Let $\alpha$ be an algebraic number. Obviously, one can find $d\in\mathbb N$ and $\beta$ an algebraic integer such that $\alpha=\frac{\beta}d$. But can one choose $d$ and $\beta$ such that for every maximal ideal $\mathfrak M$ of $O_\alpha$, the ring of intgers of $\mathbb Q(\alpha)$, such that $v_{\mathfrak M}(d)>0$, one has $v_{\mathfrak M}(\beta)=0$?
Thanks in advance for any answer.