quadratic reciprocity rule

Theorem (Law of Quadratic Reciprocity).

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

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

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	2013-03-22 11:42:27
Last modified on	2013-03-22 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