# Legendre symbol

Let $p$ be an odd prime. The Legendre symbol $\left(\frac{a}{p}\right)$ or $(a|p)$ is defined as:

 $\left(\frac{a}{p}\right)=\begin{cases}1&\text{if }a\text{ is a quadratic % residue }\pmod{p}\\ -1&\text{if }a\text{ is a quadratic nonresidue }\pmod{p}\\ 0&\text{if }p\text{ divides }a\end{cases}$

The Legendre symbol can be computed by means of Euler’s criterion or Gauss’ lemma.

Generalizations of this symbol are the Jacobi Symbol and the Kronecker symbol.

 Title Legendre symbol Canonical name LegendreSymbol Date of creation 2013-03-22 11:44:46 Last modified on 2013-03-22 11:44:46 Owner alozano (2414) Last modified by alozano (2414) Numerical id 15 Author alozano (2414) Entry type Definition Classification msc 11-00 Classification msc 97U20 Related topic JacobiSymbol Related topic EulersCriterion Related topic QuadraticResidue Related topic KroneckerSymbol Related topic QuadraticReciprocityRule Related topic QuadraticCongruence