If ζ is a primitive nth root of unity and d divides n, then ζᵈ is a primitive (n/d)th root of unity

Let \zeta be a primitive nth root of unity and let d|n. Prove that \zeta^d is a primitive n/dth root of unity.


Certainly (\zeta^d)^{n/d} = 1. Now if (\zeta^d)^t = 1, we have n|dt, and so n/d divides t. So n/d is the order of \zeta^d, and thus \zeta^d is a primitive n/dth root of unity.

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