Exercise 4, § 6

Let $K$ be a field of characteristic 0. Show that if $\alpha_i \ (1 \leq i \leq n)$ are $n$ elements of $K$ such that $\alpha_1^k + \alpha_2^k + \ldots + \alpha_n^k = 0$ for $n$ consecutive values $h, h+1, \ldots, h+n-1$ of $k$, then $a_1 = a_2 = \cdots = a_n = 0$. Show by an example that the property no longer holds if the $n$ values of $k$ are not consecutive or if $K$ is not of characteristic 0. \*