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.
