viii PREFAC E

F(G) fo r the function field of the algebraic group with coordinate ring F[G].) Thu s if

one could establish Kolchin's theorem in the Picard-Vessiot cas e from firs t principle s

(meaning differential algebr a without th e fundamental theore m of differential Galoi s

theory), the n on e could deduc e differentia l Galoi s theor y i n th e Picard-Vessiot cas e

as a consequence .

I ha d th e opportunit y t o follo w thi s approac h t o differentia l Galoi s theor y i n

the sprin g o f 1992, an d th e presen t volum e o f lectur e note s i s th e result . Thes e

lectures wer e first presented i n a course a t th e Hebrew Universit y i n Jerusalem fro m

February throug h Jun e o f 1992, and the n reprise d i n a semina r a t th e Universit y o f

Oklahoma fro m Septembe r through November of 1992. I am grateful t o my student s

in Jerusalem, particularl y Assaf Wool and Yaa'cov Kapelovich, an d my colleagues in

Oklahoma, particularl y Richar d Resco , Bren t Gordon , E d Cline , Mura d Ozaydin ,

and Leoni d Dickey , for thei r comment s an d suggestion s o n these lectures. I am als o

grateful t o Michae l Singe r fo r som e helpfu l suggestion s an d references .

The prerequisit e fo r th e course , an d henc e fo r followin g thes e notes , i s a back -

ground in algebra, especially field theory and commutative algebra, in afflne algebrai c

geometry, an d i n affin e algebrai c groups , suc h a s typicall y acquire d b y graduat e

students who have had a years course in algebra and a semesters exposure to algebraic

groups. Fo r th e latter , Bore l [B ] is a complet e reference .

Ellis Kolchin , who m I ha d know n sinc e my day s a s a J . F . Rit t Instructo r a t

Columbia i n 1969-72, passe d awa y i n th e fal l o f 1991 . I deepl y regre t no t havin g

had th e opportunit y t o discus s thi s projec t wit h him , an d hop e i t i s worth y o f hi s

memory, t o whic h I dedicate d th e course .

Outline o f approac h

This approac h t o differentia l Galoi s theor y vi a th e structur e theor y o f Picard -

Vessiot extension s ca n b e regarde d a s a sequenc e o f si x theorems , whic h w e no w

present. (Thi s outlin e doe s no t correspon d exactl y t o th e chapter-by-chapte r sum -

mary o f topic s whic h follows. )

We fix the followin g notation :

F i s a differential field with derivation DF. denote d D when there is no confusion .

(A derivation i s an additiv e map D : F — F suc h that D(ab) = D(a)b + aD{b) fo r

elements a.b o f F. an d a differential field i s a pair {F,D F) consistin g o f a field and

a designate d derivation. )

We als o sometime s denot e D !(y) b y y^\

L = Y^ + ai-\ y^

- 1)

+ .. . + aoY^, wher e a\ E F, i s a linea r homogeneou s

differential operato r ove r F.

E D F i s a differential extensio n field of F\ tha t is , a differential field containin g

F suc h tha t D

E

restricte d t o F i s D

F

.

For y i n E, L(y) = D ly + ae-\D e~l + \- a$y an d th e solution s o f L = 0 in

E ar e {y e E\L{y) = 0} .

The first proposition show s that, unlik e the situation with polynomial equations ,

no matte r wha t F i s and n o matte r wha t L i s there i s alway s a prope r extensio n E

of F generate d b y solution s o f L = 0 .

PROPOSITION. Given L and F there exists a differential extension field E ~D F in

which L = 0 has £ solutions algebraically independent over F.