Let $\alpha = \sqrt[4]{2}$.
My question is: ${\rm Polmin}(\alpha,\mathbb{Q}(i))=X^4-2$? Because: $${\rm Polmin}(\alpha,\mathbb{Q}(i)) \mid {\rm Polmin}(\alpha,\mathbb{Q})=X^4-2.$$
But $[\mathbb{Q}(i)(\alpha):\mathbb{Q}(i)]=4$, so, ${\rm Polmin}(\alpha,\mathbb{Q}(i))=X^4-2$. Is this correct?
Thank you all.