GALOIS THEORY AND ITS APPLICATIONS IN MODERN MATHEMATICS

This article examines Galois theory as one of the fundamental frameworks of modern algebra and analyzes its main theoretical principles and applications. The study focuses on the relationship between field extensions and group theory, emphasizing the role of Galois groups in determining the solvability of polynomial equations by radicals. Using a theoretical and analytical approach, the article demonstrates how Galois theory provides a structural explanation for classical algebraic problems and establishes deep connections between algebra, number theory, and geometry. Particular attention is given to the applications of Galois theory in finite field theory, which forms the mathematical foundation of coding theory and cryptography. The results of the study confirm that Galois theory is not only a cornerstone of pure mathematics but also a powerful tool with significant relevance in applied mathematics and modern technological systems.