# INTRODUCTION

These notes were first prepared for a course given at Northeastern Illinois University in the summer of 1989. Theory of Equations was a traditional course in the mathematics curriculum which dealt with the important topics of the theory and solution of polynomial and linear equations. Recently, thanks to the increased availabilty of computers, linear algebra has taken a central role in mathematics and there are specialized courses devoted to that topic. On the other hand, the theory of polynomial equations, while still important, has been scattered among various courses.

I had two motivations for preparing notes rather than using a standard text. First of all there have many changes in the way we view the material since the 1940's when such classics such as the text by Uspensky were written. Computers, and even calculators, have made much of the traditional material, which dealt with efficient hand computation schemes and separation of roots, obsolete. Different numerical methods are now prefered. Interest in dynamical systems and chaos has brought new insights into Newton's method. And there have been exciting new algorithms developed for factoring polynomials. Thus one motivation for these notes is to give a modern view of this classical subject.

My other motivation has to do with the number of different areas of mathematics in which polynomials arise. Thus by using the concept of the polynomial as a common thread I am able to cover a large area of the mathematical landscape. Generally students of mathematics are able to see only a small portion of the total of mathematics and often are not given the motivation, practical and historical, for many of the branches of modern mathematics. Thus it is my intention to use this course partly as a survey of a large chunk of pure and applied mathematics from a historical as well as modern point of view. It is my hope that students will emerge from this course not so much with specific skills in solving polynomial equations but rather with a better perspective on what mathematics is, where it came from, and where it is going.

This may be the last version of these notes, as I have retired from teaching courses such as this. In addition there are now three excellent texts covering much of the original material, the text by Fine and Rosenberger on The Fundamental Theorem of Algebra which covers the material in the first part of Chapter 3 far more thoroughly and the book by Ronald Solomon, somewhat mistitled Abstract Algebra, which covers the material in Chapter 4 in a spirit similar to these notes, but going farther into Galois theory. Also the recent text Numerical Polynomial Algebra by Hans Stetter might be considered the present, state of the art, text on Theory of Equations.

On the other hand, I have been told that my notes are one of the more accessible discussions of the numerical aspects of polynomial solving, and the material on the connection between the Fundamental Theorem and Newton's method is, at this level, as far as I know, unique to these notes. Since numerical methods are my present area of interest it is possible that new chapters and/or versions of Chapter 2 may appear, the appendix to Chapter 1 gives a small sample of how linear algebra will replace some of the classical methods.

I wish to thank various students who have helped motivate me and make these notes possible. First of all, my Summer 1989 class for their patience in receiving these notes a few pages at a time and who pointed out many typographical and other errors. I also wish to acknowledge my debt to students James Moor and Ken Touff whose masters projects provided the main motivation and much of the material for Chapter 3. And I wish to thank Professor Zhonggang Zeng who has given, and continues to give, me inspiration for the numerical aspects of this subject.

October, 2004