I need to prove this
Prove that a Gauss integer $z = a +ib$, with $a \ne 0, b \ne 0$ is irreducible if and only if $a^2 + b^2$ is a prime element in ℤ.
I've already proved that $N(z)$ prime implies, $z$ irreducible element:
If $N(z)$ is a prime, then $z$ is irreducible since if we write it as $z=xy$, it follows that $N(z) = N(x)N(y)$, thus either $x$ or $y$ is a unit.
I'm not able to prove the converse: if z is irreducible then $N(z)$ is prime , which is no generally true, but it is for this particular case with $a \ne 0, b \ne 0$.
Note The unique factorization theorem was not still introduce so far in the course and I'd like not to resort on it.
