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| {{About|Gauss's lemma in number theory||Gauss's lemma (disambiguation)}}
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| '''Gauss's lemma''' in [[number theory]] gives a condition for an integer to be a [[quadratic residue]]. Although it is not useful computationally, it has theoretical significance, being involved in some [[proofs of quadratic reciprocity]].
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| It made its first appearance in [[Carl Friedrich Gauss]]'s third proof (1808)<ref>"Neuer Beweis eines arithmetischen Satzes"; pp 458-462 of ''Untersuchungen uber hohere Arithmetik''</ref> of [[quadratic reciprocity]] and he proved it again in his fifth proof (1818).<ref>"Neue Beweise und Erweiterungen des Fundalmentalsatzes in der Lehre von den quadratischen Reste"; pp 496-501 of ''Untersuchungen uber hohere Arithmetik''</ref>
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| == Statement of the lemma ==
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| For any odd prime ''p'' let ''a'' be an integer that is [[coprime]] to ''p''.
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| Consider the integers
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| :<math>a, 2a, 3a, \dots, \frac{p-1}{2}a</math>
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| and their least positive residues modulo ''p''. (These residues are all distinct, so there are (''p''−1)/2 of them.)
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| Let ''n'' be the number of these residues that are greater than ''p''/2. Then
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| :<math>\left(\frac{a}{p}\right) = (-1)^n</math>
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| where (''a''/''p'') is the [[Legendre symbol]].
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| == Example ==
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| Taking ''p'' = 11 and ''a'' = 7, the relevant sequence of integers is
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| : 7, 14, 21, 28, 35.
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| After reduction modulo 11, this sequence becomes
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| : 7, 3, 10, 6, 2.
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| Three of these integers are larger than 11/2 (namely 6, 7 and 10), so ''n'' = 3. Correspondingly Gauss's lemma predicts that
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| : <math>\left(\frac
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| {7}{11}\right) = (-1)^3 = -1.</math>
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| This is indeed correct, because 7 is not a quadratic residue modulo 11.
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| The above sequence of residues
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| : 7, 3, 10, 6, 2
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| may also be written
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| : -4, 3, -1, -5, 2.
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| In this form, the integers larger than 11/2 appear as negative numbers. It is also apparent that the absolute values of the residues are a permutation of the residues
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| : 1, 2, 3, 4, 5.
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| == Proof ==
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| A fairly simple proof<ref>Any textbook on elementary number theory will have a proof. The one here is basically Gauss's from "Neuer Beweis eines arithnetischen Satzes"; pp 458-462 of ''Untersuchungen uber hohere Arithmetik''</ref> of the lemma, reminiscent of one of the simplest [[proofs of Fermat's little theorem]], can be obtained by evaluating the product
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| : <math>Z = a \cdot 2a \cdot 3a \cdot \cdots \cdot \frac{p-1}2 a</math>
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| modulo ''p'' in two different ways. On one hand it is equal to
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| : <math>Z = a^{(p-1)/2} \left(1 \cdot 2 \cdot 3 \cdot \cdots \cdot \frac{p-1}2 \right)</math>
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| The second evaluation takes more work. If ''x'' is a nonzero residue modulo ''p'', let us define the "absolute value" of ''x'' to be
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| : <math>|x| = \begin{cases} x & \mbox{if } 1 \leq x \leq \frac{p-1}2, \\ p-x & \mbox{if } \frac{p+1}2 \leq x \leq p-1. \end{cases}</math>
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| Since ''n'' counts those multiples ''ka'' which are in the latter range, and since for those multiples, −ka is in the first range, we have
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| : <math>Z = (-1)^n \left(|a| \cdot |2a| \cdot |3a| \cdot \cdots \cdots \left|\frac{p-1}2 a\right|\right).</math>
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| Now observe that the values |''ra''| are ''distinct'' for ''r'' = 1, 2, ..., (''p''−1)/2. Indeed, if |''ra''| = |''sa''|, then ''ra'' = ±''sa'', and therefore ''r'' = ±''s'' (because ''a'' is invertible modulo ''p''), so ''r'' = ''s'' because they are both in the range 1 ≤ ''r'' ≤ (''p''−1)/2. But there are exactly (''p''−1)/2 of them, so they must just be some rearrangement of the integers 1, 2, ..., (''p''−1)/2. Therefore
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| : <math>Z = (-1)^n \left(1 \cdot 2 \cdot 3 \cdot \cdots \cdot \frac{p-1}2\right).</math>
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| Comparing with our first evaluation, we may cancel out the nonzero factor
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| : <math>1 \cdot 2 \cdot 3 \cdot \cdots \cdot \frac{p-1}2</math>
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| and we are left with
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| : <math>a^{(p-1)/2} = (-1)^n.\ </math>
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| This is the desired result, because by [[Euler's criterion]] the left hand side is just an alternative expression for the Legendre symbol (''a''/''p'').
