What polynomial in $\mathbb{F}_{p^m}[x]$ has splitting field $\mathbb{F}_{p^n}$?
Suppose $m \mid n$. I know that the extension $\mathbb{F}_{p^n} / \mathbb{F}_{p}$ can be obtained as the splitting field of $x^{p^n} -x$. Similarly for $\mathbb{F}_{p^m} / \mathbb{F}_{p}$. Am I right in thinking that there should be some polynomial $f(x) \in \mathbb{F}_{p^m}[x]$, such that it's splitting field is $\mathbb{F}_{p^n}$? Is $f(x) = (x^{p^n} -x)/(x^{p^m} -x)$? Does this polynomial really have coefficients in $\mathbb{F}_{p^m}$? (Because it seems like it has coefficients in $\mathbb{F}_{p}$.)
Thank you!