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 1883 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 19th 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 1948 Annals paper .

Kolchin's 1973 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 , 1976 [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

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