Whereas algebraic
Galois theory studies extensions of
algebraic fields, differential Galois theory studies extensions of
differential fields, i.e. fields that are equipped with a
derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix
Lie groups, as compared with the finite groups often encountered in algebraic Galois theory.
Beukers, Frits (1992), "8. Differential Galois theory", in Waldschmidt, Michel; Moussa, Pierre; Luck, Jean-Marc; Itzykson, Claude (eds.), From number theory to physics. Lectures of a meeting on number theory and physics held at the Centre de Physique, Les Houches (France), March 7–16, 1989, Berlin:
Springer-Verlag, pp. 413–439,
ISBN3-540-53342-7,
Zbl0813.12001