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Difference between revisions of "Law of quadratic reciprocity"
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| − | The '''law of quadratic reciprocity''' predicts whether an odd [[prime | + | 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== | ||
| − | *[ | + | *[[Wikipedia:Quadratic_reciprocity|Wikipedia]] |
[[Category:Math]] | [[Category:Math]] | ||
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]\displaystyle{ q }[/math] if we know whether [math]\displaystyle{ q }[/math] is a quadratic residue or non-residue modulo [math]\displaystyle{ p }[/math].
- If at least one of [math]\displaystyle{ p }[/math] or [math]\displaystyle{ q }[/math] are congruent to 1 mod 4: [math]\displaystyle{ p }[/math] is a quadratic residue modulo [math]\displaystyle{ q }[/math] if and only if [math]\displaystyle{ q }[/math] is a quadratic residue modulo [math]\displaystyle{ p }[/math].
- If both of [math]\displaystyle{ p }[/math] or [math]\displaystyle{ q }[/math] are congruent to 3 mod 4: [math]\displaystyle{ p }[/math] is a quadratic residue modulo [math]\displaystyle{ q }[/math] if and only if [math]\displaystyle{ q }[/math] is a quadratic non-residue modulo [math]\displaystyle{ 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]\displaystyle{ p }[/math].
- 2 is a quadratic residue modulo [math]\displaystyle{ p }[/math] if and only if [math]\displaystyle{ p }[/math] is congruent to 1 or 7 (mod 8).
- -1 is a quadratic residue modulo [math]\displaystyle{ p }[/math] if and only if [math]\displaystyle{ p }[/math] is congruent to 1 (mod 4).
