Integers, Polynomials, and Rings: A Course in Algebra (Undergraduate Texts in Mathematics)

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. Contents......Page 14 Preface......Page 8 1 Introduction: The McNugget Problem......Page 18 Part I: Integers......Page 24 2.1 The Method of Induction......Page 26 2.2 The Tower of Hanoi......Page 32 2.3 The Division Theorem......Page 34 3.1 Greatest Common Divisors......Page 40 3.2 The Euclidean Algorithm......Page 44 3.3 Bézout’s Theorem......Page 48 3.4 An Application of Bézout’s Theorem......Page 51 3.5 Diophantine Equations......Page 53 4.1 Congruences......Page 58 4.2 Solving Congruences......Page 63 4.3 Congruence Classes and McNuggets......Page 67 5.1 Prime Numbers and Generalized Induction......Page 74 5.2 Uniqueness of Prime Factorizations......Page 78 5.3 Greatest Common Divisors Revisited......Page 80 6.1 Numbers......Page 86 6.2 Number Rings......Page 94 6.3 Fruit Rings......Page 100 6.4 Modular Arithmetic Rings......Page 105 6.5 Congruence Rings......Page 108 7.1 Units......Page 112 7.2 Roots of Unity......Page 116 7.3 The Theorems of Fermat and Euler......Page 118 7.4 The Euler φ-Function......Page 122 7.5 RSA Encryption......Page 127 8.1 Pascal’s Triangle......Page 132 8.2 The Binomial Theorem......Page 137 Part II: Polynomials......Page 142 9.1 Polynomial Equations......Page 144 9.2 Rings of Polynomials......Page 145 9.3 Factoring a Polynomial......Page 147 9.4 The Roots of a Polynomial......Page 150 9.5 Minimal Polynomials......Page 153 10.1 Quadratic Polynomials......Page 158 10.2 Cubic Polynomials......Page 163 10.3 The Discriminant of a Cubic Polynomial......Page 170 10.4 Quartic Polynomials......Page 176 10.5 A Closer Look at Quartic Polynomials......Page 181 10.6 The Discriminant of a Quartic Polynomial......Page 184 10.7 The Fundamental Theorem of Algebra......Page 188 11.1 Polynomials over Q......Page 194 11.2 Gauss’s Lemma......Page 198 11.3 Eisenstein’s Criterion......Page 201 11.4 Polynomials with Coefficients in F[sub(p)]......Page 204 12.1 Unique Factorization for Integers Revisited......Page 210 12.2 The Euclidean Algorithm......Page 213 12.3 Bézout’s Theorem......Page 215 12.4 Unique Factorization for Polynomials......Page 216 13.1 Square Roots......Page 218 13.2 The Quadratic Formula......Page 221 13.3 Square Roots in Finite Fields......Page 226 13.4 Quadratic Field Constructions......Page 231 14.1 A Construction of New Rings......Page 238 14.2 Polynomial Congruences......Page 243 14.3 Polynomial Congruence Rings......Page 247 14.4 Equations and Congruences with Polynomial Unknowns......Page 250 14.5 Polynomial Congruence Fields......Page 253 Part III: All Together Now......Page 256 15.1 Factoring Elements in Rings......Page 258 15.2 Euclidean Rings......Page 262 15.3 Unique Factorization......Page 266 16.1 The Irreducible Gaussian Integers......Page 272 16.2 Gaussian Congruence Rings......Page 276 16.3 Fermat’s Theorem......Page 279 17.1 Primitive Roots......Page 284 17.2 Quadratic Reciprocity......Page 288 17.3 Classification......Page 294 C......Page 298 G......Page 299 P......Page 300 Z......Page 301

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 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. This introduction to modern algebra aimsto have the reader learn to work with mathematics through reading, writing, speaking, and listening. 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 Problem 1.1. McDonald's sells Chicken McNuggets in boxes of 6, 9, and 20.