Integers, Polynomials, and Rings

1. Front Matter

   Pages i-xv

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2. Introduction: The McNugget Problem

   1. Introduction: The McNugget Problem

      Pages 1-5

3. Integers

   1. Front Matter

      Pages 7-7

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   2. Induction and the Division Theorem

      Pages 9-21

   3. The Euclidean Algorithm

      Pages 23-40

   4. Congruences

      Pages 41-55

   5. Prime Numbers

      Pages 57-67

   6. Rings

      Pages 69-93

   7. Euler’ Theorem

      Pages 95-113

   8. Binomial Coefficients

      Pages 115-124

4. Polynomials

   1. Front Matter

      Pages 125-125

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   2. Polynomials and Roots

      Pages 127-139

   3. Polynomials with Real Coefficients

      Pages 141-175

   4. Polynomials with Rational Coefficients

      Pages 177-191

   5. Polynomial Rings

      Pages 193-200

   6. Quadratic Polynomials

      Pages 201-220

   7. Polynomial Congruence Rings

      Pages 221-237

5. All Together Now

   1. Front Matter

      Pages 239-239

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   2. Euclidean Rings

      Pages 241-253

   3. The Ring of Gaussian Integers

      Pages 255-265

This book began life as a set of notes that I developed for a course at the University of Washington entitled Introduction to Modern Algebra for Tea- ers. Originally conceived as a text for future secondary-school mathematics teachers, it has developed into a book that could serve well as a text in an - dergraduatecourseinabstractalgebraoracoursedesignedasanintroduction to higher mathematics. This book di?ers from many undergraduate algebra texts in fundamental ways; the reasons lie in the book’s origin and the goals I set for the course. The course is a two-quarter sequence required of students intending to f- ?ll the requirements of the teacher preparation option for our B.A. degree in mathematics, or of the teacher preparation minor. It is required as well of those intending to matriculate in our university’s Master’s in Teaching p- gram for secondary mathematics teachers. This is the principal course they take involving abstraction and proof, and they come to it with perhaps as little background as a year of calculus and a quarter of linear algebra. The mathematical ability of the students varies widely, as does their level of ma- ematical interest.

• Abstract algebra
• Representation theory
• algebra
• binomial
• ring theory

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)

"Originallyconceived 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)

• College of Arts and Sciences, University of Washington, Seattle, USA

  Ronald S. Irving

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