I will present some recent work, joint with with Spencer Hamblen, Bjorn Poonen, and Laura Walton. Let f(z) = z^d-c be a polynomial in K[z], and let a be in K, where K is some (local) field. The set \cup_{n=0}^\infty f^{-n}(a) = \cup_{n=0}^\infty \{x : f^n(x) = a\} of all preimages of a under f form a d-ary rooted tree, where a is the root and each x is connected to f(x). Let K_n = K(f^{-n}(a)) be the field extension obtained by adjoining to K the nth preimages of a under f. The Galois group of K_n over K acts on the preimage tree by permuting the elements at each level in the tree in a way that preserves the structure of the tree. I will discuss how the size and structure of these fields and their Galois groups depend on the valuations of the constants c and a. No prior knowledge of dynamics, Galois theory, or local fields will be assumed, and students are encouraged to attend.