General Reducibility and Solvability of Polynomial Equations

Abstract

A complete work on general reducibility and solvability of polynomial equations by algebraic meansradicals is developed. These equations called, reanegbèd and vic-emmeous are designed by using simple algebraic principles on how systems of equations and polynomials behave. Reanegbèd equations are capable of reducing or rewriting all polynomial equations to desired forms while vic-emmeous equations are capable of extracting out a root or roots from two or more algebraically closed polynomials. Unlike quadratic, cubic, and quartic polynomials, the general quintic and higher degree polynomials cannot be solved algebraically in terms of finite number of additions, subtractions, multiplications, divisions, and root extractions as rigorously demonstrated by Abel (1802 –1829) and Galois (1811 –1832). However, allowing the use of reanegbèd equations and vic-emmeous equations make reducing and solving all polynomial equations possible algebraically in terms of finite number of additions, subtractions, multiplications, divisions, and root extractions.

Keywords: Roots of polynomial equations, Polynomials (irreducibility, etc.), Galois Theory, Solving Polynomial Systems, Polynomial factorization, Polynomial Ring