Inverse Galois Problem and Significant Methods

TL;DR

This paper reviews key results and methods related to the inverse Galois problem, which asks if every finite group can be realized as a Galois group over the rationals, highlighting progress and techniques like Hilbert irreducibility and rigidity.

Contribution

It provides a comprehensive overview of significant results and methods in the study of the inverse Galois problem, emphasizing recent progress and key techniques.

Findings

Progress in realizing finite groups as Galois groups over Q

Discussion of methods like Hilbert irreducibility and rigidity

Compilation of significant results in inverse Galois theory

Abstract

The inverse problem of Galois Theory was developed in the early 1800 s as an approach to understand polynomials and their roots. The inverse Galois problem states whether any finite group can be realized as a Galois group over Q (field of rational numbers). There has been considerable progress in this as yet unsolved problem. Here, we shall discuss some of the most significant results on this problem. This paper also presents a nice variety of significant methods in connection with the problem such as the Hilbert irreducibility theorem, Noether s problem, and rigidity method and so on.