Preface Differential Galoi s theory is the theory of solutions of differential equation s over a differential bas e field, or rather the nature of the differential field extension generated by th e solutions , i n muc h th e sam e wa y tha t ordinar y Galoi s theor y i s th e theor y of fiel d extension s generate d b y solution s o f (on e variable ) polynomia l equations , with th e additiona l featur e tha t th e correspondin g differentia l Galoi s group s (o f automorphisms o f the extension fixin g th e base and commuting with the derivation ) are algebrai c groups . This book, despit e its title, deals only with the differential Galoi s theory of linear homogeneous differentia l equations , whos e differentia l Galoi s group s ar e algebrai c matrix groups. Thi s branch of the theory is known as the Picard-Vessiot theory, afte r its founders C . Picar d (1856-1941 ) an d E . Vessio t (1865-1952 ) whos e fundamenta l papers on the subject appeared in 188 3 and 1892 , respectively. (Thes e historical notes are du e t o Phylli s Cassidy. ) Thus differential Galoi s theory ha s a history datin g back t o the 19t h century I t was subsequentl y advance d b y J. F . Ritt a t Columbia . However , i t wa s onl y put o n its present soli d footing i n the postwar wor k o f Ellis Kolchin, whic h saw publicatio n beginning wit h hi s 194 8 Annals paper . Kolchin's 197 3 Differential algebra and algebraic groups, Ne w York , Academi c Press [Ko2] , comprehensivel y codifie s hi s work , althoug h hi s orgina l paper s fro m the 40 s an d 50 s remain equall y accessible . Reader s wh o fin d Kolchin' s idiosyncrati c language difficul t t o di p int o briefl y wil l appreciat e Irvin g Kaplansky' s Introduction to differential algebra, 2nd edition , Hermann , Paris , 197 6 [K], which treat s th e topi c marvelously concisel y (althoug h fo r som e reaso n Kaplansk y choose s t o omi t th e construction o f the Picard-Vessiot extensions, which play the same role in differentia l Galois theor y tha t Galoi s extension s pla y i n ordinar y Galoi s theory , referrin g th e reader t o Kolchin' s origina l paper s instead) . Ther e ar e also many othe r account s o f differential Galoi s theor y i n th e literature , fo r exampl e b y Delign e [D] , Fahim [F] , Levelt [L], Katz [Ka], Pommaret [P], and Takeuchi [T] to name some recent examples. Kolchin's book end s with a theorem tha t describe s the structur e o f a differentia l Galois extension as a twisted form of the function fiel d of the differential Galoi s group (remember thi s is an algebrai c group s o it has a function field ) wit h scalar s the bas e differential field . I n 1986 , Michae l Singe r [S ] gave a proo f o f thi s theore m fo r th e Picard-Vessiot cas e base d o n differentia l Galoi s theory , an d henc e understandable , say, t o a reade r wh o ha s mastere d Kaplansky' s boo k (Kolchin' s proof , o f a mor e widely applicabl e theorem , use s cohomology). O n th e othe r hand , i f one know s th e Kolchin theorem, then it should be easy to establish the fundamental correspondenc e theorem of differential Galoi s theory: i f E = F(G), the n it is not very surprising that subfield extension s K betwee n F an d E correspon d t o subgroups of G. (Her e we use vii
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