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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 secondaryschool 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.
 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 nonmathematicians, and as a guide to selfeducation beyond academic training, too." (Werner Kleinert, Zentralblatt MATH, Vol. 1046 (2), 2004)
"Originally conceived as a text for a course for future secondaryschool 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 (12), 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)
 Table of contents (17 chapters)


Introduction: The McNugget Problem
Pages 15

Induction and the Division Theorem
Pages 921

The Euclidean Algorithm
Pages 2340

Congruences
Pages 4155

Prime Numbers
Pages 5767

Table of contents (17 chapters)
 Download Preface 1 PDF (72.2 KB)
 Download Sample pages 1 PDF (128.7 KB)
 Download Table of contents PDF (63.8 KB)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Integers, Polynomials, and Rings
 Book Subtitle
 A Course in Algebra
 Authors

 Ronald S. Irving
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2004
 Publisher
 SpringerVerlag New York
 Copyright Holder
 SpringerVerlag New York
 eBook ISBN
 9780387218311
 DOI
 10.1007/b97633
 Hardcover ISBN
 9780387403977
 Softcover ISBN
 9780387201726
 Series ISSN
 01726056
 Edition Number
 1
 Number of Pages
 XVI, 288
 Topics