The Complete Picard Vessiot Closure

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.