Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics.
To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.
The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can—and cannot—represent the roots of polynomials.
The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory.
The presentation includes the following features:
- Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters.
- The text can be used for a one, two, or three-term course.
- Each new topic is motivated with a question.
- A collection of projects appears in Chapter 23.
Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.
Table of Contents
1.Review of Sets, Functions, and Proofs
2.Introduction: A Number Game
9.Special Types of Homomorphisms
12.Results on Commutative Rings
17.Direct Sums and Direct Products
18.The Structure of Finite Abelian Groups
20.Learning from Z
21.The Problems of the Ancients
22.Solvability of Polynomial Equations by Radicals
Steve Rosenberg is a professor in the Mathematics and Computer Science Department at the University of Wisconsin-Superior. He received his Ph.D. from the Ohio State University. As an educator, Dr. Rosenberg has both developed and taught a wide array of courses in mathematics and computer science. As a researcher, he has published results in the areas of algebraic number theory, cryptographic protocols, and combinatorial designs, among others. As a software developer, his clients included Coca-Cola Enterprises and the pension agency of Cook County Illinois. He has extensive experience in computer science and software engineering.