INTRODUCTION

The aim of this article is to survey some constructive aspects of differential Galois theory and to indicate some analogies between ordinary Galois theory and differential Galois theory in characteristic zero and nonzero. We hope it may serve as an appetizer for people who work in ordinary Galois theory but are not familiar with the differential analogue. In the first part we start with a constructive foundation of the Picard-Vessiot theory in characteristic zero mimicking Kronecker's construction of root fields. This leads to a smallest differential field extension (with no new constants) containing a full system of solutions of a (system of) linear differential equation(s) with a linear algebraic group as differential Galois group. Then we explain the Galois correspondence between the intermediate differential fields of a Picard- Vessiot extension and the Zariski closed subgroups of the differential Galois group. On the way we deal with the question of solvability by elementary functions, comparable to the question of solvability by radicals in ordinary Galois theory. In Chapter 3 we describe the link between the differential Galois group and the monodromy group over the complex numbers generalizing the effective version of Riemann's existence theorem used in (ordinary) inverse Galois theory [MM]. Further we recall the solution of the inverse differential Galois problem over C in the case of monodromy groups (Riemann-Hilbert problem) given by Plemelj (1908) and its completion by Tretkoff and Tretkoff [TT] for differential Galois groups.