Given an integer $n$, a monic polynomial $f\in \mathbb{Z}[x]$ and one of its complex roots $\alpha$, I'm trying to show that the number of roots $\beta$ of $f$ such that $\alpha^n = \beta^n$ is
- independent of the choice of $\alpha$
- a factor of $\deg f$.
How can I show this? I get that $\alpha$ and $\beta$ differ by an $n$th root of unity here, but I'm not sure how to proceed.
