Let $u_k = \alpha_1^k + \cdots + \alpha_n^k$, where $\alpha_i \in \mathbb{C}$ and $n \ge 1$.
How do I find $\alpha_i$ given that $u_k = k$ for $k = 1, \dots, n$?
If it helps, I suspect that $\alpha_i$ are roots of $P(x) = x^n - \left( \frac{1}{1!}x^{n-1} + \frac{1}{2!}x^{n-2} + \cdots + \frac{1}{n!} \right)$.
EDIT: As noted in the comments by @Servaes, I suspected wrongly.