El icosaedro y la irreducibilidad de la quíntica

Abstract

According to Galois theory, every irreducible quintic whose Galois group is isomorphic to A5 can not be solved by radicals, due to this group is not solvable. Since the symmetry group of the icosahedron is also isomorphic to A5, it is natural to think that there is any connection between the solutions of the irreducible quintic and the icosahedron. In this dissertation we will show up this connection. One of the first things we will do is to build a polyhedral equation associated to the icosahedral Möbius group (the icosahedral equation) and we will study a method based on hypergeometric functions to solve it. After that, we will reduce the general quintic to a simplest form, the so called canonical form. Using the symmetries of the icosahedron, we will be able to build a suitable quintic resolvent whose roots coincide with those of the canonical quintic and can be expressed as a function of the solution of the icosahedral equation.