Analytical solutions to the general quintic equation using elliptic functions

Abstract

At the end of the eighteen century solutions by radicals for the quadratic, cubic and quartic equation were well known, but there was no formula for the general quintic and higher degree equations. Thanks to the theory developed by Evariste Galois at the beginning of the nineteenth century, we know that it is not possible to find a general solution by radicals for such equations. In this work we will put our attention on the general polynomial equation of degree 5 , which since it cannot be solved in a general way by radicals, we will look for another way to find its solutions. This is, by means of elliptic functions, especially the $\wp(z)$-Weierstrass elliptic function. In order to be possible to make this relationship between complex analysis and algebra, we will first have to reduce the general polynomial of degree 5 to its one-parameter Bring Jerrard form, through the use of Tschirnhausen transformations and Newton's identities. Once there, thanks to the differential equation that $\wp(z)$ satisfies, it is possible to identify the solutions of a particular elliptic function with the solutions of the one-parameter Bring Jerrard.