Context: Let $\omega = e^{\frac{2\pi i}{p}}$ for some prime $p$. Kummer’s Lemma states that if $u$ is a unit in $\mathbb{Z}[\omega]$ and $\overline{u}$ is its complex conjugate, then $\frac{u}{\overline{u}}$ is a power of $\omega$.
I'm looking at the following proof. The author argues that there are only finitely many algebraic integers $\alpha$ of degree $\leq n$ such that $\alpha$ and all of the algebraic conjugates have absolute value $1$. Then, he makes a claim that if $\alpha$ is such an algebraic integer, then so is $\alpha ^k$ for any $k \in \mathbb{N}$, eventually yielding the desired result.
My question is why? Why is it true that if all the algebraic conjugates of $\alpha$ have absolute value $1$, then all of the algebraic conjugates of $\alpha^k$ also have absolute value $1$? The explanation on the linked site makes zero sense to me.
Any help is appreciated. Thanks!