# Prime roots of unity linearly independent over the reals

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 $$K=\mathbb{R}$$?

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.