Computation of Galois groups of rational polynomials

TL;DR

This paper introduces a new practical algorithm for computing Galois groups of rational polynomials by developing efficient methods for finding group invariants, improving the applicability of computational Galois theory.

Contribution

It presents a novel method for computing large-degree invariants and applies these to create the first degree-independent practical algorithm for Galois group computation.

Findings

New invariants enable computation of Galois groups for higher degrees.

The algorithm is the first practical degree-independent method.

Improved invariants outperform previous approaches in efficiency.

Abstract

Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial is a very old problem. Currently, the best algorithmic solution is Stauduhar's method. Computationally, one of the key challenges in the application of Stauduhar's method is to find, for a given pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant, but not G-invariant. While generic, theoretical methods are known to find such F, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.