I was led by this question to the following problem:
Find $n$ complex numbers $\lambda_1\dots\lambda_n\in\mathbb{C}$ that satisfy
$\begin{align} \sum_i\lambda_i & =0\\ \sum_i\lambda_i^2 & =0\\ \sum_i\lambda_i^3 & =0\\ &\vdots\\ \sum_i\lambda_i^{n-1} & =0\\ \end{align}$
I'm pretty sure that the only solution for these equations is $\lambda_k = w \zeta^k$ where $w$ is some complex number and $\zeta$ is the primitive $n^{\text{th}}$ root of unity. By "only" I mean of course that you can permute the $\lambda_i$'s because the equations are all symmetric polynomials, but that's it.
Is there a proof for this?