Let $p$ be a prime and $a > 1$ be a squarefree positive integer. We wish to understand (at least to some extent) the Galois correspondence in the Galois group $G$ of $x^p - a$. The splitting field is $\mathbb{Q}(\omega, \sqrt[p]{a})$ where $\omega = e^{\frac{2\pi i}{p}}$ - one can easily show it is of degree $p(p-1)$. I can see that the Galois group is non-Abelian and generated by $\sigma$ (fix $\omega$, send $\sqrt[p]{a}$ to $\omega \sqrt[p]{a}$) and $\theta$ (fix $\sqrt[p]{a}$, send $\omega$ to $\omega^g$ where $g$ is a primitive root mod $p$), of orders $p$ and $p-1$. I guess with some effort one can find suitable relations like $\theta^{-1}\sigma\theta = .... $.
What I am particularly interested in - is there a nice way to further the computation and understand which are the normal subgroups of $G$ and how many are they in number?