# Definition:Legendre Symbol

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## Definition

Let $p$ be an odd prime.

Let $a \in \Z$ be an integer.

### Definition 1

The **Legendre symbol** $\paren {\dfrac a p}$ is defined as:

- $\paren {\dfrac a p} := \begin{cases} +1 & : a^{\frac {\paren {p - 1} } 2} \bmod p = 1 \\ 0 & : a^{\frac {\paren {p - 1} } 2} \bmod p = 0 \\ -1 & : a^{\frac {\paren {p - 1} } 2} \bmod p = p - 1 \end{cases}$

### Definition 2

The **Legendre symbol** $\paren {\dfrac a p}$ is defined as:

\(\ds 0 \) | if $a \equiv 0 \pmod p$ | ||||||||

\(\ds +1 \) | if $a$ is a quadratic residue of $p$ | ||||||||

\(\ds -1 \) | if $a$ is a quadratic non-residue of $p$ |

## Also presented as

For a given $p$, the **Legendre symbol** can be treated as a mapping:

- $f_p: \Z \to \set {-1, 0, 1}$

## Also known as

The **Legendre symbol** for fixed prime $p$ is also known as the **quadratic character** modulo $p$.

## Also see

## Source of Name

This entry was named for Adrien-Marie Legendre.

## Historical Note

The **Legendre symbol** was introduced by Adrien-Marie Legendre 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Legendre symbol** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Legendre symbol**