Gauss sum of a finite-field character ($G(\chi )={\sum }_{x\ne 0}\chi (x)\psi (x)$)

Here $\psi$ is a nontrivial additive character of the finite field. For nontrivial $\chi$, substituting $x=ty$ in the squared absolute value givesThe inner sum is $q-1$ at $t=1$ and $-1$ otherwise; multiplicative character orthogonality completes the identity.