$\mathbb{Z}[\omega]$ is a UFD, $\Rightarrow x+y\omega=u\alpha^{p}$

Question

I am aware this is the same question as If $\mathbb{Z}[\omega]$ is a UFD, then $x+y\omega=u\alpha^{p}$.. But it doesn't have an answer yet (or even any comments) and I am also interested in finding the solution.