Preface The aim of this book is to present the Galois theory of homogeneous lin- ear differential equations. This theory goes back to the work of Picard and Vessiot at the end of the 19th century and bears their names. It paral- lels the Galois theory of algebraic equations. The notions of splitting field, Galois group, and solvability by radicals have their counterparts in the no- tions of Picard-Vessiot extension, differential Galois group, and solvability by quadratures. The differential Galois group of a homogeneous linear dif- ferential equation has a structure of linear algebraic group hence it is en- dowed, in particular, with the Zariski topology. The fundamental theorem of Picard-Vessiot theory establishes a bijective correspondence between inter- mediate differential fields of a Picard-Vessiot extension and closed subgroups of its differential Galois group. Solvability by quadratures is characterized by means of the differential Galois group. Picard-Vessiot theory was clari- fied and generalized in the work of Kolchin in the mid-20th century. Kolchin used the differential algebra developed by Ritt and also built the foundations of the theory of linear algebraic groups. Kaplansky’s book “Introduction to Differential Algebra” made the theory more accessible, although it omits an important point, namely the construction of the Picard-Vessiot extension. The more recent books by Magid and van der Put and Singer assume that the reader is familiar with algebraic varieties and linear algebraic groups, although the latter book compiles the most important topics in an appen- dix. We point out that not all results on algebraic varieties and algebraic groups needed to develop differential Galois theory appear in the standard books on these topics. For our book we have decided to develop the theory of algebraic varieties and linear algebraic groups in the same way that books on classical Galois theory include some chapters on group, ring, and field xi
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