A Brief Account of Klein's Icosahedral Extensions

TL;DR

This paper offers a simplified approach to solving quintic equations with icosahedral Galois groups, utilizing hypergeometric functions and algebraic transformations to develop a practical zero-finding algorithm.

Contribution

It introduces a new, more accessible method for understanding and solving icosahedral quintic equations by leveraging hypergeometric functions and algebraic transformations.

Findings

Developed a practical algorithm for computing zeros of the quintic.

Provided a complete explanation of the icosahedral equation and its solutions.

Simplified Klein's original algebraic procedures using Heymann's theory.

Abstract

We present an alternative relatively easy way to understand and determine the zeros of a quintic whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous \textit{Vorlesungen \"uber das Ikosaeder und die Aufl\"osung der Gleichungen vom f\"unften Grade} are here shortened via Heymann's theory of transformations. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series. Within this framework, we develop a practical algorithm to compute the zeros of the quintic.