I know that, in general, for field extensions $F(\alpha)$ and $F(\beta)$ of a field $F$ with the minimal polynomial of $\alpha$ over $F$ equal to that of $\beta$, it is not necessarily true that $F(\alpha)=F(\beta)$.
Now I was wondering, are there certain extra conditions on $F$ for which this statement would hold?
More specifically, is this, for example, true if $F=\mathbb{Q}$ or does the statement depend on $\alpha$ and $\beta$?