For those, like me, who forgot why this is true: writing $\zeta_n$ for a primitive $n$-th root of unity, $1-\zeta_n$ is a unit in $\mathbb{Z}[\zeta_n]$ when $n$ has at least two distinct prime factors (Washington, Introduction to Cyclotomic Fields (1982) Springer GTM 83, proposition 2.8), so then by Galois action $1-\zeta_n^i$ is a unit if $i$ is prime to $n$, and therefore $\zeta_n^i-\zeta_n^j$ is a unit if $i-j$ is prime to $n$.
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