Are fields $Q[i\sqrt[4]{5}]$ and $Q[\sqrt[4]{5}]$ isomorphic?
I tried to prove they are not simiralry like one can show that $Q[i\sqrt{5}]$ and $Q[\sqrt{5}]$ are not isomorphic because if they were then $(f(i\sqrt{5})^2=f(-5)=-5$ but in $Q[\sqrt{5}]$ there is no element such that it's square is negative, but in my case this fourth degree root matches with powers of $i$ and I think I can't do anything similar.
They are both isomorphic to the field $$\mathbb{Q}[X]/(X^4-5).$$ So, they are isomorphic. The basic idea is that, their minimal polynomials are the same, as pointed in comment.