Constructions using Galois Theory

TL;DR

This paper introduces algorithms for computing fixed fields, splitting fields, and radical extension towers using Galois group computations, avoiding polynomial factorization, and extends these methods to geometric Galois groups for absolute factorizations.

Contribution

It presents novel algorithms that extend Galois group computations to efficiently determine field extensions and geometric Galois groups without polynomial factorization.

Findings

Algorithms for fixed and splitting fields without polynomial factorization

Extension of Galois group computations to radical towers

Methods for computing geometric Galois groups and absolute factorizations

Abstract

We describe algorithms to compute fixed fields, splitting fields and towers of radical extensions without using polynomial factorisation in towers or constructing any field containing the splitting field, instead extending Galois group computations for this task. We also describe the computation of geometric Galois groups (monodromy groups) and their use in computing absolute factorizations.