Let K be a perfect field of characteristic p >= 0. Consider the problem of classifying all non-constant rational functions f(x) in K(x)\K(x^p) for which the normal closure of K(x)/K(f(x)) has genus 0 or 1. In this talk we will demonstrate how this problem arises naturally in the study of certain problems in number theory. We will show how one can obtain a complete classification in a special case of K = Q, and then we will apply our results to resolve said problems in number theory. This is joint work with M. Satriano and M. Zieve.