\begin{definition}[Definition:Legendre Symbol/Definition 1]
Let $p$ be an odd prime.
Let $a \in \Z$.
The '''Legendre symbol''' $\paren {\dfrac a p}$ is defined as:
:$\paren {\dfrac a p} := \begin{cases} +1 & : a^{\frac {\paren {p - 1} } 2} \bmod p = 1 \\
0 & : a^{\frac {\paren {p - 1} } 2} \bmod p = 0 \\
-1 & : a^{\frac {\paren {p - 1} } 2} \bmod p = p - 1
\end{cases}$
where $x \bmod y$ denotes the modulo operation.
\end{definition}