Here is the question. $R$ is a UFD and $F$=Frac$R$, $E=F(α)$ is an algebraic extension of $F$.
Prove of disprove that, if the minimal polynomial of $α$ over $F$ belongs to $R[x]$, then $α$ is a root of some monic polynomial in $R[x]$.
As far as I consider, I thought the statement is false. Here is my counter example.
Consider $R=\mathbb{Z}$, $F=\mathbb{Q}$ and $α=cos(π/9)$, it corresponds to the minimal polynomial $8x^3-6x-1$. I have no way to multiple a new polynomial in $\mathbb{Z}[x]$ to make it becomes a monic polynomial.
Am I on the right track?