Definition (Gauss sum)

Let $p$ and $q$ be two distinct primes. Let $\zeta$ be any primitive $q$-th root of unity. That is, $\zeta$ is a $q$-th root of unity with order $q$. The Gauss Sum, $\mathscr{G}$, is defined as $$\mathscr{G}:=\sum_{j=0}^{q-1}\left( \frac{j}{q} \right)\zeta^{j}$$