Semi-topological Galois theory and the inverse Galois problem

TL;DR

This paper introduces semi-topological Galois groups and splitting coverings to provide a novel perspective on the inverse Galois problem, bridging field theory and topology.

Contribution

It develops semi-topological Galois groups for Weierstrass polynomials and establishes a Galois correspondence, advancing the theoretical framework for the inverse Galois problem.

Findings

Defined splitting coverings analogous to splitting fields

Proved the existence of a Galois correspondence in this setting

Provided a new approach to studying the inverse Galois problem

Abstract

We enhance the analogy between field extensions and covering spaces by introducing the concept of splitting covering which correspondences to the splitting field in Galois theory. We define semi-topological Galois groups for Weierstrass polynomials and prove the existence of a Galois correspondence. This new tool enables us to study the inverse Galois problem from a new viewpoint.