Square of Quadratic Gauss Sum

Theorem
Let $p$ be an odd prime.

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

Let $\map g {a, p}$ denote the quadratic Gauss sum of $a$ and $p$.

Then:
 * $\map g {a, p}^2 = \paren {\dfrac {-1} p} \cdot p$