Extensions of local fields and truncated power series

Let K be a finite tamely ramified extension of Qp and let L/K be a totally ramified (Z/pnZ)-extension. Let πL be a uniformizer for L, let σ be a generator for Gal(L/K), and let f(X) be an element of OK[X] such that σ(πL)=f(πL). We show that the reduction of f(X) modulo the maximal ideal of OK 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.