quadratic reciprocity rule
Theorem (Law of Quadratic Reciprocity).
Let $p$ and $q$ be two distinct odd primes. Then:
$$\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)={(1)}^{(p1)(q1)/4}$$ 
where $\mathrm{\left(}\frac{\mathrm{\cdot}}{\mathrm{\cdot}}\mathrm{\right)}$ is the Jacobi (http://planetmath.org/JacobiSymbol) symbol (or Legendre symbol^{}).
The following is an equivalent^{} formulation of the Law of Quadratic Reciprocity:
Theorem (Quadratic Reciprocity (second form)).
Let $p\mathrm{,}q$ be distinct odd primes. Then:

1.
$\left({\displaystyle \frac{p}{q}}\right)=\left({\displaystyle \frac{q}{p}}\right)$ if one of $p$ or $q$ is congruent^{} to $1$ modulo $4$;

2.
$\left({\displaystyle \frac{p}{q}}\right)=\left({\displaystyle \frac{q}{p}}\right)$ if both $p$ and $q$ are congruent to $3$ modulo $4$.
Title  quadratic reciprocity rule 

Canonical name  QuadraticReciprocityRule 
Date of creation  20130322 11:42:27 
Last modified on  20130322 11:42:27 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  33 
Author  alozano (2414) 
Entry type  Theorem 
Classification  msc 11A15 
Synonym  quadratic reciprocity 
Related topic  EulersCriterion 
Related topic  CubicReciprocityLaw 
Related topic  QuadraticReciprocityForPolynomials 
Related topic  LegendreSymbol 