Abstract Algebra: An Interactive Approach, Second Edition

550 Pages·2016·3.095 MB·English

by  William Paulsen

Abstract Algebra
An Interactive Approach
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TEXTBOOKS in MATHEMATICS
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ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH
William Paulsen
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INTRODUCTION TO ABSTRACT ALGEBRA
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LINEAR ALBEBRA: A FIRST COURSE WITH APPLICATIONS
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TEXTBOOKS in MATHEMATICS
Abstract Algebra
An Interactive Approach
William Paulsen
Arkansas State University
Jonesboro, Arkansas, U.S.A.
C4521_FM.indd 3 6/26/09 11:08:12 AM
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Contents
List of Figures ix
List of Tables xi
Preface xiii
Acknowledgments xv
About the Author xvii
Symbol Description xix
Mathematica(cid:13)R vs. GAP xxiii
1 Understanding the Group Concept 1
1.1 Introduction to Groups . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Prime Factorizations . . . . . . . . . . . . . . . . . . . . . . 10
1.4 The Definition of a Group . . . . . . . . . . . . . . . . . . . 15
Problems for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . 21
2 The Structure within a Group 27
2.1 Generators of Groups . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Defining Finite Groups in Mathematica and GAP . . . . . . 31
2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Problems for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Patterns within the Cosets of Groups 53
3.1 Left and Right Cosets . . . . . . . . . . . . . . . . . . . . . . 53
3.2 How to Write a Secret Message . . . . . . . . . . . . . . . . . 58
3.3 Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . 71
Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Mappings between Groups 79
4.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 The Three Isomorphism Theorems . . . . . . . . . . . . . . . 93
v
vi Contents
Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Permutation Groups 107
5.1 Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 Cayley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4 Numbering the Permutations . . . . . . . . . . . . . . . . . . 127
Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Building Larger Groups from Smaller Groups 135
6.1 The Direct Product . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 The Fundamental Theorem of Finite Abelian Groups . . . . 141
6.3 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.4 Semi-Direct Products . . . . . . . . . . . . . . . . . . . . . . 161
Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 171
7 The Search for Normal Subgroups 175
7.1 The Center of a Group . . . . . . . . . . . . . . . . . . . . . 175
7.2 The Normalizer and Normal Closure Subgroups . . . . . . . 179
7.3 Conjugacy Classes and Simple Groups . . . . . . . . . . . . . 183
7.4 The Class Equation and Sylow’s Theorems . . . . . . . . . . 190
Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 203
8 Solvable and Insoluble Groups 209
8.1 Subnormal Series and the Jordan-H¨older Theorem . . . . . . 209
8.2 Derived Group Series . . . . . . . . . . . . . . . . . . . . . . 217
8.3 Polycyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . 224
8.4 Solving the PyraminxTM . . . . . . . . . . . . . . . . . . . . 232
Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . 239
9 Introduction to Rings 245
9.1 Groups with an Additional Operation . . . . . . . . . . . . . 245
9.2 The Definition of a Ring . . . . . . . . . . . . . . . . . . . . 252
9.3 Entering Finite Rings into GAP and Mathematica . . . . . . 256
9.4 Some Properties of Rings . . . . . . . . . . . . . . . . . . . . 264
Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . 269
10 The Structure within Rings 273
10.1 Subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
10.2 Quotient Rings and Ideals . . . . . . . . . . . . . . . . . . . 277
10.3 Ring Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . 284
10.4 Homomorphisms and Kernels . . . . . . . . . . . . . . . . . . 292
Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . 302
Contents vii
11 Integral Domains and Fields 309
11.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . 309
11.2 The Field of Quotients . . . . . . . . . . . . . . . . . . . . . 318
11.3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 324
11.4 Ordered Commutative Rings . . . . . . . . . . . . . . . . . . 338
Problems for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . 345
12 Unique Factorization 351
12.1 Factorization of Polynomials . . . . . . . . . . . . . . . . . . 351
12.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . 362
12.3 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 373
12.4 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . 379
Problems for Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . 385
13 Finite Division Rings 391
13.1 Entering Finite Fields in Mathematica or GAP . . . . . . . . 391
13.2 Properties of Finite Fields . . . . . . . . . . . . . . . . . . . 396
13.3 Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . . . . 405
13.4 Finite Skew Fields . . . . . . . . . . . . . . . . . . . . . . . . 417
Problems for Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . 423
14 The Theory of Fields 429
14.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 429
14.2 Extension Fields . . . . . . . . . . . . . . . . . . . . . . . . . 436
14.3 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Problems for Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . 455
15 Galois Theory 459
15.1 The Galois Group of an Extension Field . . . . . . . . . . . . 459
15.2 The Galois Group of a Polynomial in Q . . . . . . . . . . . . 468
15.3 The Fundamental Theorem of Galois Theory . . . . . . . . . 479
15.4 Solutions of Polynomial Equations Using Radicals . . . . . . 486
Problems for Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . 491
Answers to Odd-Numbered Problems 497
Bibliography 517
Index 519

List of Figures
1.1 Terry’s animated dance steps . . . . . . . . . . . . . . . . . . 2
1.2 Circle graphs for modulo 10 operations . . . . . . . . . . . . . 7
2.1 Visualizing arrangements of three books . . . . . . . . . . . . 34
2.2 Rotations of the octahedron . . . . . . . . . . . . . . . . . . . 37
2.3 PyraminxTM puzzle . . . . . . . . . . . . . . . . . . . . . . . 48
3.1 Circle graphs revealing cosets of Terry’s group. . . . . . . . . 54
3.2 Circle graph for squaring in Z∗ . . . . . . . . . . . . . . . . . 59
33
3.3 Circle graph for cubing in Z∗ . . . . . . . . . . . . . . . . . . 60
33
3.4 Circle graph for cubing modulo 33 . . . . . . . . . . . . . . . 61
4.1 Diagram of a typical homomorphism . . . . . . . . . . . . . . 89
4.2 Commuting diagram for first isomorphism theorem . . . . . . 95
4.3 Commuting diagram for second isomorphism theorem . . . . 98
4.4 Commuting diagram for third isomorphism theorem . . . . . 102
5.1 Circle graph for a typical permutations. . . . . . . . . . . . . 110
5.2 Circle graph of a typical cycle . . . . . . . . . . . . . . . . . . 111
5.3 Circle graphs for multiplying elements of Q by i. . . . . . . . 121
5.4 Multiplying cosets of D by elements . . . . . . . . . . . . . . 124
4
6.1 Circle graph for multiplying by 3 in Z . . . . . . . . . . . . . 152
8
6.2 Proof without words that Aut(Q)≈S . . . . . . . . . . . . . 158
4
8.1 Example of two subnormal series of different lengths . . . . . 213
8.2 Diagram showing the strategy of the refinement theorem . . . 213
8.3 PyraminxTM without corners gone . . . . . . . . . . . . . . . 233
8.4 PyraminxTM with numbered faces . . . . . . . . . . . . . . . 237
9.1 Plot depicting the rational numbers . . . . . . . . . . . . . . 246
9.2 Sample path going through every rational . . . . . . . . . . . 247
10.1 Commuting diagram for first ring isomorphism theorem . . . 298
11.1 Polar coordinates for a complex number . . . . . . . . . . . . 331
11.2 The eight roots of unity . . . . . . . . . . . . . . . . . . . . . 334
ix