  Let $K$ be a finite tamely ramified extension of $\Q_p$ and let $L/K$ be a
totally ramified $(\Z/p^n\Z)$-extension. Let $\pi_L$ be a uniformizer for $L$,
let $\sigma$ be a generator for $\Gal(L/K)$, and let $f(X)$ be an element of
$\O_K[X]$ such that $\sigma(\pi_L)=f(\pi_L)$. We show that the reduction of
$f(X)$ modulo the maximal ideal of $\O_K$ determines a certain subextension of
$L/K$ up to isomorphism. We use this result to study the field extensions
generated by periodic points of a $p$-adic dynamical system.
