# Integers, Polynomials, and Rings

## A Course in Algebra

• Contains numerous theorems but only a few proofs, allowing this book to be used for a course where the proofs will serve as exercises

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

### 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

1. 1.College of Arts and SciencesUniversity of WashingtonSeattleUSA

### Bibliographic information

• Book Title Integers, Polynomials, and Rings
• Book Subtitle A Course in Algebra
• Authors Ronald S. Irving
• Series Title Undergraduate Texts in Mathematics
• DOI https://doi.org/10.1007/b97633
• Copyright Information Springer-Verlag New York, Inc. 2004
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-40397-7
• Softcover ISBN 978-0-387-20172-6
• eBook ISBN 978-0-387-21831-1
• Series ISSN 0172-6056
• Edition Number 1
• Number of Pages XVI, 288
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

From the reviews:

"The book focuses mainly on the ‘doing’ of algebra. … The chief aim of the author is for students ‘to master such skills as learning what a mathematical statement is, what a mathematical argument or proof is, how to present an argument orally … and how to converse effectively about mathematics.’ … the author strives to motivate students, gradually developing their insights and abilities. … It is an excellent primer for beginners in the field of abstract algebra, especially for future school teachers." (P. Shiu, The Mathematical Gazette, Vol. 89 (516), 2005)

"This is an instructional exposition which treats some elementary number theory … . It is apparent that the author has made every effort to motivate students resp. to put them in the right way. ‘I love algebra. I want my students to love algebra’ – I believe that the author succeeded even in this regard." (G. Kowol, Monatshefte für Mathematik, Vol. 144 (2), 2005)

"This is a very elementary introduction to elementary number theory and some related topics in algebra … . The topics chosen are well suited for a student’s first exposure to ‘serious’ mathematics (much more so, in the reviewer’s opinion, than the calculus course that is the norm in almost all curricula almost everywhere)." (S. Frisch, Internationale Mathematische Nachrichten, Issue 196, 2004)

"The book … represents a very special introduction to modern algebra … . focuses less on contents and more on the ‘doing’ of algebra. … Many proofs are left as exercises, together with detailed hints or outlines, and these exercises actually form the heart of the entire text. … Summing up, this book is a great primer for beginners in the field … . could serve well in an undergraduate course for non-mathematicians, and as a guide to self-education beyond academic training, too." (Werner Kleinert, Zentralblatt MATH, Vol. 1046 (2), 2004)

"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 … . The topics studies should be of interest to all mathematics students and are especially appropriate for future teachers. … 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." (Zentralblatt für Didaktik der Mathematik, November, 2004)

"Mathematics is often regarded as the study of calculation … . 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. … 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." (L’Enseignement Mathematique, Vol. 50 (1-2), 2004)

"The book is meant to be a structurally different abstract algebra textbook. … the book is very unitary and it has a good flow. … Integers, Polynominals and Rings is a unique book, and should be extremely useful for an audience of future high school teachers. It would also be a valuable supplement for students taking a traditional abstract algebra course, especially since it is very readable." (Ioana Mihaila, MathDL, January, 2004)