We give the first polynomial-time algorithm for checking whether the Galois group Gal ( f ) of an input polynomial f (X) ε f[X] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f. Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f (X) ε f[X] with nilpotent Galois group, computes for each prime factor p of #Gal (f), a polynomial gp(X) ε f[X] whose Galois group of is the p-Sylow subgroup of Gal (f). © 2012 ACM.