Element satisfying cubic equation in degree $5$ extension

Consider the extension $\mathbb{Q}(α) : \mathbb{Q}$, where $α^5 −α−1 = 0$. Suppose that $f (x) \in \mathbb{Q}[x]$ is of degree $3$ and has a root in $\mathbb{Q}(α)$. Prove that $f (x)$ has a root in $\mathbb{Q}$. Is there a way to show this without long calculations? (For example, using the short tower law). More generally, is it true that, for natural numbers $m, n$, with $m<n$, there is no element satisfying a degree $m$ equation in a degree $n$ extension? Is it only true when $m, n$ are coprime, and why?


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