Extensions of local fields and truncated power series

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.