I'm asked to proved that $x^n-11$ is irreducible in $\mathbb{Q}(\sqrt{-5})[x]$. I have deduced to a point where I have to prove that if $x^n-11$ is reducible in $\mathbb{Q}(\sqrt{-5})[x]$ then it is also reducible in $\mathbb{Z}[\sqrt{-5}][x]$.
Is this fact true? If it is, can someone tell me how to prove that. If not, how else should I approach the original question.
Note that $\mathbb{Z}[\sqrt{-5}][x]$ is not a Unique Factorization Domain! Thanks!