Emma Aho: The impossibility of solving a quintic equation

Abstract

One of the main purposes of algebra is to study algebraic equations and their solutions. This paper will show how it is impossible to solve the general quintic equation by the use of radicals, but also how a soluble quintic equation must have either one real and four complex conjugate roots or five real roots. The paper also gives an account of the history that lead to the solving of the general quadratic, cubic and quartic equations and provides methods for solving those. In those methods it is also shown how in order to solve an equation of degree n, an auxiliary equation of degree n−1 needs to be solved as well.