CHAPTER-BY-CHAPTER SUMMAR Y xii i

Sophus Lie , Influence de Galois sur le developpement des mathematiques, Gesam .

Abh., Volum e 6 (quote d b y Ut a Mertzbac h i n Lie's Galois theory of differential

equations I: Historical background (preliminary report); Special Session on History of

Mathematics, Amer . Math . Soc . meetin g numbe r 882 , DeKalb, IL , Ma y 22 , 1993).

For the reader who is willing to project a lot into this brief assertion o f Lie's, we

may say that he is asserting that the space V = L~ l (0 ) of solutions of the differentia l

equation L = 0 is a homogeneous space for a finite (dimensional ) Li e group G. Whil e

this is obviously false if taken literally, if we take Lie's remark about general solution s

to mea n tha t V contain s a Zarisk i dens e subse t whic h i s a principa l homogeneou s

space fo r G, the n th e Picard-Vessio t extensio n E fo r L ove r th e fiel d F o f rationa l

functions will be isomorphic to F(G). The n Lie's comment about the striking analogy

of this theory with that o f Galois could mean tha t th e correspondence theory for th e

differential subfield s o f th e extensio n E = F(G) D F wil l b e a consequenc e o f thi s

isomorphism, whic h i s also th e mai n ide a o f th e presen t work .

Chapter-by-chapter summar y

Chapter 1: Differentia l Ideal s

Topics include general introductio n t o differentia l rings , differentia l polynomia l

algebra, characterizatio n o f ideal s differentially generate d b y a linea r homogeneou s

differential operator , an d th e fac t tha t th e quotien t o f th e differentia l polynomia l

algebra b y suc h a n idea l i s an (ordinary ) polynomia l ring . Theore m 1 is proven i n

this chapter .

Chapter 2 : Th e Wronskia n

Topics covere d ar e th e propertie s o f th e Wronskian . Th e propositio n followin g

Theorem 1 and it s corollary ar e prove n i n thi s chapter .

Chapter 3 : Picard-Vessio t Extension s

Topics covered are the definition o f Picard-Vessiot extensions, their construction,

and thei r uniqueness . Theorem s 2 , 3, and 4 are prove n i n thi s chapter .

Chapter 4 : Automorphism s o f Picard-Vessio t Extension s

Topics covere d ar e th e structur e o f th e grou p o f automorphism s o f a Picard -

Vessiot extensio n a s a n algebrai c group . Theore m 5 is proven i n thi s chapter .

Chapter 5 : Th e Structur e o f Picard-Vessio t Extension s

Topics covered include the structure of a Picard-Vessiot extension as the quotient

field o f a n affin e domain . Theore m 6 is proven i n thi s chapter .

Chapter 6 : Th e Galoi s Correspondenc e an d it s Consequence s

Topics covere d includ e th e fundamenta l theore m o f differentia l Galoi s theor y

and som e applications , includin g equation s wit h solvabl e (connecte d componen t o f

their) Galois group and equations solvable by quadratures, and equations with Galoi s

group SL n. Th e corollar y t o Theore m 6 is proven i n thi s chapter .

Chapter 7 : Th e Invers e Galoi s Proble m

Topics covere d includ e th e invers e proble m an d derivation s o f th e coordinat e

ring o f a n algebrai c group , an d th e constructiv e solutio n o f th e invers e problem fo r

various groups , includin g solvabl e groups .