## 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 $n$th root of unity and let $d|n$. Prove that $\zeta^d$ is a primitive $n/d$th 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 $n/d$th root of unity.