I need to show that the splitting field for a polynomial of the form $f(x) = x^p - 1$, where $p$ is prime and the coefficients of $f$ are in $\mathbb{Q}$, can be expressed as $\mathbb{Q}(\gamma)$, where $\gamma \in \mathbb{C}$. I have to do this by computing the root $\gamma$ explicitly.
I know intuitively that this will have to be the primitive p-th root of unity, i.e. $\gamma = e^{\frac{2\pi i}{n}}$, but I'm not sure how to formally justify this or how to derive this root. I think it will have something to do with the p-th cyclotomic polynomial $\Phi_p(x) = \frac{x^p - 1}{x - 1} = x^{p-1}+x^{p-2}+...+x+1$.
All you need to do from here is show that all of the roots of $x^p-1$ lie within $\Bbb Q(\gamma) = \Bbb Q(e^{2\pi i/n})$. This is done by noting that the roots are given by $\gamma, \gamma^2, \ldots, \gamma^{p-1}, \gamma^p = 1$.