Let $ a_1, a_2, a_3, \ldots , a_n $ be complex number satisfying $ \displaystyle \sum_{j=1}^n a_j ^k= k $ where $ k =2,3,\ldots, n+1 $.
Prove (or disprove) that the least degree polynomial with integer coefficients that has a root $\displaystyle \sum_{j=1}^n a_j $ is $ n+1 $.
I've made up this question. I have shown that it is true for $n=1,2,3,4$. I'm not sure whether it holds for all positive integer $n$.