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| == Applications ==
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| Gauss's lemma is used in many,<ref>Lemmermeyer, ch. 1</ref><ref>Lemmermeyer, p. 9, "like most of the simplest proofs [ of QR], [Gauss's] 3 and 5 rest on what we now call Gauss's Lemma</ref> but by no means all, of the known proofs of quadratic reciprocity.
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| For example, [[Gotthold Eisenstein|Eisenstein]]<ref>Lemmermeyer, p. 236, Prop 8.1 (1845)</ref> used Gauss's lemma to prove that if ''p'' is an odd prime then
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| <!--
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| If ''p'' is an odd prime and ''a'' is not a multiple of ''p'' then
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| this is not needed. If p|a the numerator is 0.
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| -->
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| :<math>\left(\frac{a}{p}\right)=\prod_{n=1}^{(p-1)/2}\frac{\sin{(2\pi an/p)}}{\sin{(2\pi n/p)}},</math>
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| and used this formula to prove quadratic reciprocity, (and, by using [[Elliptic function|elliptic]] rather than [[Trigonometric functions|circular]] functions, to prove the [[cubic reciprocity|cubic]] and [[quartic reciprocity]] laws.<ref>Lemmermeyer, ch. 8</ref>)
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| [[Kronecker]]<ref>Lemmermeyer, ex. 1.34 (The year isn't clear because K. published 8 proofs, several based on Gauss's lemma, between 1875 and 1889)</ref> used the lemma to show that
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| :<math>\left(\frac{p}{q}\right)=\sgn\prod_{i=1}^{\frac{q-1}{2}}\prod_{k=1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right).</math>
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| Switching ''p'' and ''q'' immediately gives quadratic reciprocity.
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|
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| It is also used in what are probably the simplest proofs of the "second supplementary law"
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| :<math>\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8} = \begin{cases} +1\text{ if }p\equiv \pm 1\pmod {8}\\-1\text{ if }p\equiv \pm 3\pmod {8}\end{cases}</math>
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| <!--
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| Gauss's lemma can be used to prove the following statement, and the reverse is true too.
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| eh? are you implyng that one can deduce GL from the formula?
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| I'm removing this statement till someone can provide a reference, and moving the formula to the top of the section | |
| -->
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| ==Higher powers==
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| Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity,<ref>Gauss, BQ, §§ 69–71</ref> Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + ''i'' in '''Z'''[''i''], the ring of [[Gaussian integers]]. Subsequently,<ref>Lemmermeyer, Ch. 8</ref> Eisenstein used third- and fourth-power versions to prove [[cubic reciprocity|cubic]] and [[quartic reciprocity]].
