A Brief Account of Klein's Icosahedral Extensions

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.