Let $K/k$ be Galois and $\zeta$ be primitive $n$-th root of unity. I want to prove that $K(\zeta)/k(\zeta)$ is also Galois.
Proof: Recall that an extension is Galois iff it is a splitting field of separable polynomial. If $K$ is the splitting field of $f(x) \in k[x]$ then $K(\zeta)$ is the splitting field of $\frac{f(x)}{ \operatorname{gcd}(f(x), \Phi_n(x))} \in k(\zeta)$ and that's it.
Am I right? If I am wrong, please, don't right your solution.