###### Theorem (Law of Quadratic Reciprocity).

Let $p$ and $q$ be two distinct odd primes. Then:

 $\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{(p-1)(q-1)/4}$

where $\left(\frac{\cdot}{\cdot}\right)$ 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 $p,q$ be distinct odd primes. Then:

1. 1.

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

2. 2.

$\displaystyle\left(\frac{p}{q}\right)=-\left(\frac{q}{p}\right)$ if both $p$ and $q$ are congruent to $3$ modulo $4$.

Title quadratic reciprocity rule QuadraticReciprocityRule 2013-03-22 11:42:27 2013-03-22 11:42:27 alozano (2414) alozano (2414) 33 alozano (2414) Theorem msc 11A15 quadratic reciprocity EulersCriterion CubicReciprocityLaw QuadraticReciprocityForPolynomials LegendreSymbol