Let K be a number field or a function field of characteristic 0, and let f \in K(x) and a \in K. I study the tower of extensions given by the preimage fields K_n := K(f^{-n}(a)). I prove that an analogue of ZsigmondyA's theorem holds for ramified primes in this tower; for number fields, this requires assuming the abc conjecture. This result has applications to the Galois theory of these extensions, and in some cases I show that the groups Gal(K_n/K) are as large as possible. This is joint work with Tom Tucker.