Difference between revisions of "Law of quadratic reciprocity"

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The '''law of quadratic reciprocity''' predicts whether an odd [[prime number]] ''p'' is a [[quadratic residue]] or non-residue modulo another odd prime number <math>q</math> if we know whether <math>q</math> is a quadratic residue or non-residue modulo <math>p</math>.
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The '''law of quadratic reciprocity''' predicts whether an odd [[prime]] number ''p'' is a [[quadratic residue]] or non-residue modulo another odd prime number <math>q</math> if we know whether <math>q</math> is a quadratic residue or non-residue modulo <math>p</math>.
  
 
*If at least one of <math>p</math> or <math>q</math> are congruent to 1 mod 4: <math>p</math> is a quadratic residue modulo <math>q</math> if and only if <math>q</math> is a quadratic residue modulo <math>p</math>.
 
*If at least one of <math>p</math> or <math>q</math> are congruent to 1 mod 4: <math>p</math> is a quadratic residue modulo <math>q</math> if and only if <math>q</math> is a quadratic residue modulo <math>p</math>.
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==External links==
 
==External links==
*[https://en.wikipedia.org/wiki/Quadratic_reciprocity Wikipedia]
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*[[Wikipedia:Quadratic_reciprocity|Wikipedia]]
 
[[Category:Math]]
 
[[Category:Math]]

Latest revision as of 10:40, 6 February 2019

The law of quadratic reciprocity predicts whether an odd prime number p is a quadratic residue or non-residue modulo another odd prime number [math]q[/math] if we know whether [math]q[/math] is a quadratic residue or non-residue modulo [math]p[/math].

  • If at least one of [math]p[/math] or [math]q[/math] are congruent to 1 mod 4: [math]p[/math] is a quadratic residue modulo [math]q[/math] if and only if [math]q[/math] is a quadratic residue modulo [math]p[/math].
  • If both of [math]p[/math] or [math]q[/math] are congruent to 3 mod 4: [math]p[/math] is a quadratic residue modulo [math]q[/math] if and only if [math]q[/math] is a quadratic non-residue modulo [math]p[/math].

This theorem was first proved by Carl Friedrich Gauss in 1801.

This does not cover the cases where we want to know whether -1 or 2 are quadratic residues or non-residues modulo [math]p[/math].

  • 2 is a quadratic residue modulo [math]p[/math] if and only if [math]p[/math] is congruent to 1 or 7 (mod 8).
  • -1 is a quadratic residue modulo [math]p[/math] if and only if [math]p[/math] is congruent to 1 (mod 4).

External links