The Complete Picard Vessiot Closure

Abstract

Let F be a differential field with field of constants C. We assume C to be algebraically closed and of characteristic 0. The complete Picard--Vessiot closure of F is a differential field extension of F with the same constants C as F, which has no Picard--Vessiot extensions, and is minimal over F with these properties. There is a correspondence between subfields of the complete Picard--Vessiot closure and subgroups of its differential automorphism group, which arises because the complete Picard--Vessiot closure comes from F via repeated Picard--Vessiot extensions. This correspondence also obtains for certain normal subfields of the complete Picard--Vessiot closure, fields which can be characterized independently of their embedding in the complete Picard--Vessiot closure.