  The classical Galois theory deals with certain finite algebraic extensions
and establishes a bijective order reversing correspondence between the
intermediate fields and the subgroups of a group of permutations called the
Galois group of the extension. It has been the dream of many mathematicians at
the end of the nineteenth century to generalize these results to systems of
algebraic partial differential (PD) equations and the corresponding finitely
generated differential extensions, in order to be able to add the word
differential in front of any classical statement. The achievement of the
Picard-Vessiot theory by E. Kolchin between 1950 and 1970 is now well known.
The purpose of this paper is to sketch the general theory for such differential
extensions and algebraic pseudogroups by means of new methods mixing
differential algebra, differential geometry and algebraic geometry. As already
discovered by E. Vessiot in 1904 through the use of automorphic systems, a
concept never acknowledged, the main point is to notice that the Galois theory
(old and new) is mainly a study of principal homogeneous spaces (PHS) for
algebraic groups or pseudogroups. Hence, all the formal theory of PD equations
developped by D.C. Spencer around 1970 must be used together with modern
algebraic geometry, in particular tensor products of rings and fields. However,
the combination of these new tools is not sufficient and we have to create the
analogue for Lie pseudogroups of the so-called invariant derivations introduced
by A. Bialynicki-Birula after 1960 in the study of algebraic groups and fields
with derivations. We shall finally prove the usefulness of the resulting
differential Galois theory through striking applications to mechanics,
revisiting shell theory, chain theory, the Frenet-Serret formulas and the
integration of Hamilton-Jacobi equations.
