lnsolvability of the Quintic via Galois Theory and Abel's Method

Abstract

The main focus of my Senior Individualized Project was to study and examine a particular field of abstract algebra known as Galois Theory. Much of the project involved reading and studying lan Stewart's text, Galois Theory. Studying from this book included the reading of selected chapters, focusing on the theorems, not only the understanding of them and their proofs, but also the ability to use them to get to the algebraic insolubility of the quintic equation. In addition to this text I also read an article by Michael I. Rosen entitled "Niels Hendrik Abel and Equations of the Fifth Degree." In this article, Rosen presents Abel's proof of why the quintic equation is insoluable, yet no Galois Theory is used in his article. Abel's work precedes that of Galois'. The remainder of this paper will be to look at how both of these mathematicians used the tools available to them to prove the insolubility of the quintic equation. I am attempting to write this paper so that anyone with a first year semester algebra class could read it and gain a very basic understanding of Galois Theory and how Abel proved the insolubility of the general quintic equation.