A Comparative Study of Galois Theory and Its Computational and Engineering Applications

Galois theory provides a fundamental algebraic framework for understanding the solvability and symmetry properties of polynomial equations and has played a significant role in the development of modern algebra. In recent years, its relevance has extended beyond pure mathematics into computational techniques and engineering-oriented applications. This paper presents a comparative study of the theoretical foundations of Galois theory and its integration with computational methods, emphasizing its applicability in contemporary engineering and scientific problems. The study highlights how algebraic structures and symmetry principles derived from Galois theory can support symbolic computation, algorithmic problem solving, and simplified modeling approaches in applied systems. Particular attention is given to its relevance in mathematical modeling and computational analysis, where structural insights contribute to efficient solution strategies. The paper aims to provide a concise and accessible overview that bridges classical theory with computational and engineering perspectives, making it useful for researchers and practitioners working in applied mathematics, computational engineering, and interdisciplinary scientific domains.