A Polynomial Time Nilpotence Test for Galois Groups and Related Results

TL;DR

This paper presents a deterministic polynomial-time algorithm for testing if the Galois group of a polynomial is nilpotent, and generalizes solvability tests to determine if it belongs to certain group classes, with efficient computation of group order factors.

Contribution

It introduces a polynomial-time algorithm to determine nilpotence of Galois groups and extends solvability tests to broader group classes, including prime factorization of group order.

Findings

Polynomial-time nilpotence test for Galois groups

Generalization of Landau-Miller solvability test

Efficient computation of prime factors of Galois group order

Abstract

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 $# \Gal{f}$.