Testing Nilpotence of Galois Groups in Polynomial Time
We give the first polynomial-time algorithm for checking whether the Galois group Gal (f) of an input polynomial f (X) is an element of Q[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) is an element of Q[X] with nilpotent Galois group, computes for each prime factor p of #Gal (f), a polynomial g(p)(X) is an element of Q[X] whose Galois group of is the p-Sylow subgroup of Gal (f).
nilpotence galois group polynomial time univariate polynomials