# Integers, Polynomials, and Rings

## A Course in Algebra

• Ronald S. Irving
Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xv

1. Pages 1-5
3. ### Integers

1. Front Matter
Pages 7-7
2. Pages 9-21
3. Pages 23-40
4. Pages 41-55
5. Pages 57-67
6. Pages 69-93
7. Pages 95-113
8. Pages 115-124
4. ### Polynomials

1. Front Matter
Pages 125-125
2. Pages 127-139
3. Pages 141-175
4. Pages 177-191
5. Pages 193-200
6. Pages 201-220
7. Pages 221-237
5. ### All Together Now

1. Front Matter
Pages 239-239
2. Pages 241-253
3. Pages 255-265
4. Pages 267-279
6. Back Matter
Pages 281-284

### Introduction

Mathematics is often regarded as the study of calculation, but in fact, mathematics is much more. It combines creativity and logic in order to arrive at abstract truths. This book is intended to illustrate how calculation, creativity, and logic can be combined to solve a range of problems in algebra. Originally conceived as a text for a course for future secondary-school mathematics teachers, this book has developed into one that could serve well in an undergraduate course in abstract algebra or a course designed as an introduction to higher mathematics. Not all topics in a traditional algebra course are covered. Rather, the author focuses on integers, polynomials, their ring structure, and fields, with the aim that students master a small number of serious mathematical ideas. The topics studied should be of interest to all mathematics students and are especially appropriate for future teachers.

One nonstandard feature of the book is the small number of theorems for which full proofs are given. Many proofs are left as exercises, and for almost every such exercise a detailed hint or outline of the proof is provided. These exercises form the heart of the text. Unwinding the meaning of the hint or outline can be a significant challenge, and the unwinding process serves as the catalyst for learning.

Ron Irving is the Divisional Dean of Natural Sciences at the University of Washington. Prior to assuming this position, he served as Chair of the Department of Mathematics. He has published research articles in several areas of algebra, including ring theory and the representation theory of Lie groups and Lie algebras. In 2001, he received the University of Washington's Distinguished Teaching Award for the course on which this book is based.

### Keywords

Abstract algebra Representation theory algebra binomial ring theory

#### Authors and affiliations

• Ronald S. Irving
• 1
1. 1.College of Arts and SciencesUniversity of WashingtonSeattleUSA

### Bibliographic information

• DOI https://doi.org/10.1007/b97633
• Copyright Information Springer-Verlag New York, Inc. 2004
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-40397-7
• Online ISBN 978-0-387-21831-1
• Series Print ISSN 0172-6056
• Buy this book on publisher's site