A good place to start would be looking at the algebraic theory of integration and differentiation.  James Davenport's 1981 book would be a reasonable first read: On the Integration of Algebraic Functions.  After that, reading up on computer algebra systems would be a way to follow up your studies.

Hello, you can read all that is related to differential algebra.

Liouville : formal integration (about 1850s).

Ritt-Kolchin theories.

More recently : J. Davenport, W. Sit, M.Singer works and French researchers : Boulier, Lazard, Petitot, Hubert. There are also many articles on Galois theory and differential equations  (start : Picar-Vessiot around 1900). Today there are both theoretical researchs and also many papers on effectivity.

Differential equations is probably about as far as you can get from abstract algebra in the realm of analysis.

For example :  subject of D-modules .

Dear Prof  R. Shakthivel 

Algebraic differential equation is a differential equation that can be expressed by means of differential algebra .

A simple concept is that of a polynomial vector field .

The theory of D-modules is a global theory of linear differential equations, and has been developed to include substantive results in the algebraic theory.

See

Mikhalev, A.V.; Pankrat'ev, E.V. (2001), "Differential algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Best Regards ,

Dr/Tarek Ibrahim ( T. F. Ibrahim )

Associate professor

¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia

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