viii Contents §3.7. Quotients 67 Exercises 72 Chapter 4. Lie Algebras and Algebraic Groups 75 §4.1. Lie algebras 75 §4.2. The Lie algebra of a linear algebraic group 79 §4.3. Decomposition of algebraic groups 86 §4.4. Solvable algebraic groups 91 §4.5. Correspondence between algebraic groups and Lie algebras 95 §4.6. Subgroups of SL(2, C) 104 Exercises 115 Part 3. Differential Galois Theory Chapter 5. Picard-Vessiot Extensions 121 §5.1. Derivations 121 §5.2. Differential rings 122 §5.3. Differential extensions 124 §5.4. The ring of differential operators 125 §5.5. Homogeneous linear differential equations 126 §5.6. The Picard-Vessiot extension 127 Exercises 133 Chapter 6. The Galois Correspondence 141 §6.1. Differential Galois group 141 §6.2. The differential Galois group as a linear algebraic group 145 §6.3. The fundamental theorem of differential Galois theory 151 §6.4. Liouville extensions 158 §6.5. Generalized Liouville extensions 160 Exercises 162 Chapter 7. Differential Equations over C(z) 165 §7.1. Fuchsian differential equations 165 §7.2. Monodromy group 173 §7.3. Kovacic’s algorithm 176 Exercises 211

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