Legendre symbol

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