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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?

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    By sum, do you mean any linear combination?2011-05-18
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    @Joel: the answer would be no if that were the intended question, since plenty of elements of $\mathbb{Q}(i)$, for example, have absolute value $1$ but are not roots of unity.2011-05-18
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    @Qiaochu: I think Joel meant integer linear combinations (as opposed to only sums of distinct roots).2011-05-18
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    Can anyone explain the question my giving an example. Please2011-05-18
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    @Qiaochu, @ joriki : Indeed, I meant "integer linear combination". Thanks for pointing the omission.2011-05-18
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    Why does this question have 10 upvotes if no one even knows what it's asking? It was only asked 41 minutes before this comment and it has 10 upvotes already?2011-05-18
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    @Chandru: $\mathrm e^{2\pi\mathrm i/3}+e^{-2\pi\mathrm i/3}=-1$, that is, two cube roots of unity add up to a square root of unity.2011-05-18
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    @Numth: Because it's a one-liner that evokes the immediate intuition that the answer must be "yes" but on second thought appears not quite as trivial to prove as one might have thought. Also it seems like something that might be useful in applications.2011-05-18
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    @Joriki: OK. got it Thanks a lot.2011-05-18
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    @Joriki: Thanks for the insight. I had the same thought that @Numth did!2011-05-18
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    @Name12345: Could you edit/confirm the meaning of "sum" in the question as "integer linear combination"?2011-05-18
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    @Name12345: OK.2011-05-18
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    @Name12345: As I point out in my answer, you've changed the statement and it's now false, because the sum, though a root of unity, doesn't have to be an $n$-th root of unity. (Also there shouldn't be a magnitude around the second occurrence of the sum.)2011-05-19

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