Going over a past exam in my elementary number theory course, I noticed this question that caught my attention. The question asked for the conditions that allowed $-3$ to be a quadratic residue mod $p$. Doing some experimentation, I found that this was possible when $p \equiv 1 \pmod 3$. So I guess I have answered part of the question. But the proof is obviously nagging me:
Prove $-3$ is a quadratic residue in $\Bbb Z_p$ if and only if $p \equiv 1\pmod 3$.
I've done a bit of work on this, but haven't been able to come up with anything close to elegant nor conclusive. Any help would be appreciated.