## The conjugates of a root of unity are roots of unity

Let $\zeta$ be a root of unity. Show that the conjugates of $\zeta$ are also roots of unity.

If $\zeta$ is a root of 1, then $\zeta$ is a root of $p(x) = x^n - 1$ for some $n$. Thus the minimal polynomial of $\zeta$ over $\mathbb{Q}$ is also a root of $p(x)$, so the conjugates of $\zeta$ are roots of unity.