Definition:Legendre Symbol

Definition
The Legendre symbol is a function introduced by in Paris in 1798, during his partly successful attempt to prove the Law of Quadratic Reciprocity.

The function was later expanded into the Jacobi Symbol, the Kronecker Symbol, the Hilbert Symbol and the Artin Symbol.

Legendre defined the symbol as follows:

Let $p$ be an odd prime.

Let $a \in \Z$.

Then:
 * $\left({\dfrac a p}\right) := a^{\frac{(p - 1)} 2} \pmod p$

From Values of Legendre Symbol, we have that:
 * $\left({\dfrac a p}\right) = \begin{cases}

0& : a \equiv 0 \pmod p \\ +1 & : a \not \equiv 0 \pmod p \text{ and for some integer }x, x^2 \equiv \;a \pmod p \\ -1 & : \text{if there is no such } x \end{cases}$

From Euler's Criterion, this can be seen to be equivalent to:


 * $\left({\frac a p}\right) = \begin{cases}

0 & : a \equiv 0 \pmod p \\ +1 & : a \text{ is a quadratic residue of } p \\ -1 & : a \text{ is a quadratic non-residue of } p \end{cases}$

For a given $p$, the Legendre symbol is usually treated as a function $f_p: \Z \to \left\{{-1, 0, 1}\right\}$.

Applications
The Legendre Symbol can be used as a tool to finding whether a number is a quadratic residue (mod $p$) through the following conditions:


 * $\left({\dfrac a p}\right) = 1$ $a$ is a quadratic residue mod $p$
 * $\left({\dfrac a p}\right) = -1$ $a$ is a Quadratic non-residue mod $p$.

This follows directly from the definitions of quadratic residue and quadratic non-residue, and Euler's Criterion.

Also see

 * Law of Quadratic Reciprocity
 * Properties of Legendre Symbol