquisitiones_Arithmeticae/VQ38ReTh">Disquisitiones Arithmeticae</a></i> and his papers, writing </p> <i>The fundamental theorem must certainly be regarded as one of the most elegant of its type.</i> (Art. 151) <p>Privately, Gauss referred to it as the "golden theorem". He published six <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a> for it, and two more were found in his posthumous papers. There are now over 240 published proofs. The shortest known proof is included below, together with short proofs of the law's supplements (the Legendre symbols of −1 and 2). </p><p>Generalizing the <a href="/facts/Reciprocity_law/s4YLAgn5">reciprocity law</a> to higher powers has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of <a href="/facts/Abstract_algebra/YNNE1CNz">modern algebra</a>, number theory, and <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, culminating in <a href="/facts/Artin_reciprocity/pNkpYrHz">Artin reciprocity</a>, <a href="/facts/Class_field_theory/4cvKbn6a">class field theory</a>, and the <a href="/facts/Langlands_program/koK9PhID">Langlands program</a>. </p>