Contents
Preface vi i
Outline o f approac h vii i
Chapter-by-chapter summar y xii i
CHAPTER
1. Differentia l Ideal s 1
Basic conventions 1
Linear ideal s 3
Examples 6
Constants 10
CHAPTER 2 . Th e Wronskia n 15
Definition o f th e Wronskia n 15
Linear independenc e ove r constant s 17
The genera l equatio n o f orde r n 2 0
CHAPTER
3 . Picard-Vessio t Extension s 2 3
Definition o f Picard-Vessio t extension s 2 3
Existence an d uniquenes s 2 5
Characterization o f Picard-Vessio t extension s 2 7
Examples 3 0
Infinite extension s 3 8
CHAPTER
4 . Automorphism s o f Picard-Vessio t Extension s 4 3
Automorphisms o f ful l universa l solutio n algebra s 4 4
Algebraic grou p structure s o n automorphism s 4 8
Examples 5 3
CHAPTER
5 . Th e Structur e o f Picard-Vessio t Extension s 6 1
Examples 7 0
Reduction t o a n algebraicall y close d bas e 7 3
CHAPTER
6 . Th e Galoi s Correspondenc e an d it s Consequence s 7 5
The fundamenta l theore m 7 5
Virtually solvabl e extension s 7 8
Integration i n finite term s 8
Abelian extension s 8 3
Examples 8 4
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