Algebraic D-groups and differential Galois theory

We discuss various relationships between the algebraic D-groups of Buium, 1992, and differential Galois theory. In the first part we give another exposition of our general differential Galois theory, which is somewhat more explicit than Pillay, 1998, and where generalized logarithmic derivatives on algebraic groups play a central role. In the second part we prove some results with a "constrained Galois cohomological flavor". For example, if G and H are connected algebraic D-groups over an algebraically closed differential field F, and G and H are isomorphic over some differential field extension of F, then they are isomorphic over some Picard - Vessiot extension of F. Suitable generalizations to isomorphisms of algebraic D-varieties are also given.