Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients

Abstract

We show under a mild hypothesis that given field elements $a_0, \dots, a_m \in K$, there always exists a degree-$m$ polynomial whose $n$th power whose degree-$jn$ coefficient is equal to $a_j$ for $0 \leq j \leq m$. We provide an alternate proof for the $n = 2$ case which is more constructive.