Resolvent Method for Computing Galois Groups of Polynomials of Degrees Four and Five

We describe a general algorithm for the computation of Galois groups of polynomials over global fields from the point of view of using it to compute Galois groups of polynomials over function fields with prime characteristic, including characteristic 2 in which some invariants which are efficient to use in other characteristics are invariant for too large a group. We state new invariants for most of these situations when the characteristic is 2. We also describe the use of this algorithm for computing Galois groups of reducible polynomials over both number fields and function fields.