Introduction This book has been conceived as a self-contained introduction to differential Galois theory. The self-teaching reader or the teacher wanting to give a course on this subject will find complete proofs of all results included. We have chosen to make a classical presentation of the theory. We refer to the Picard-Vessiot extension as a field rather than introducing the notion of Picard-Vessiot ring so as to keep the analogy with the splitting field in the polynomial Galois theory. We also refer to differential equations rather than differential systems, although the differential systems setting is given in the exercises. The chapters on algebraic geometry and algebraic groups include all questions which are necessary to develop differential Galois theory. The differential Galois group of a linear differential equation is a linear algebraic group, hence affine. However, the construction of the quotient of an algebraic group by a subgroup needs the notion of abstract affine variety. Once we introduce the notion of geometric space, the concept of algebraic variety comes naturally. We also consider it interesting to include the notion of projective variety, which is a model for algebraic varieties, and present a classical example of an algebraic group which is not affine, namely the elliptic curve. The chapter on Lie algebras aims to prove the equivalence between the solvability of a connected linear algebraic group and the solvability of its Lie algebra. This fact is used in particular to determine the algebraic subgroups of SL(2, C). We present the characterization of differential equations solv- able by quadratures. In the last chapter we consider differential equations defined over the field of rational functions over the complex field and present xiii
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