When is $-2$ a quadratic residue in a finite field?

Question

If $F$ is a finite field of order $q=p^{\alpha}$ , where $q$ is an odd prime power, then when is $-2$ a quadratic residue in $F$?

Answer 1

This is basically a combination of quadratic reciprocity and the uniqueness of finite fields of a given order.

For the QR part, you just compute

$$\left({-2\over p}\right)=\left({-1\over p}\right)\left({2\over p}\right) = (-1)^{p-1\over2}(-1)^{p^2-1\over 8}$$

If this is $1$, then all $p^\alpha$ has $-2$ as a residue for all $\alpha$, if it is $-1$, then all fields of order $p^{2\alpha}$ do, because the finite field of order $p^2$ always does--it contains all roots of quadratics by uniqueness of finite fields of a given orde.