Legendre symbol LegendreSymbol 2001-10-08 17:21:40 2008-01-25 16:50:41 Definition Legendre Character Jacobi \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \textbf{Legendre Symbol.}\\ Let $p$ be an odd prime. The \emph{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.