A formula to solve sextic degree equation

TL;DR

This paper introduces a specialized formula for solving certain sixth-degree equations using Martinelli's polynomial, providing a method to find solutions when specific coefficient conditions are met, despite the general impossibility of solving all degree 5 or higher equations by radicals.

Contribution

The paper presents a new formula for solving specific sixth-degree equations based on coefficient criteria and resolvent equations, expanding solution methods within algebra.

Findings

A formula for specific sixth-degree equations using Martinelli's polynomial.

Identification of coefficient conditions for solvability.

Demonstration of solving an example sixth-degree equation.

Abstract

According to the Abel-Ruffini theorem, equations of degree equal to or greater than 5 cannot, in most cases, be solved by radicals. Due of this theorem we will present a formula that solves specific cases of sixth degree equations using Martinellis polynomial as a base. To better understand how this formula works, we will solve a sixth degree equation as an example. We will also see that all sixth degree equations that meet the coefficient criterion have a resolvent of fifth degree that can be splitted into a second degree and a third degree equation. Throughout the paper we will see a demonstration of the ratio of the coefficients of a sixth degree equation that can be solved with the formula that will be presented this paper