If the integral linear combination of some $n$th roots of unity has magnitude 1, does this necessarily imply that this linear combination is some root of unity as well? More precisely,
Let $\zeta_1, \ldots \zeta_k$ be $n$th roots of unity. If $$|\sum_{i=1}^k n_i \zeta_i| = 1,$$ where $n_i \in \mathbb{Z}$, does this imply that $\sum_{i=1}^k n_i \zeta_i$ is an $n$th root of unity? What about if the $n_i$ are Gaussian integers?