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Let $p$ be an odd prime and let $\omega \in \mathbb{C}$ be a $p$th root of unity.

I am investigating the solutions to $$\sum_{i=0}^{p-1} a_i\omega^i = k$$ where $k\neq 0$ and each of the $a_i$ belong to a field $K$.

It's well known that if $K=\mathbb{Z}$ or $\mathbb{Q}$, then there is a unique solution for a given fixed $k$.

Can the same be said if we instead take $K=\mathbb{R}$?

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No. The $\omega^i$ are in $\mathbb C$, which has dimension two as an $\mathbb R$ vector space. So no more than two $\omega^i$ can be linearly independent.

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  • $\begingroup$ Great, thank you! $\endgroup$ Aug 26 '21 at 4:58

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