OUTLINE O F APPROAC H

IX

PROOF.

Le t R = F\y\$, ye.t-i], a polynomia l rin g ove r F i n £ 2 indetermi -

nates. Defin e a derivatio n o n R b y

DR{yij) = j/j+ i fo r j £ - 1

and

AR(JV./-I )

= - y ^ o ^ -

0

i? i s a differentia l integra l domai n ove r F an d j/.o , 1 i £, ar e algebraicall y

independent solution s o f L = 0 in R. W e let ^ = Q(R), th e fractio n field o f R. •

Consider th e followin g example :

EXAMPLE.

Le t C denot e th e complexes , le t F = C(e x), le t D = ^ , an d le t

L = y(i ) _ y(o)

#

The n (omittin g the subscripts) th e proposition yields E = C(e x)(y)

with D(y) — y.

Notice what happened: F alread y contained solution s of L = 0 , for exampl e e

x,

and th e constructio n adjoine d a n additiona l independen t solution , namel y y. W e

also not e tha t i f z = £ , the n D(z ) = 0 .

Elements o f a differentia l field ar e calle d constants i f the y hav e derivativ e zero .

In th e example , th e adjunctio n o f a superfluou s solutio n adde d a new constant. A s

we wil l see , thi s i s generic . Fo r th e record , w e not e tha t th e se t o f constant s o f a

differential field is a subfield, an d that the set of solutions in the extension E o f L = 0

is a vecto r spac e ove r th e field o f constants .

It turn s ou t tha t th e appropriat e extension s t o conside r fo r differentia l Galoi s

theory, th e Picard-Vessiot extensions , ar e thos e wit h n o ne w constants .

DEFINITION. E D F i s a Picard-Vessio t extensio n o f F fo r L i f

(1) E i s generated ove r F a s a differential field b y the solution s o f L = 0 in E.

(2) Th e constant s o f E ar e th e constant s o f F.

(3) L = 0 has £ solution s i n E linearl y independen t ove r constants .

REMARK.

Suppos e that E D F i s a Picard-Vessiot extensio n o f F fo r L an d tha t

E 5 K 5 F i s a n intermediat e differentia l field. The n i t follow s tha t E i s als o a

Picard-Vessiot extensio n o f K fo r L also .

For convenience , w e refer t o a n extensio n E D F satisfyin g onl y part (2 ) o f th e

definition a s a "n o ne w constants " extension .

For th e res t o f thi s outline , w e make th e followin g conventions :

Let C denot e th e field o f constant s o f F.

Assume tha t F i s characteristic zer o an d tha t C i s algebraicall y closed .

The followin g resul t give s a conditio n fo r n o ne w constants :

THEOREM

1. Suppose that R D F is a differential integral domain. If Q(R), the

fraction field of R, has a new constant, then R contains a non-zero prime differential

ideal.

Combining Theore m 1 with th e propositio n abov e the n yields :

COROLLARY. Let P C F[y 7;] = R be a maximal prime differential ideal, where

R is as in the proposition. Then E = Q(R/P) D F (where E is the fraction field of

R/P) satisfies (1) and (2 ) of the definition of a Picard-Vessiot extension.