Square of Quadratic Gauss Sum

Theorem
Let $p$ be an odd prime.

Let $a$ be an integer coprime to $p$.

Let $g \left({a, p}\right)$ denote the quadratic Gauss sum of $a$ and $p$.

Then:
 * ${g \left({a, p}\right)}^2 = \left({\dfrac {-1} p}\right) \cdot p$