We give a deterministic polynomial-time algorithm to check whether the Galois group Gal (f) of an input polynomial f(X) ∈ ℚ[X] is nilpotent: the running time is polynomial in size (f). Also, we generalize the Landau-Miller solvability test to an algorithm that tests if Gal (f) is in Τd: this algorithm runs in time polynomial in size (f) and n d and, moreover, if Gal (f) ∈ Τd it computes all the prime factors of #Gal (f). © Springer-Verlag Berlin Heidelberg 2006.