Integers, Polynomials, and Rings: A Course in Algebra / Edition 1

Integers, Polynomials, and Rings: A Course in Algebra / Edition 1

by Ronald S. Irving
     
 

ISBN-10: 0387403973

ISBN-13: 9780387403977

Pub. Date: 01/08/2004

Publisher: Springer New York

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

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Overview

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.

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Product Details

ISBN-13:
9780387403977
Publisher:
Springer New York
Publication date:
01/08/2004
Series:
Undergraduate Texts in Mathematics Series
Edition description:
2004
Pages:
288
Product dimensions:
9.21(w) x 6.14(h) x 0.75(d)

Related Subjects

Table of Contents

Preface
1Introduction: The McNugget Problem1
2Induction and the Division Theorem9
3The Euclidean Algorithm23
4Congruences41
5Prime Numbers57
6Rings69
7Euler's Theorem95
8Binomial Coefficients115
9Polynomials and Roots127
10Polynomials with Real Coefficients141
11Polynomials with Rational Coefficients177
12Polynomial Rings193
13Quadratic Polynomials201
14Polynomial Congruence Rings221
15Euclidean Rings241
16The Ring of Gaussian Integers255
17Finite Fields267
Index281

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