Calculating the Galois group of L1(L2(y))=0,L1,L2 completely reducible operators Original Research Article

In Calculating Galois groups of completely reducible linear operators, Compoint and Singer describe a decision procedure that computes the Galois group of a completely reducible linear differential operator with rational or algebraic function coefficients (i.e., a linear differential operator that is the least common left multiple of irreducible operators or, equivalently, one whose Galois group is a reductive group). At present, it is unknown how to calculate the Galois group of a general operator. In this paper, we push beyond the completely reducible case by showing how to compute the Galois group of an operator of the form L1ring operatorL2 where L1 and L2 are completely reducible and have rational function coefficients. We begin by showing how to compute the Galois group of an equation of the form L(y)=b with L completely reducible. This corresponds to the case of L1ring operatorL2 where L1=D−b′/b. We then show how one can reduce the general case to the above case and give several examples.