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.
Author(s): Ronald S. Irving
Series: Undergraduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 2004
Language: English
Pages: 301
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
