If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.

Question

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.

Answer 1

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.