Let $F$ be a field of characteristic $0$, and $K$ a finite extension of $F$. Then it is well known (see this) that $K$ can be obtained by attaching an element $\alpha \in K$ to $F$, i.e. $K=F(\alpha)$.
Question: Is it always possible to choose $\alpha$ such that $\alpha^m\in K$ for some $m\in \mathbb{N}$?