On differential Galois groups of strongly normal extensions

TL;DR

This paper extends Kolchin's differential Galois theory to strongly normal extensions over fields with non-algebraically closed constants, including ordered and p-valued differential fields, providing a partial Galois correspondence.

Contribution

It generalizes existing results by removing the algebraically closed constants assumption and establishes a relative Galois correspondence in ordered differential fields.

Findings

Partial Galois correspondence established

Extension of Kolchin's results to broader constant fields

Use of elimination of imaginaries in ordered fields

Abstract

We give a detailed proof of Kolchin's results on differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. We closely follow former works due to Pillay and his co-authors which were written under the assumption that the field of constant is algebraically closed. In the present setting, which encompasses the cases of ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in the theory of closed ordered fields, we establish a relative Galois correspondence for definable subgroups of the group of differential order automorphisms.