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Let $0＜ \rho＜1$ be an irrational and there exists a positive integer $n$ such that $\rho^n \in \mathbb{Q}$. Let $r$ be the smallest positive integer such that $u=\rho^r\in \mathbb{Q}$.

Question: can we show that $f(x)=x^r-u$ is the minimal polynomial of $\rho$ in $\mathbb{Q} [x]$?

wzstrong
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    I would use this as a starting point. – Jyrki Lahtonen Jan 03 '20 at 07:35
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    Unless my bad night of sleep dulled me severely, it seems to me that the cited result answers your question in the affirmative. $u$ won't be a $p$th power of a rational for any prime factor $p$ of $r$ for then $\rho^{r/p}$ would also be rational. And because $u>0$ it won't be $-4$ times a fourth power either. – Jyrki Lahtonen Jan 03 '20 at 07:46

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