Introduction to Differential Galois Theory

Abstract

[en] Galois theory is one of the most beautiful areas of mathematics presented in undergraduate studies. Most of its success is due to the brilliant idea by the French mathematician Évariste Galois of associating a group to every algebraic equation in a way that its solvability can be studied through the language of group theory. Later on, this same idea was pursued by mathematicians Émile Picard and Ernest Vessiot in the field of linear differential equations. The resulting theory of linear differential equations mirrors in great part that of polynomial equations. The purpose of this work is twofold. On the one hand, we are going to give an alternative definition of the Galois group associated with a polynomial equation and prove its equivalence with the usual definition. Although somewhat more laborious to deal with, this definition allows for a deeper intuition of what the Galois group is about. On the other hand, we want to study the extension of Galois theory applied to linear differential equations. We will develop the theory of differential algebra in a way that will enable us to translate the alternative definition of the Galois group of a polynomial given in the first section to that of a linear differential equation. We will also prove its equivalence with the usual definition for the differential Galois group. Finally, we will comment on some ways in which these ideas are used to tackle the representation of solutions to differential equations in terms of their coefficients.