Connections Between Abstract Algebra and Polynomial Equations: The Legacy of Lagrange

This paper investigates the evolution and theory of polynomial factorization, tracing a path from classical solutions of quadratics to modern techniques for analyzing quintic polynomials. We begin with foundational methods for factoring quadratics and cubics, including Cardano’s formula and its algebraic extensions. Moving into quartic equations, we compare the approaches of Ferrari and Descartes, and introduce Lagrange’s revolutionary perspective on root symmetries. This sets the stage for a deeper exploration into the structure of polynomials through Group Theory and Gröbner bases. While traditional methods fail to provide general solutions for the quintic, we demonstrate how modern algebraic tools and computational techniques allow us to investigate its structure through the lens of the symmetry.