I want to show the only solution to the following system of equations is the trivial one ($x_{i} = 0$). I don't know if this is true, but I think it should be.
Let $x_{i} \in \mathbb{C}$ for $1 \le i \le n$
$$x_{1}+\ldots+x_{n} = 0 \\ x_{1}^{2}+\ldots+x_{n}^{2}=0 \\ \ldots \\ x_{1}^{n}+ \ldots+x_{n}^{n}=0$$
I worked out the case for $n=2$ and $3$, with basic algebraic manipulation but I can't see a way to more systematically show $x_{i} = 0$ for $\forall i$ for an arbitrary $n$. It seems there should be a neat manipulation to make this work out, or perhaps not. Any help or suggestions would be appreciated thanks!