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Katie asks further about the case where we have a subset $A$ of $\mathbb{Z}/n$ and asks for $\sum_{k \in A} \zeta^k$ to have integer absolute value. She writes that, for $n$ prime, the only solutions …

UPDATE: Previous argument was flawed. Here is what can salvage. I can show there is no solution with $n$ an odd prime, or with $n$ odd and $\omega$ cyclotomic. Let $\sigma$ denote complex conjuga …

$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$The main result of this answer will be that, if $V$ is a representation defined over $\RR$ and over $\QQ(\zeta_M)$, then t …

This is a vague thought: is there some simple recurrence for the characteristic polynomials of the charctertistic polynomials of the corresponding matrices. For example, if you look at the A_n chains, …

Certainly not in general: If $p \equiv 1 \bmod n$ (but $n \neq \pm 1$) then $\mathbb{Q}_p(\zeta_n) = \mathbb{Q}_p$) so $Tr(\zeta_n) = \zeta_n$ which is not in $\mathbb{Q}$. To give another example, ta …

Sorry, you are out of luck. Let $\phi_n$ denote the $n$-th cyclotomic polynomial. I will show that (1) $\phi_n$ factors completely into linear factors modulo $p$ (a prime) if and only if $n$ is of th …

The solution to the Dipohantine equation: You have, indeed, found all the solutions. Put $\alpha = e^{2 \pi i x}$, $\beta = e^{2 \pi i y}$, $\gamma = e^{2 \pi i z}$, so you want $$(\alpha+\alpha^{-1}) …