Before presenting it, we should first say a few words about Sophus Lie. Whilst the Galois mechanism is transparent throughout his work, it is never explicit. Furthermore, there is no hint of the introduction of any kind of differential field. As such a concept is crucial in the establishment of any differential Galois theory, we do not intend to give further details of his work. Nevertheless, it is clear that his results on differential invariants have been used by Vessiot in a fundamental way.
We shall now explain the main error committed by Vessiot and Drach which led them and many of their successors in the wrong direction.
The basic idea of the classical Galois theory is to associate a group of permutations with any given polynomial equation in one unknown with coefficients in a given ground field. Such a group is called the Galois group of the equation. One should also notice that there exists a bijective dual correspondence between the subgroups of the Galois group and the intermediate fields between the ground field and an extension of it, called a Galois extension, that depends only on the given equation. The latter extension is also called a normal extension because an intermediate extension of the ground field is normal if, and only if, its corresponding Galois group is normal in the initial Galois group.
Once the Galois theory is known, only two lines can be followed.
One of them, adopted by Drach in his thesis, deals with the possibility of adding the word "differential" before every statement of the algebraic formulation. As a result, we may say that Drach was the first, quite in advance of his time, to introduce such new concepts as differential fields, differential extensions, differential specializations, differential basis, formal translations, .... However the method of using Lie pseudogroups was not really familiar to him. Moreover, he adopted a wrong definition of an " irreducible system." but such a misunderstanding can only be seen in the framework of differential algebra, which was introduced later by Joseph Fels Ritt. Roughly speaking, Drach was looking for "maximal differential ideals" instead of "prime differential ideals." The reason for this confusion arises from the fact that the concepts coincide in the non-differential Galois case used as a model. The same error was committed by Elie Cartan in the only paper he wrote about a generalization of the Galois theory. Finally, one should notice a fundamental error in Drach's thesis, whereby the wrong crucial result is not even proved. This error was known to Picard and Vessiot, as well as to Paul Painleve, as is clear from their private letters.
The second line was adopted by Vessiot and one may say that he worked for thirty years to correct the latter error of Drach. He did succeed in the end, but did not change the wrong definition of irreducibility. His idea was to notice that, when one is dealing with a polynomial equation in Galois theory, one may transform it into a system of algebraic equations with nice invariant
properties by introducing the so-called symmetric functions of the roots. At that time, Vessiot pointed out in his extraordinary 1904 memoire, which won him the Grand prix de l'Academie Francaise, that the same method could be used with success for both the Picard-Vessiot theory and the Drach-Vessiot theory by introducing new dependent variables. Then it was possible, exactly as in the non-differential case, to associate with any linear ODE or any linear homogeneous PDE in one unknown, a system of PDE with nice invariant properties. Vessiot called such a system an "automorphic system." Briefly stated, one may say that a system of PDE is an automorphic system if its space of solutions is a principal homogeneous space or PHS for a certain Lie pseudogroup. Unfortunately, the same confusion as discussed above was introduced by saying that the given ODE or PDE was irreducible if the corresponding automorphic system was irreducible in the above sense. Meanwhile, Vessiot was sure that the contemporary work of Cartan on Lie pseudogroups was not applicable to a differential Galois theory because he already knew that the generalization of the famous Maurer-Cartan equations for Lie groups was not at all the one proposed by Cartan, but the one he proposed in accordance with the ideas of Lie.
Summarizing what we have said up to now, we may say that the Galois theory cannot be applied to all extensions, but only to those called Galois extensions. Similarly, Drach and Vessiot noticed that the Picard- Vessiot theory and the Drach-Vessiot theory were not associated with all systems of PDE, but only with irreducible automorphic systems.
The great success of Kolchin in 1953 was that he established a link between these two points of view and applied it to the Picard-Vessiot theory. Indeed, he noticed that the differential extension associated with any given linear ODE with coefficients in a certain differential ground field was a particular case of more general differential extensions called strongly normal extensions for the same reason as in the Galois case. Moreover, it was possible to associate such a differential extension with a PHS for a (linear) algebraic group, called again the Galois Group of the differential extension. Once he had this result, he obtained a similarly bijective dual correspondence between the algebraic subgroups of the Galois group and the intermediate differential fields between the differential ground field and a strongly normal extension of it determined by the PHS. The achievement of this theory is presented in a dense book published in 1973 under the title Differential algebra and algebraic groups, which is a "bible" on these topics. Other generalizations have been proposed by many authors and we draw particular attention to a very interesting paper published in 1962 by A. Bialynicki-Birula.
It is now somewhat natural to wonder if the same kind of generalization can also be applied to the Drach-Vessiot theory, or indeed perhaps to any kind of irreducible automorphic system in the sense of differential algebra.