TITLE: Sums of roots of unity
QUESTION [41 upvotes]: 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?

REPLY [8 votes]: Much the same problem was just posted on mathoverflow (https://mathoverflow.net/questions/75753), and before David Speyer made the connection with this Stackexchange problem I posted an answer using the Dirichlet unit theorem instead of Kronecker's characterization of roots of unity.  Like the approach David used, this one generalizes to an arbitrary "CM field" (i.e. a totally imaginary quadratic extension of a totally real number field, which by Dirichlet is the only way that a nontrivial extension of number fields can have the same unit-group rank).