If $E/F$ is a field extension of characteristic 0 with $E=F[\alpha]$ with $\alpha \notin F$, $\alpha^p \in F$. Here $p$ is a prime number. Let $E^* = E[\epsilon]$ where $\epsilon$ is a primitive $p$-th root of unity. We need to show that $E^*$ is Galois over $F$, and if $E$ is also Galois over $F$, then $E=E^*$.
I can show that $E^*$ is Galois over $F$. I am stuck on the second question. Here since $\alpha$ is a root of $x^p-q$ for some $q \in F$, $\alpha$ should be in the form of $q^{\frac{1}{p}}\epsilon^k$ for some $k \le p$. Also I know that $[E^*:F]=[E^*:E][E:F]$, so if $[E^*:E]=1$ or $Gal(E^*/E)=\{id\}$, we are done. But I don't know how to compute $[E:F]$, or in other words, the degree of the minimal polynomial of $\alpha$. Field $F$ may already contain some roots of $x^p-q$, so the minimal polynomial may not be $x^p-q$. And the same problem for $[E^*:F]$. I think the main problem is how to know the minimal polynomial for $\alpha$. Any hint will be helpful.
