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A relation connecting the values of the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043520/g0435201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043520/g0435202.png" /> for different odd prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043520/g0435203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043520/g0435204.png" /> (cf. [[Quadratic reciprocity law|Quadratic reciprocity law]]). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation
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A relation connecting the values of the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. [[Quadratic reciprocity law|Quadratic reciprocity law]]). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043520/g0435205.png" /></td> </tr></table>
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$$\left(\frac pq\right)\left(\frac qp\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$
  
 
there are two more additions to this law, viz.:
 
there are two more additions to this law, viz.:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043520/g0435206.png" /></td> </tr></table>
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$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}\quad\text{and}\quad\left(\frac2p\right)=(-1)^{(p^2-1)/8}.$$
  
 
The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [[#References|[1]]]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime.
 
The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [[#References|[1]]]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime.
  
Attempts to establish the reciprocity law for cubic and biquadratic residues led Gauss to introduce the ring of complex integers.
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Attempts to establish the reciprocity law for [[Cubic residue|cubic]] and [[biquadratic residue]]s led Gauss to introduce the ring of [[Gaussian integer]]s.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  "Disquisitiones Arithmeticae" , Yale Univ. Press  (1966)  (Translated from Latin)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. [I.M. Vinogradov] Winogradow,  "Elemente der Zahlentheorie" , R. Oldenbourg  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Hasse,  "Vorlesungen über Zahlentheorie" , Springer  (1950)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  "Disquisitiones Arithmeticae" , Yale Univ. Press  (1966)  (Translated from Latin)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. [I.M. Vinogradov] Winogradow,  "Elemente der Zahlentheorie" , R. Oldenbourg  (1956)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Hasse,  "Vorlesungen über Zahlentheorie" , Springer  (1950)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of [[Algebraic number theory|algebraic number theory]] and [[Class field theory|class field theory]]. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law.
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Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of [[algebraic number theory]] and [[class field theory]]. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law.
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[[Category:Number theory]]

Latest revision as of 17:43, 19 December 2014

A relation connecting the values of the Legendre symbols (cf. Legendre symbol) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. Quadratic reciprocity law). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation

$$\left(\frac pq\right)\left(\frac qp\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$

there are two more additions to this law, viz.:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}\quad\text{and}\quad\left(\frac2p\right)=(-1)^{(p^2-1)/8}.$$

The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [1]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime.

Attempts to establish the reciprocity law for cubic and biquadratic residues led Gauss to introduce the ring of Gaussian integers.

References

[1] C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)
[2] I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)
[3] H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)


Comments

Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of algebraic number theory and class field theory. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law.

How to Cite This Entry:
Gauss reciprocity law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_reciprocity_law&oldid=13461
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article