Differential Field Arithmetic

This thesis gives some results concerning an analogue of field arithmetic in the setting of differential fields. Our perspective is that of the model theory of differential fields. In the first chapter we give some background material and generalities on definable Galois theory. In the second chapter we describe some exact sequences allowing computations with definable Galois cohomology. In the third chapter we describe differential field arithmetical results and consequences for existence theorems for various kinds of differential Galois extensions. In the fourth chapter we give a model theoretic treatment and generalization of a certain extension of the Picard-Vessiot theory. Some of the material is solo work, some of it is joint with Anand Pillay and some is joint with both Anand Pillay and Omar Leon Sanchez.