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| ===''n''th power residue symbol===
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| {{main|Power residue symbol}}
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| Let ''k'' be an [[algebraic number field]] with [[ring of integers]] <math>\mathcal{O}_k,</math> and let <math>\mathfrak{p} \subset \mathcal{O}_k </math> be a [[Number_field#Prime_ideals|prime ideal]]. The [[ideal norm]] of <math>\mathfrak{p} </math> is defined as the cardinality of the residue class ring (since <math>\mathfrak{p} </math> is prime this is a [[finite field]]) <math> \mathcal{O}_k / \mathfrak{p}\;:\;\;\; \mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}|.</math>
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| Assume that a primitive ''n''th [[root of unity]] <math>\zeta_n\in\mathcal{O}_k,</math> and that ''n'' and <math>\mathfrak{p} </math> are [[coprime]] (i.e. <math>n\not\in \mathfrak{p}.</math>) Then
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| '''No two distinct ''n''th roots of unity can be congruent '''<math>\pmod{\mathfrak{p}}.</math>
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| The proof is by contradiction: assume otherwise, that <math>\zeta_n^r\equiv\zeta_n^s\pmod{\mathfrak{p}}, \;\;0 <r<s\le n.</math> Then letting <math>t=s-r,\;\;\zeta_n^t\equiv 1 \pmod{\mathfrak{p}}, </math> and <math> 0 < t< n.\ </math> From the definition of roots of unity,
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| :<math>x^n-1=(x-1)(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}),</math> and dividing by ''x'' − 1 gives
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| :<math>x^{n-1}+x^{n-2}+\dots +x + 1 =(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}).</math>
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| Letting ''x'' = 1 and taking residues <math>\pmod{\mathfrak{p}},</math>
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| :<math>n\equiv(1-\zeta_n)(1-\zeta_n^2)\dots(1-\zeta_n^{n-1})\pmod{\mathfrak{p}}.</math>
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| Since ''n'' and <math> \mathfrak{p}</math> are coprime, <math> n\not\equiv 0\pmod{\mathfrak{p}},</math> but under the assumption, one of the factors on the right must be zero. Therefore the assumption that two distinct roots are congruent is false.
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| Thus the residue classes of <math> \mathcal{O}_k / \mathfrak{p}</math> containing the powers of ζ<sub>''n''</sub> are a subgroup of order ''n'' of its (multiplicative) group of units, <math>(\mathcal{O}_k/\mathfrak{p}) ^\times = \mathcal{O}_k /\mathfrak{p}- \{0\}.</math> Therefore the order of <math>(\mathcal{O}_k/\mathfrak{p})^ \times</math> is a multiple of ''n'', and
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| :<math>\mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}| = |(\mathcal{O}_k / \mathfrak{p} )^\times| + 1 \equiv 1 \pmod{n}.</math>
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| There is an analogue of Fermat's theorem in <math>\mathcal{O}_k:</math> If <math>\alpha \in \mathcal{O}_k,\;\;\; \alpha\not\in \mathfrak{p},</math> then<ref>Lemmermeyer, Ch. 4.1</ref>
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| :<math>\alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \pmod{\mathfrak{p} },
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| </math> and since <math>\mathrm{N} \mathfrak{p} \equiv 1 \pmod{n},</math>
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| :<math>\alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\pmod{\mathfrak{p} }
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| </math> is well-defined and congruent to a unique ''n''th root of unity ζ<sub>''n''</sub><sup>''s''</sup>.
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| This root of unity is called the '''''n''th-power residue symbol for <math>\mathcal{O}_k,</math>''' and is denoted by
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| :<math>
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| \left(\frac{\alpha}{\mathfrak{p} }\right)_n= \zeta_n^s \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}.
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| </math>
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| It can be proven that<ref>Lemmermeyer, Prop 4.1</ref>
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| :<math>
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| \left(\frac{\alpha}{\mathfrak{p} }\right)_n= 1 \mbox{ if and only if there is an } \eta \in\mathcal{O}_k\;\;\mbox{ such that } \;\;\alpha\equiv\eta^n\pmod{\mathfrak{p}}.</math>
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| ===1/''n'' systems===
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| Let <math>\mu_n = \{1,\zeta_n,\zeta_n^2,\dots,\zeta_n^{n-1}\} </math> be the multiplicative group of the ''n''th roots of unity, and let <math>A=\{a_1, a_2,\dots,a_m\}</math> be representatives of the cosets of <math>(\mathcal{O}_k / \mathfrak{p})^\times/\mu_n.</math> Then ''A'' is called a '''1/''n'' system ''' <math>\pmod\mathfrak{p}.</math><ref>Lemmermeyer, Ch. 4.2</ref>
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| In other words, there are <math>mn=\mathrm{N} \mathfrak{p} -1 </math> numbers in the set <math>A\mu=\{ a_i \zeta_n^j\;:\; 1 \le i \le m, \;\;\;0 \le j \le n-1\},</math> and this set constitutes a representative set for <math>(\mathcal{O}_k / \mathfrak{p})^\times.</math>
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| The numbers 1, 2, ..., (''p'' − 1)/2, used in the original version of the lemma, are a 1/2 system (mod ''p'').
