When does adjoining a single root of an irreducible polynomial generate the splitting field?

Question

Suppose for example we were in $\mathbb{Q}$, the rationals, and wanted to find the splitting field of some polynomial $f(x) \in \mathbb{Q}[x]$ for which we know there exists an algebraic root $\alpha$. When is the extension field $\mathbb{Q}(\alpha)$ a splitting field for $f$? For example, the polynomial $x^p-1$ has algebraic solutions $\beta = e^{\frac{2\pi i}{p}}$ as well as any power less than $p-1$ of $\beta$ ($p$ prime). The field extension $\mathbb{Q}(\beta)$ is therefore a splitting field of $x^p-1$ achieved by adjoining a single root of $x^p-1$ to $\mathbb{Q}$. What conditions must hold on $f$ for this to be the case?