We present a new algorithm for computing the centre of the Galois group of a given polynomial f ∈ Q[x] along with its action on the set of roots of f , without previously computing the group. We show that every element in the centre is representable by a family of polynomials in Q[x]. For computing such polynomials, we use quadratic Newton-lifting and truncated expressions of the roots of f over a p-adic number field. As an application we give a method for deciding the nilpotency of the Galois group. If f is irreducible with nilpotent Galois group, an algorithm for computing it is proposed.

/pdf/deciding-the-nilpotency-of-the-galois-group-by-computing-48nkyu9qje.pdf

Deciding the nilpotency of the Galois group by computing elements in the centre

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).

/pdf/a-polynomial-time-nilpotence-test-for-galois-groups-and-6sa0mp5or9.pdf

A polynomial time nilpotence test for galois groups and related results

We give a deterministic polynomial-time algorithm to check whether the Galois group $\Gal{f}$ of an input polynomial $f(X) \in \Q[X]$ 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 $\Gamma_d$: this algorithm runs in time polynomial in $\size{f}$ and $n^d$ and, moreover, if $\Gal{f}\in\Gamma_d$ it computes all the prime factors of $# \Gal{f}$.

A Polynomial Time Nilpotence Test for Galois Groups and Related Results

We give the first polynomial-time algorithm for checking whether the Galois group Gal(f) of an input polynomial f(X) ∈ 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) ∈ Q[X] with nilpotent Galois group, computes for each prime factor p of n Gal(f), a polynomial gp(X)∈ Q[X] whose Galois group of is the p-Sylow subgroup of Gal(f).

Testing nilpotence of galois groups in polynomial time

This paper shows a probabilistic algorithm to decide whether the Galois group of a given irreducible polynomial with rational coefficients is the generalized symmetric group Cp o Sm or the generalized alternating group Cp o Am. In the affirmative case, we give generators of the group with their action on the set of roots of the polynomial.

Explicit determination of generalized symmetric and alternating Galois groups

We propose a polynomial time algorithm to decide whether the Galois group of an irreducible polynomial ƒ ∈ Q[x] is abelian, and, if so, determine all its elements along with their action on the set of roots of ƒ. This algorithm does not require factorization of polynomials over number fields. Instead we shall use the quadratic Newton—Lifting and the truncated expressions of the roots of ƒ over a p—adic number field Qp, for an appropriate prime p in Z.

A method for deciding whether the Galois group is abelian

This paper is devoted to show, first, how to easily determine, when it exists, a non-trivial element in the centre of the Galois group of an irreducible polynomial in ℤ[x] and, second, how to deal in an efficient way with solvability by radicals when a non-trivial element in the centre of the Galois group of the considered polynomial is available.

Pilar Fernandez-Ferreiros

Papers

Deciding the nilpotency of the Galois group by computing elements in the centre

A method for deciding whether the Galois group is abelian

Partial solvability by radicals