Descent for differential modules and skew fields

In this paper we study rationality questions for differential modules and differential operators. If a differential operator L is equivalent to its conjugates over k, is it then equivalent to an operator defined over k? We will show how counterexamples to this question correspond to skew fields, and we will make this correspondence explicit in both directions. Similar questions are studied for projective equivalence of differential operators. The main tool is the study of differential modules over skew fields.