Synopses & Reviews
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.
Review
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)
Review
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)
Synopsis
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.
Synopsis
This introduction to modern algebra differs from texts in this area in fundamental ways. The author's primary goal is to have the reader learn to work with mathematics through reading, writing, speaking, and listening. The choice of content is important, but he regards it as a vehicle, not as an end in itself. It is the raw material through which the readers develop the ability to understand and communicate mathematics. One non-standard feature of the book is that the author proves only a few of the theorems. Most proofs are left as exercises, and these exercises can form the core of a course based on this book.
Table of Contents
Introduction: The McNugget Problem.- Induction and the Division Theorem.- The Euclidean Algorithm.- Congruences.- Prime Numbers.- Rings.- Euler's Theorem.- Binomial Coefficients.- Polynomials and Roots.- Polynomials with Real Coefficients.- Polynomials with Rational Coefficients.- Polynomial Rings.- Quadratic Polynomials.- Polynomial Congruence Rings.- Euclidean Rings.- The Ring of Gaussian Integers.- Finite Fields.- Index.