If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.
I suppose this would not be true if p $\equiv$ 1 (modulo 4).
To prove something is a non-square I find to be tricky. It's difficult to see any straightforward way to do this using only the definition of congruence for example.
Best Answer
Assume $$x^2\equiv-1\pmod{p}$$
$$(x^2)^{2k+1}\equiv - 1\pmod{p}$$
$$x^{4k+2}\equiv -1\pmod{p}$$
For all $k$. Now $\exists k$ such that $4k+2=p-1$
Thus,
$$x^{p-1}\equiv-1\pmod{p}$$
Which is in clear violation of fermat's little theorem.