Let $K$ be a number field, let $a \in K$, and let $n > 0$ be an integer. If $\alpha^n = a$, for $\alpha$ in some algebraic closure of $K$, then the simple radical extension $K(\alpha) / K$ may or may not be normal. For example, $K = \mathbb{Q}$, $a = 2$, $n = 3$, $\alpha = \sqrt[3] 2$, gives the non-normal extension $\mathbb{Q}(\sqrt[3] 2) / \mathbb{Q}$, while $K = \mathbb{Q}$, $a = 1$, $\alpha = \zeta_n$ (primitive $n$th root of $1$) gives the normal extension $\mathbb{Q}(\zeta_n) / \mathbb{Q}$.
My question is if there is some criterion to establish when $K(\alpha) / K$ is normal and when it is not. (Of course, I mean a criterion easier than finding the minimal polynomial of $\alpha$ over $K$ and checking if all its roots belong to $K(\alpha)$.)