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| Constructing a 1/''n'' system is straightforward: let ''M'' be a representative set for <math>(\mathcal{O}_k / \mathfrak{p})^\times.</math> Pick any <math>a_1\in M </math> and remove the numbers congruent to <math>a_1, a_1\zeta_n, a_1\zeta_n^2, \dots, a_1\zeta_n^{n-1}</math>
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| from ''M''. Pick ''a''<sub>2</sub> from ''M'' and remove the numbers congruent to <math>a_2, a_2\zeta_n, a_2\zeta_n^2, \dots, a_2\zeta_n^{n-1}</math> Repeat until ''M'' is exhausted. Then {''a''<sub>1</sub>, ''a''<sub>2</sub>, ... ''a''<sub>m</sub>} is a 1/''n'' system <math>\pmod\mathfrak{p}.</math> | |
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| ===The lemma for ''n''th powers===
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| Gauss's lemma for the ''n''th power residue symbol is<ref>Lemmermeyer, Prop. 4.3</ref>
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| Let <math>\zeta_n\in \mathcal{O}_k </math> be a primitive ''n''th root of unity, <math>\mathfrak{p} \subset \mathcal{O}_k </math> a prime ideal, <math>\gamma \in \mathcal{O}_k, \;\;n\gamma\not\in\mathfrak{p},</math> (i.e. <math>\mathfrak{p}</math> is coprime to both γ and ''n'') and let ''A'' = {''a''<sub>1</sub>, ''a''<sub>2</sub>,..., ''a''<sub>''m''</sub>} be a 1/''n'' system <math>\pmod{\mathfrak{p}}.</math>
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| Then for each ''i'', 1 ≤ ''i'' ≤ ''m'', there are integers π(''i''), unique (mod ''m''), and ''b''(''i''), unique (mod ''n''), such that
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| :<math>\gamma a_i \equiv \zeta_n^{b(i)}a_{\pi(i)} \pmod{\mathfrak{p}}, | |
| </math>
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| and the ''n''th-power residue symbol is given by the formula
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| :<math>
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| \left(\frac{\gamma}{\mathfrak{p} }\right)_n = \zeta_n^{b(1)+b(2)+\dots+b(m)}.
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| </math>
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| The classical lemma for the quadratic Legendre symbol is the special case ''n'' = 2, ζ<sub>2</sub> = −1, ''A'' = {1, 2, ..., (''p'' − 1)/2}, ''b''(''k'') = 1 if ''ak'' > ''p''/2, ''b''(''k'') = 0 if ''ak'' < ''p''/2.
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| ===Proof===
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| The proof of the ''n''th-power lemma uses the same ideas that were used in the proof of the quadratic lemma.
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| The existence of the integers π(''i'') and ''b''(''i''), and their uniqueness (mod ''m'') and (mod ''n''), respectively, come from the fact that ''A''μ is a representative set.
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| Assume that π(''i'') = π(''j'') = ''p'', i.e.
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| :<math>\gamma a_i \equiv \zeta_n^r a_p \pmod{\mathfrak{p}}</math> and <math>\gamma a_j \equiv \zeta_n^s a_p \pmod{\mathfrak{p}}.</math> | |
| Then
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| :<math>\zeta_n^{s-r}\gamma a_i \equiv \zeta_n^s a_p \equiv \gamma a_j\pmod{\mathfrak{p}}</math>
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| Because γ and <math>\mathfrak{p}</math> are coprime both sides can be divided by γ, giving
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| :<math>\zeta_n^{s-r} a_i \equiv a_j\pmod{\mathfrak{p}},</math>
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| which, since ''A'' is a 1/''n'' system, implies ''s'' = ''r'' and ''i'' = ''j'', showing that π is a permutation of the set {1, 2, ..., ''m''}.
