As the summary says we have $ f(x) = x^n - \theta \in \mathbb{Q}[x] $. We will call the pth primitive root $ \omega $ and we denote $[\mathbb{Q}(\omega) : \mathbb{Q}] = j$. We want to show that the Galois group is generated by $\sigma, \tau$ such that
$$ \sigma^j = \tau^p = 1, \sigma^k\tau = \tau\sigma$$
I know that the splitting field of $ f $ is going to be $Q(t,\omega)$ and that the degree of this extension is going to be $[Q(t,\omega): :Q(\omega)][Q(\omega : Q)] $ where $ t^p = \theta $, further as minimal polynomial of $ \omega $ is going to be $ p^{th} $ cyclotomic I have the second multiple being (p-1) and I can prove that the whole extension will have degree p(p-1). Now my idea is to define the morphisms as:
$$\sigma(t) = t, \sigma(\omega) = \omega^2$$ and $$\tau(t) = t\omega, \tau(\omega) = \omega$$
I can show that order of these two groups are p-1 and p but I don't know how to show that they generate my group.
I suspect that I am meant to construct the group as a semidirect product of $<\tau>$ and $<\omega>$ but I can't figure it out completely.
