The Complete Picard Vessiot Closure

TL;DR

This paper introduces the concept of the complete Picard--Vessiot closure of a differential field, a minimal extension with no further Picard--Vessiot extensions, and explores its automorphism group and subfield correspondence.

Contribution

It defines and characterizes the complete Picard--Vessiot closure, establishing a correspondence between its subfields and automorphism subgroups, extending classical Galois theory.

Findings

The complete Picard--Vessiot closure has no further Picard--Vessiot extensions.

There is a Galois-type correspondence between subfields and automorphism groups.

Normal subfields can be characterized independently of their embedding.

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.