Series vanishes for all powers $k$ implies each term vanishes

Question

Suppose we have that for $n$ fixed and $\{a_i\}$ fixed that $$ \sum_{i = 1}^n a_i^k = 0 $$ for all $k \in \mathbb{Z}^+$. Under what conditions does this imply that $a_i = 0$ for all $i$? I'd like to know what conditions on the underlying field there are (I'm suspicious that finite characteristic would mess things up) and if this holds for infinite sums.