Computing subfields of number fields and applications to Galois group computations

TL;DR

This paper introduces a faster algorithm for computing all subfields of a number field by using Galois-generating subfields, which significantly reduces computational effort and aids Galois group calculations.

Contribution

It presents a novel concept of Galois-generating subfields and a set of criteria for early termination, improving the efficiency of subfield computation algorithms.

Findings

Significant speedup over previous algorithms

Effective identification of key subfields for Galois group computation

Enhanced methods for early termination of subfield searches

Abstract

A polynomial time algorithm to give a complete description of all subfields of a given number field was given in an article by van Hoeij et al. This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields. In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method from van Hoeij et al. For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields. Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.