Describe the Galois group of the polynomial $x^5 -3 ∈ \mathbb{Q}(\zeta)[x]$ over $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive fifth root of unity.
Here is what I attempted:
Since the splitting field of $x^5-3$ over $\mathbb{Q}$ is $\mathbb{Q}(\zeta, \sqrt[5]{3})$ and $[\mathbb{Q}(\zeta, \sqrt[5]{3}):\mathbb{Q}]=20$ and $[\mathbb{Q}(\zeta):\mathbb{Q}]=4$ , it follows that $[\mathbb{Q}(\zeta, \sqrt[5]{3}):\mathbb{Q}(\zeta)]=5$.
And it is separable over $\mathbb{Q}(\zeta)$, so the galois group of $x^5-3$ over $\mathbb{Q}(\zeta)$ is isomorphic to a group of order $5$,
which has to be a cyclic group $C_5$.
I'd like to know whether there is a gap in the argument.
And any comment will be appreciated.