I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra.

Looking at the wikipedia page, I have never studied Lie groups. Is it at all possible to pick it up while I study Differential Galois theory?

  • $\begingroup$ Also, what would be the best introductory book on this subject? $\endgroup$ – Hubble Apr 4 '14 at 19:56
  • $\begingroup$ @iHubble There are some other threads where several good references (books, introductory articles) are mentioned: 1, 2, 3, 4 $\endgroup$ – Martin Sleziak Nov 9 '14 at 7:31

Luckily, I am studying this forgotten branch of mathematics. The following references are must-read according to my knowledge:

  1. Irving Kaplansky, Introduction to differential algebra, Hermann Paris,1957

  2. Hyman Bass and Alexandru Buium et.al, Selected Works of Ellis Kolchin with Commentary, American Mathematical Society, 1999

  3. Buium, Differential Algebra and Diophantine Geometry, Hermann, 1994

These three references are ordered according to their difficulty. The first book is the only book which can be read with bare knowledge of Lie theory. Since you mentioned you are familiar with algebraic geometry, the second book is a good guide towards the papers written by Kolchin which are obscure due to the evolution of notions and concepts throughout these years. And the third book points out the recent trends in this branch and requires more prerequisite and mathematical maturity. If you are not sure whether you want to put so much effort into this branch this will be a good read: A first look at differential algebra

  1. Ellis R. Kolchin, Existence theorems connected with the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Bull. Amer. Math. Soc. 54 (1948), 927–932

    References E. R. Kolchin, Galois Theory of Differential Fields , American Journal of Mathematics, Vol. 75, No. 4 (Oct., 1953), pp. 753-824

    References E.R.Kolchin, Algebraic Matric Groups and the Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential Equations , Annals of Mathematics, Second Series, Vol. 49, No.1 (Jan., 1948), pp.1-42

Hopefully my answer will help, and my question is here. Why differential Galois theory is not widely used?

For more advanced discussions on this topic, there is a MO post Why differential Galois theory is not widely used?(MO version).

An interesting related MO post is Does a theory of stochastic differential algebras exist?. In this post I briefly explained difference between differential algebra and stochastic calculus.

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