#### Date Approved

2016

#### Date Posted

12-12-2016

#### Degree Type

Open Access Senior Honors Thesis

#### Department

Mathematics

#### First Advisor

Dr. Kenneth Shiskowski

#### Second Advisor

Dr. Carla Tayeh

#### Abstract

Many peoples' first exploration into more rigorous and formalized mathematics is with their early explorations in algebra. Much of their time and effort is dedicated to finding roots of polynomials-a challenge that becomes more increasingly difficult as the degree of the polynomials increases, especially if no real number roots exist. The Fundamental Theorem of Algebra is used to show that there exists a root, particularly a complex root, for any nth degree polynomial. After struggling to prove this statement for over 3 centuries, Carl Friedrich Gauss offered the first fairly complete proof of the theorem in 1799. Further proofs of the theorem were later developed, which included the short proof by contradiction of Charles Fefferman. First published in the American Mathematical Monthly in 1967, this complete proof offers a fairly elementary explanation that only requires an undergraduate understanding of Real Analysis to work through.

This project is a proof analysis of Fefferman's proof for the Fundamental Theorem of Algebra. In this analysis, rigorous detail of the proof is offered as well as an explanation of the purpose behind certain sections and how they help to show the existence of a complex root for nth degree polynomials. It is the goal of this project to work with Fefferman's proof to develop a clearer explanation of the theorem and how it is able to show this property.

#### Recommended Citation

Linford, Kyle O., "An analysis of Charles Fefferman's proof of the Fundamental Theorem of Algebra" (2016). *Senior Honors Theses*. 504.

https://commons.emich.edu/honors/504