నైరూప్య

On Solvability of Higher Degree Polynomial Equations

Samuel Bonaya Buya

I present methods that can be used to obtain algebraic solution of polynomial equations of degree five and above. In this contribution I look into methods by which higher degree polynomial equations can be factorized to obtain lower degree solvable auxiliary equations. The factorization method has been used to successfully solve quartic equations. A careful selection of the appropriate factorized form can bear much fruit in solving higher degree polynomial. Ehrenfried Walter von Tschirnhaus (1651-1708) invented the Tschirnhaus transformation. The Swedish algebraist Erland Bring (1736-1798) showed by a Tschirnhaus transformation that the general quintic equation can be transformed to the trinomial form. The English mathematician George Jerrard (1804-1863) generalized this result to higher degree polynomial. The possibility of solvability of higher degree polynomials would pave way for transformations that can reduce higher degree polynomials to their trinomial form. The Newton Identity relates the roots of polynomials with their coefficients. It is possible to introduce an instantiation of this formula where a root of polynomial is correlated to its coefficient. This is in order to facilitate easy reduction of polynomials to lower degrees for solvability. Once a polynomial is reduced to solvable lower degree forms and there consequent roots it is possible covert it as a root of the degree of original polynomial. The paper will seek to address briefly on the things highlighted in this abstract. Solvability of higher degree polynomials will of necessity call for a re-examination of the Abel-Ruffini impossibility theorem and the Galois Theory at large.