The differential Galois group of the rational function field

We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |C|. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about C(x) also applies to rational function fields over more general fields of coefficients.