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.

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