Abstract

Abstract: The Inverse Galois Problem, a longstanding challenge in mathematics, seeks to determine which finite groups can be realized as Galois groups over a given field. While progress has been made in solving this problem for certain groups, many fundamental questions remain unanswered. This research paper delves into the algebraic number theory perspective of the Inverse Galois Problem, examining the connection between algebraic number fields and the corresponding Galois groups. By exploring the properties of algebraic number fields and their associated Galois extensions, we aim to shed light on the possible groups that can arise as Galois groups, as well as the underlying algebraic structures that govern these connections. We survey recent advancements in the field, including the application of class field theory, cohomology theory, and modular forms, to tackle the Inverse Galois Problem from an algebraic number theory perspective. We also discuss the interplay between the Inverse Galois Problem and other areas of mathematics, such as representation theory and arithmetic geometry. Through this comprehensive analysis, we hope to provide a deeper understanding of the Inverse Galois Problem and contribute to the ongoing research in this fascinating area of mathematics.Keywords: Inverse Galois Problem, Galois groups, algebraic number theory, algebraic number fields, class field theory, cohomology theory, modular forms, representation theory, arithmetic geometry