Extensions of local fields and truncated power series

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