Suppose that $p$ and $q = 2p + 1$ are both odd primes. Show that the $p − 1$ primitive roots of $q$ are precisely the quadratic non-residues of $q$, other than the quadratic non-residue $2p$ of $q$.
I think I probably need to use the fact that $q$ is congruent to $3 \rm\, mod\, 4$, but I've been fiddling around with definitions and can't quite seem to get anywhere. Any help would be greatly appreciated. Thanks!