International Journal of Science, Engineering and TechnologyAuthors: Assistant Professor Rajkumar Soni, Assistant Professor Rahul Kaushik
Abstract: Early successes in solving polynomial equations up to degree four by radicals most famously Cardano’s solution to the cubic and Ferrari’s to the quartic demonstrate the power of adjoin-and-solve techniques in classical algebra (Dummit and Foote 765). However, the general quintic and higher-degree cases elude such formulas: Abel’s impossibility theorem proves that no expression in a finite combination of radicals can capture the roots of an arbitrary fifth-degree polynomial (Abel “Mémoire” 12). This elusion finds its true explanation in the language of field extensions and group theory. By considering a polynomial’s splitting field and the automorphisms that permute its roots, one constructs the Galois group a measure of the equation’s intrinsic symmetries (Stewart 34). The Fundamental Theorem of Galois Theory then establishes a one-to-one correspondence between intermediate fields and subgroups of this Galois group, yielding a precise criterion: a polynomial is solvable by radicals if and only if its Galois group is a solvable group (Rotman 216; Artin 52).This paper first reviews the foundations of field extensions and Abel’s theorem, then develops Galois’s structural framework. It next applies the Galois correspondence to characterize solvable cases, illustrating cubic and quartic examples before showing why the symmetric group S5S_5S5 defies solvability. Subsequent sections examine special higher-degree families such as cyclotomic and trinomial cases and modern algorithms for computing Galois groups and constructing number fields (Cohen; Neumann 142). Through case studies, we compare classical formulaic methods with contemporary computational approaches, highlight open problems, and discuss implications for number theory, cryptography, and algebraic geometry. In conclusion, we underscore the enduring relevance of Galois theory and outline future directions that integrate algorithmic techniques with group-theoretic insights.
