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# Def/Legendre symbol

Definition of Legendre Symbol: Suppose that $p$ is a prime number. Suppose that $a$ is an integer. The Legendre symbol, denoted $\left( {a \over p} \right)$, depends on the residue $\bar a$, mod $p$, and is defined by: $$\left( {a \over p} \right) = \bar a^{(p-1)/2} \mbox{ mod } p.$$

By Euler's criterion, the Legendre symbol $\left( {a \over p} \right)$ has the following properties:

• $\left( {a \over p} \right) = 1$ if $a$ is a square, mod $p$, and $a$ is not a multiple of $p$.
• $\left( {a \over p} \right) = -1$ if $a$ is not a square, mod $p$.
• $\left( {a \over p} \right) = 0$ if $a$ is a multiple of $p$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Prime number, Def/Residue, State/Eulers criterion

The following statements and definitions logically rely on the material of this page: State/Quadratic reciprocity, and State/Zolotarevs lemma

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Theory of quadratic residues