See quadratic reciprocity

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