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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 Γd: this algorithm runs in time polynomial in size(f) and nd and, moreover, if Gal(f)∈Γd it computes all the prime factors of # Gal(f).

A Polynomial Time Nilpotence Test for Galois Groups and Related Results