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

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 $0\le j\le m$. We provide an alternate proof for the $n=2$ case which is more constructive.