Algebraic theory of differential equations

Abstract

In this final chapter, I want to explain how certain aspects of the theory of linear differential equations with, say, polynomial coefficients, can be viewed in an algebraic setting. There is in fact a full “Galois theory of differential equations” of which I try to convey some ideas. I conclude with a theorem due to Liouville, a particular case of which is the fact that the function ∫ exp(x²) dx has no elementary algebraic expression.