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| Then on the one hand, by the definition of the power residue symbol,
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| :<math>
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| \begin{align}
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| (\gamma a_1)(\gamma a_2)\dots(\gamma a_m) &= \gamma^{\frac{\mathrm{N} \mathfrak{p} -1}{n}} a_1 a_2\dots a_m \\&\equiv \left(\frac{\gamma}{\mathfrak{p} }\right)_n a_1 a_2\dots a_m \pmod{\mathfrak{p}},
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| \end{align}
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| </math>
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| and on the other hand, since π is a permutation,
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| :<math>
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| \begin{align}
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| (\gamma a_1)(\gamma a_2)\dots(\gamma a_m)
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| &\equiv
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| {\zeta_n^{b(1)}a_{\pi(1)}} {\zeta_n^{b(2)}a_{\pi(2)}}\dots{\zeta_n^{b(m)}a_{\pi(m)}} \\
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| &\equiv
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| \zeta_n^{b(1)+b(2)+\dots+b(m)}a_{\pi(1)} a_{\pi(2)}\dots a_{\pi(m)}\\
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| &\equiv
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| \zeta_n^{b(1)+b(2)+\dots+b(m)} a_1 a_2\dots a_m
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| \pmod{\mathfrak{p}},
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| \end{align}
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| </math>
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| so
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| :<math>
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| \left(\frac{\gamma}{\mathfrak{p} }\right)_n a_1 a_2\dots a_m \equiv \zeta_n^{b(1)+b(2)+\dots+b(m)} a_1 a_2\dots a_m
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| \pmod{\mathfrak{p}},
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| </math>
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| and since for all 1 ≤ ''i'' ≤ ''m'', ''a''<sub>''i''</sub> and <math>\mathfrak{p}</math> are coprime, ''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>m</sub> can be cancelled from both sides of the congruence,
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| :<math>\left(\frac{\gamma}{\mathfrak{p} }\right)_n \equiv \zeta_n^{b(1)+b(2)+\dots+b(m)}
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| \pmod{\mathfrak{p}},
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| </math>
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| and the theorem follows from the fact that no two distinct ''n''<sup>th</sup> roots of unity can be congruent (mod <math>\mathfrak{p}</math>).
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| == Relation to the transfer in group theory ==
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| Let ''G'' be the multiplicative group of nonzero residue classes in '''Z'''/''p'''''Z''', and let ''H'' be the subgroup {+1, −1}. Consider the following coset representatives of ''H'' in ''G'',
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| :<math>1, 2, 3, \dots, \frac{p-1}{2}.</math>
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| Applying the machinery of the [[transfer (group theory)|transfer]] to this collection of coset representatives, we obtain the transfer homomorphism
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| :<math>\phi : G \to H,</math>
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| which turns out to be the map that sends ''a'' to (−1)<sup>''n''</sup>, where ''a'' and ''n'' are as in the statement of the lemma. Gauss's lemma may then be viewed as a computation that explicitly identifies this homomorphism as being the quadratic residue character.
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| == See also ==
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| Two other characterizations of squares modulo a prime are [[Euler's criterion]] and [[Zolotarev's lemma]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § ''n''".
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | title = Theoria residuorum biquadraticorum, Commentatio prima
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| | publisher = Comment. Soc. regiae sci, Göttingen 6
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| | location = Göttingen
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| | year = 1828}}
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | title = Theoria residuorum biquadraticorum, Commentatio secunda
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| | publisher = Comment. Soc. regiae sci, Göttingen 7
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| | location = Göttingen
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| | year = 1832}}
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| These are in Gauss's ''Werke'', Vol II, pp. 65–92 and 93–148
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| German translations of the above are in the following, which also has the [[Disquisitiones Arithmeticae]] and Gauss's other papers on number theory, including the six proofs of quadratic reciprocity.
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Maser | first2 = H. (translator into German)
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| | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)
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| | publisher = Chelsea
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| | location = New York
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| | year = 1965
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| | isbn = 0-8284-0191-8}}
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| *{{citation
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| | last1 = Lemmermeyer | first1 = Franz
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| | title = Reciprocity Laws: from Euler to Eisenstein
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = Berlin
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| | year = 2000
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| | isbn = 3-540-66957-4}}
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| [[Category:Modular arithmetic]]
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| [[Category:Lemmas]]
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| [[Category:Articles containing proofs]]
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| [[Category:Quadratic residue]]
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