One of the aims of classical Galois theory of polynomial equations is to give criteria and algorithms that allow one to solve these equations in terms of radicals. A similar theory has been developed for linear differential equations and this allows one to decide if a linear differential equation can be solved in finite terms, that is in terms of exponentials, integrals and algebraic functions. In the differential case, the Galois group is a group of matrices that can be described either algebraically (in terms of fields and automorphisms) or analytically (in terms of analytic continuation and Stokes phenomena).

Recently, this theory has been applied to the problem of showing that certain Hamiltonian systems are not completeley integrable, that is do not have a maximum system of first integrals in involution.

I will give an elementary introduction to both the algebraic and analytic approaches to the Galois theory of linear differential equations and describe how this was used in the above application.