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?

2. Can $X$ be something different from $\root^p\of c$ (for some $c\in\mathbb C$ and prime $p$)?