One might ask the following natural question: Can the automorphism group of an infinite field K be so large that its action on the elements of K has only finitely many orbits? In a 2005 paper, Kiran S. Kedlaya and Bjorn Poonen show that no, a field K with finitely many Aut(K)-orbits is itself finite. Their proof utilizes a peculiar "trace map" construction for such fields. Note that since each element of the prime field of K forms its own orbit, this question is easily settled for characteristic zero fields. In this case, the authors formulate and make progress on a (conjectural) relative version of this question. In my talk I'll discuss their construction and their results on the relative version, give some examples (and counterexamples), and present the proof of their theorem.