From the comments, it looks like we're solving the equation $\sum_{i=0}^{n-1} a_i \omega_n^i= 0$ where $\omega_n$ is a primitive $n$th root of unity in $\mathbb{C}$ and the $a_i$s are real (if they can be complex, the problem seems pretty trivial; I'm not sure how to do it if they can only be rational). Take $\omega_n = e^{2\pi i/n}$. This splits into two equations: $\sum_{i=0}^{n-1} a_i \cos(2\pi i/n) = 0$ $\sum_{i=0}^{n-1} a_i \sin(2\pi i/n) = 0.$
$\sin 0 = 0$, so the second sum has no $a_0$ term. This means that $a_2, a_3, \ldots, a_{n-1}$ determine $a_1$: $a_1 = -\frac{\sum_{i=2}^{n-1} a_i \sin(2\pi i/n)}{\sin(2\pi/n)}.$ Now, in the same way, $a_1, a_2, \ldots, a_{n-1}$ determine $a_0$ from the first equation: $a_0 = -\sum_{i=1}^{n-1} a_i \cos(2\pi i/n) = \frac{\sum_{i=2}^{n-1} a_i \sin(2\pi i/n)}{\sin(2\pi/n)} - \sum_{i=2}^{n-1} a_i \cos(2\pi i/n).$
This means that $a_2, a_3, \ldots, a_{n-1}$ can be picked arbitrarily, and they determine $a_0$ and $a_1$ uniquely.
(Originally I had a lot of nonsense in this post, it has since been edited)