Definition

If $p$ is an odd prime, and $a\in\mathbb{Z}$, we define the Legendre Symbol, written as $\left( \frac{a}{p} \right)$ by $$\left( \frac{a}{p} \right)=\begin{cases} 0&\text{if }p|a \\ 1&\text{if }a\text{ is a quadratic residue} \\ -1 & \text{if }a\text{ is a non-residue} \end{cases}$$ There should be no confusion with fractions here. The symbol represents $a$ over $p$.

Remark

Note this equivalence $$a\text{ is a quadratic residue}\iff x^{2}\equiv a\pmod{p}\text{ is solvable}$$