Local Arboreal Representations

Let K be a field complete with respect to a discrete valuation v of residue characteristic p. Let f (z) is an element of K[z] be a separable polynomial of the form z(l) - c. Given a is an element of K, we examine the Galois groups and ramification groups of the extensions of K generated by the solutions to f(n)(z) = a. The behavior depends upon v(c), and we find that it shifts dramatically as v(c) crosses a certain value: 0 in the case p inverted iota l , and - p/(p - 1) in the case p = l.