On the decomposition of rational functions

Let f:=p/q∈K(x) be a rational function in one variable. By Lüroth’s theorem, the collection of intermediate fields K(f)⊊L⊊K(x) is in bijection with inequivalent proper decompositions f=g∘h, with g,h∈K(x) of degrees ≥2. In [Alonso, Cesar, Gutierrez, Jaime, Recio, Tomas, 1995. A rational function decomposition algorithm by near-separated polynomials. J. Symbolic Comput. 19, 527–544] an algorithm is presented to calculate such a function decomposition. In this paper we describe a simplification of this algorithm, avoiding expensive solutions of linear equations. A MAGMA implementation shows the efficiency of our method. We also prove some indecomposability criteria for rational functions, which were motivated by computational experiments.