See quadratic reciprocity

Evaluate \(J(\alpha)\) using a Gauss sum : for \(\chi: { \mathbf{F} }_q \to {\mathbf{C}}\) a multiplicative character, define \begin{align*} G(\chi) &\coloneqq\sum_{x\in { \mathbf{F} }_q} \chi(x) \psi_q(x) .\end{align*}