At present we do not know a general algorithm that will compute the Galois group of a linear differential equation with coefficients in a differential field k, even when k = Q̄(x), where Q̄ is the algebraic closure of the rational numbers. In contrast, algorithms for calculating the Galois group of a polynomial with coefficients in Q or Q̄(x) have been known for a long time (van der Waerden, 1953;...
