1. Introduction ................................................. 5
2. Differential rings ........................................... 7
2.1. Derivations ............................................. 7
2.2. Differential rings ...................................... 8
2.3. Differential extensions ................................ 10
2.4. The ring of differential operators ..................... 11
3. Picard-Vessiot extensions ................................... 13
3.1. Homogeneous linear differential equations .............. 13
3.2. Existence and uniqueness of the Picard-Vessiot
extension .............................................. 14
4. Differential Galois group ................................... 21
4.1. Examples ............................................... 21
4.2. The differential Galois group as a linear algebraic
group .................................................. 23
5. Fundamental theorem ......................................... 30
6. Liouville extensions ........................................ 39
6.1. Liouville extensions ................................... 39
6.2. Generalized Liouville extensions ....................... 40
7. Appendix on algebraic varieties ............................. 42
7.1. Affine varieties ....................................... 42
7.2. Abstract affine varieties .............................. 47
7.3. Auxiliary results ...................................... 49
8. Appendix on algebraic groups ................................ 52
8.1. The notion of algebraic group .......................... 52
8.2. Connected algebraic groups ............................. 53
8.3. Subgroups and morphisms ................................ 55
8.4. Linearization of affine algebraic groups ............... 57
8.5. Homogeneous spaces ..................................... 59
8.6. Decomposition of algebraic groups ...................... 60
8.7. Solvable algebraic groups .............................. 63
8.8. Characters and semi-invariants ......................... 67
8.9. Quotients .............................................. 68
9. Suggestions for further reading ............................. 72
10. Application to Integrability of Hamiltonian Systems Appendix
by Juan J. Morales-Ruiz .................................... 74
10.1. General non-intergrability theorems .................. 74
10.2. Hypergeometric Equation .............................. 78
10.3. Non-integability of Homogeneous Potentials ........... 79
10.4. Suggestions for further reading ...................... 83
11. Bibliography ............................................... 87
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