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.
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.