Suppose that $X$ is an inclusion-minimal finite set of non-zero complex numbers
such that $\sum\limits_{x\in X}x^n=0\ $ for infinitely many integers $n$.

  
> 1. Can the cardinality of $X$ be a composite number?
>
> <del>2. Can $X$ be something different from $\root^p\of c$ 
(for some $c\in\mathbb C$ and prime $p$)?</del>

(<i>Inclusion-minimal</i> means that the number of $n\in\mathbb Z$ such that $\sum\limits_{x\in Y}x^n=0$ is finite for any proper subset 
$Y\subset X$.)