= Gauss sum of a finite-field character
{c}
{title2=$G(\chi)=\sum_{x\ne0}\chi(x)\psi(x)$}
= Gauss sums of finite-field characters
{c}
{synonym}
Here $\psi$ is a nontrivial additive <character> of the <finite field>. For nontrivial $\chi$, substituting $x=ty$ in the squared absolute value gives
$$
|G(\chi)|^2=\sum_{t\ne0}\chi(t)\sum_{y\ne0}\psi((t-1)y)=q.
$$
The inner sum is $q-1$ at $t=1$ and $-1$ otherwise; multiplicative <character> <orthogonality> completes the identity.
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