Suppose that $\pi,\sigma \in \mathbb{Z}[i] $, that $N(\pi$)=$N(\sigma)$ and that $\pi$ is irreducible. Prove that $\sigma$ is irreducible.

Question

Suppose that $\pi,\sigma \in \mathbb{Z}[i] $, that $N(\pi$)=$N(\sigma)$ and that $\pi$ is irreducible. Prove that $\sigma$ is irreducible.

I feel like this should be a straightforward question but I'm not sure about how to start. I know that if $N(\sigma)$ is irreducible in the integers then $\sigma$ is irreducible in the Gaussian integers. Pretty sure the converse is not true.

Answer 1

This follows from the fact that $z=a+bi$ is irreducible in $\Bbb Z[i]$ if and only if $N(z)=a^2+b^2$ is prime in $\Bbb Z$, where $ab\neq 0$. The case $ab=0$ is clear.

Reference: Irreducible element iff prime norm in Gaussian integers?