I understand the proof that my professor gave in class for the most part, but there's one snag that I'm having. Here's the proof to begin with [Side note: $N(x)=a^2+b^2$, where $a+bi$ is a complex number]:
Assume not, then $p =rs $ over $\mathbb{Z}[i] $, where $r, s $ are not units.
$N(p) = N(r)N(s) \rightarrow p^2 = N(r)N(s) \rightarrow N(r) = N(s) = p $
If $r = a + bi $, then $N(r) = p \rightarrow a^2 + b^2 = p \rightarrow a^2 + b^2 \equiv 3 (\bmod{4}) $
My question is this (hopefully it isnt a silly): How does the conclusion imply (or show) that p must be a gaussian prime? Thank you for reading this and any help at all would be appreciated.