PGL acting on

Abstract

Let be the field of rational functions over and treat as the group of degree one rational functions in equipped with composition. acts on from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup H of is the stabilizer of some rational function with under this action, where is uniquely determined by H up to a left composition by an element of In this article, we determine the rational function explicitly for every