Determining that a certain prime integer is irreducible in the Gaussian integers Let $p\in \mathbb{Z}$ be prime such that $p \equiv 3 \bmod 4$. I've read in many places that $p$ must be irreducible in $\mathbb{Z}[i]$, but I can't see why.
Could someone please explain the reason?

This is because of an elementary fact , that $x^2 \equiv -1$ modulo p has only solutions iff $ p \equiv 1 $ modulo 4. You can see this proof in Niven Zuckermann's book of Number Theory.
