Minimal polynomial for $\sqrt[5]{2}$ over $\mathbb Q(\sqrt[]{3})$

Question

Find the minimal polynomial for $\sqrt[5]{2}$ over $\mathbb Q(\sqrt[]{3})$.

The natural candidate is $x^5-2$. But the problem is to prove it is irreducible. In theory, I can write down the roots explicitly and by squaring multiple times somehow show that no root lies in the field. But even then there may be a quadratic factor, and using the method of undetermined coefficients is way too long. Can I prove irreducibility easier?

Answer 1

$\Bbb Z[\sqrt3]$ is UFD, $a^2-3b^2=2$ has no solution (since it has no solution mod $3$), so $2$ is prime in $\Bbb Z[\sqrt3]$. Now use Eisenstein.