Local Arboreal Representations

Abstract

Let K be a field complete with respect to a discrete valuation v of residue characteristic p. Let f(z)∈K[z] be a separable polynomial of the form z[superscript ℓ]−c. Given a∈K, we examine the Galois groups and ramification groups of the extensions of K generated by the solutions to fn(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∤ℓ, and −p/(p−1) in the case p=ℓ.