This work explores the unsolvability of the general quintic equation through the lens of Galois theory. We begin by providing a historical perspective on the problem. This starts with the solution of the general cubic equation derived by Italian mathematicians. We then move on to Lagrange's insights on the importance of studying the permutations of roots. Finally, we discuss the critical contributions of Évariste Galois, who connected the solvability of polynomials to the properties of permutation groups. Central to our thesis is the introduction and motivation of key concepts such as fields, solvable groups, Galois groups, Galois extensions, and radical extensions.

We rigorously develop the theory that connects the solvability of a polynomial to the solvability of its Galois group. After developing this theoretical framework, we go on to show that there exist quintic polynomials with Galois groups that are isomorphic to the symmetric group S5. Given that S5 is not a solvable group, we establish that the general quintic polynomial is not solvable by radicals. Our work aims to provide a comprehensive and intuitive understanding of the deep connections between polynomial equations and abstract algebra.