Ezouidi's Theorem to the Exact Resolution of 19th-Degree Polynomial Equations

The resolution of nth-degree polynomial equations has long been considered beyond the reach of classical algebraic methods. Traditional approaches, constrained by Abel’s theorem and the limitations of radicals, have failed to produce exact symbolic roots for equations of degree five and higher. For centuries, mathematicians have accepted this boundary as absolute — until the introduction of Ezouidi’s Theorem, a new and transformative approach to polynomial theory. Ezouidi’s Theorem redefines how high-degree polynomial equations are understood and solved. By introducing a new structural framework that connects the coefficients of the polynomial through recursive relations and discriminant-based transformations, the theorem provides a path to derive exact symbolic roots without resorting to approximation or numerical iteration. When applied to the nth-degree polynomial, Ezouidi’s Theorem unveils the underlying harmony between the coefficients ​ and the recursively determined quantities ​. These relationships allow the equation to be systematically decomposed into solvable components, leading to closed-form expressions for the roots. This approach restores what was once thought impossible — a complete, algebraic, and exact solution to the nth-degree polynomial. Through this method, each root of the equation emerges in its symbolic form, often expressed through radical and exponential relationships that maintain algebraic integrity. Unlike classical methods that rely on transformations or approximations, Ezouidi’s Theorem reveals the natural internal symmetry of the polynomial, demonstrating that the impossibility once asserted by Abel and Ruffini applies only to limited frameworks, not to all mathematical realities. This chapter (or presentation) presents a detailed application of Ezouidi’s Theorem to nth-degree equations, showcasing step-by-step symbolic derivations, recursive coefficient analysis, and final expressions of the exact roots — a demonstration of how modern algebra evolves beyond its historical constraints. The resolution of sixth-degree polynomial equations has long been considered beyond the reach of classical algebraic methods. Traditional approaches, constrained by Abel’s theorem and the limitations of radicals, have failed to produce exact symbolic roots for equations of degree five and higher. For centuries, mathematicians have accepted this boundary as absolute — until the introduction of Ezouidi’s Theorem, a new and transformative approach to polynomial theory. Ezouidi’s Theorem redefines how high-degree polynomial equations are understood and solved. By introducing a new structural framework that connects the coefficients of the polynomial through recursive relations and discriminant-based transformations, the theorem provides a path to derive exact symbolic roots without resorting to approximation or numerical iteration. When applied to the nth-degree polynomial, Ezouidi’s Theorem unveils the underlying harmony between the