A unified approach to the Galois closure problem

TL;DR

This paper introduces a unified categorical framework for solving the Galois closure problem across various mathematical structures, providing algorithms with degree bounds for their outputs.

Contribution

It presents a unified approach and algorithms for Galois closures applicable to diverse settings like field extensions, graph covers, and algebraic varieties.

Findings

Algorithms successfully compute Galois closures in multiple contexts.

Degree bounds for Galois closures are established.

Unified categorical framework enhances understanding of Galois problems.

Abstract

In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a connected undirected graph, finite covering spaces of a locally connected topological space, finite \'etale covers of a smooth projective irreducible algebraic variety, and finite covers of normal varieties. We present two algorithms whose outputs are shown to be desired Galois closures. An upper bound of the degree of the Galois closure under each algorithm is also obtained.