Impossibilities proved by Galois theory

Abstract

I studied three areas in my paper:

1. The basic field theory needed to prove the impossibility of three goals of Ruler and Compass construction. Here I found the results flowed rather easily. I found that the level of material needed to prove these problems was, for the most part, not out of reach for any undergraduate mathematics student.

2. I studied the step into Galois theory and the necessary material needed to prove the impossibility of solving the general Quintic with radicals. This was the biggest step. The jump from the basic field theory I studied in 1. to the Galois theory needed for 2. is extreme. I spent most of my time studying this material so that I may understand it well enough to present it.

3. I studied the rational distance problem, specifically how its arguments are extremely similar to those of the problems in 1. and 2. I found that the main argument lies in drawing an equality between the geometry of the problem, and then using this relation to prove equivalent field extensions. This can only happen for some polygons, and that is the conclusion in the paper.

More generally, I found that independent study, no matter how expository and non novel, in the area you're interested in is extremely fascinating. I was motivated and intrigued to continue studying well beyond the normal hours, and certainly this project has cemented in me the will to pursue a Ph.D in mathematics.