I've read the following fact on my number theory textbook, there's no proof on the book of such result, I tried working it out on my own but I'm kinda lost, the lemma is the following:
Given two integers $a,b$ and a prime of the form $p=4q+3$, $p|a^2+b^2$ $\iff p|a$ and $p|b$
I would post my work but I doubt it would be of any help since I don't feel I got anywhere, I tried analyzing $x^2+y^2\equiv0 \pmod p$ since $x^2\equiv-y^2$ looked somewhat promising to me, the only thing I noticed is that $x^2 \neq-1\pmod p$ since $p\equiv3\pmod4$