Abstract algebra : an introduction to groups, rings and fields

This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya-Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.Considerable emphasis is placed on the algebraic system consisting of congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced -- via congruence classes -- to the idea of cosets and factor groups. This enables the transition to cosets and factor objects in a more abstract setting to be relatively painless. The chapters dealing with applications help to reinforce the concepts and methods developed in the context of more down-to-earth problems.Most introductory texts in abstract algebra either avoid cosets, factor objects and homomorphisms completely or introduce them towards the end of the book. In this book, these topics are dealt with early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible, even if there are alternative methods of proof.

Preface	p. vii
List of Tables	p. ix
List of Figures	p. xi
Chapter 1 Logic and Proofs	p. 1
1.1 Introduction	p. 1
1.2 Statements, Connectives and Truth Tables	p. 2
1.3 Relations Between Statements	p. 6
1.4 Quantifiers	p. 7
1.5 Methods of proof	p. 10
1.6 Exercises	p. 13
Chapter 2 Set Theory	p. 17
2.1 Definition	p. 17
2.2 Relations Between Sets	p. 19
2.3 Operations Defined on Sets Or New sets from Old	p. 20
2.4 Exercises	p. 24
Chapter 3 Cartesian Products, Relations, Maps and Binary Operations	p. 29
3.1 Introduction	p. 29
3.2 Cartesian Product	p. 29
3.3 Maps	p. 37
3.4 Binary Operations	p. 46
3.5 Exercises	p. 53
Chapter 4 The Integers	p. 59
4.1 Introduction	p. 59
4.2 Elementary Properties	p. 59
4.3 Divisibility	p. 67
4.4 The Fundamental Theorem of Arithmetic	p. 73
4.5 The Algebraic System (Zn,+, )and Congruences	p. 76
4.6 Congruences in Z and Equations in Zn	p. 85
4.7 Exercises	p. 91
Chapter 5 Groups	p. 97
5.1 Introduction	p. 97
5.2 Definitions and Elementary Properties	p. 98
5.3 Alternative Axioms for Groups	p. 106
5.4 Subgroups	p. 108
5.5 Cyclic Groups	p. 115
5.6 Exercises	p. 120
Chapter 6 Further Properties of Groups	p. 127
6.1 Introduction	p. 127
6.2 Cosets	p. 127
6.3 Isomorphisms and Homomorphisms	p. 135
6.4 Normal Subgroups and Factor Groups	p. 142
6.5 Direct Products of Groups	p. 154
6.6 Exercises	p. 158
Chapter 7 The Symmetric Groups	p. 165
7.1 Introduction	p. 165
7.2 The Cayley Representation Theorem	p. 165
7.3 Permutations as Products of Disjoint Cycles	p. 167
7.4 Odd and Even Permutations	p. 172
7.5 Conjugacy Classes of a Group	p. 178
7.6 Exercises	p. 183
Chapter 8 Rings, Integral Domains and Fields	p. 187
8.1 Rings	p. 187
8.2 Homomorphisms, Isomorphisms and Ideals	p. 194
8.3 Isomorphism Theorems	p. 199
8.4 Direct Sums of Rings	p. 201
8.5 Integral Domains and Fields	p. 206
8.6 Embedding an Integral Domain in a Field	p. 212
8.7 The Characteristic of an Integral Domain	p. 215
8.8 Exercises	p. 218
Chapter 9 Polynomial Rings	p. 229
9.1 Introduction	p. 229
9.2 Definitions and Elementary Properties	p. 230
9.3 The Division Algorithm and Applications	p. 234
9.4 Irreducibility and Factorization of Polynomials	p. 241
9.5 Polynomials Over More Familiar Fields	p. 247
9.6 Factor Rings of the Form $$$$, F a Field	p. 255
9.7 Exercises	p. 263
Chapter 10 Field Extensions	p. 269
10.1 Introduction	p. 269
10.2 Definitions and Elementary Results	p. 269
10.3 Algebraic and Transcendental Elements	p. 275
10.4 Algebraic Extensions	p. 278
10.5 Finite Fields	p. 286
10.6 Exercises	p. 291
Chapter 11 Latin Squares and Magic Squares	p. 297
11.1 Latin Squares	p. 297
11.2 Magic Squares303
11.3 Exercises	p. 306
Chapter 12 Group Actions, the Class Equation and the Sylow Theorems	p. 309
12.1 Group Actions	p. 309
12.2 The Class Equation of a Finite Group	p. 314
12.3 The Sylow Theorems	p. 315
12.4 Applications of the Sylow Theorems	p. 321
12.5 Exercises	p. 335
Chapter 13Isometries p. 341
13.1 Isometries of Rn	p. 341
13.2 Finite Subgroups of E(2)	p. 345
13.3 The Platonic Solids	p. 348
13.4 Rotations in R3	p. 353
13.5 Exercises	p. 359
Chapter 14 Polya-Burnside Enumeration	p. 363
14.1 Introduction	p. 363
14.2 A Theorem of Polya	p. 366
14.3 Exercises	p. 373
Chapter 15 Group Codes	p. 377
15.1 Introduction	p. 377
15.2 Definitions and Notation	p. 379
15.3 Group Codes	p. 384
15.4 Construction of Group Codes	p. 388
15.5 At the Receiving End	p. 390
15.6 Nearest Neighbor Decoding for Group Codes	p. 392
15.7 Hamming Codes	p. 397
15.8 Exercises	p. 399
Chapter 16 Polynomial Codes	p. 405
16.1 Definitions and Elementary Results	p. 405
16.2 BCH Codes	p. 412
16.3 Exercises	p. 420
Appendix A Rational, Real and Complex Numbers	p. 423
A.1 Introduction	p. 423
A.2 The Real and Rational Number Systems	p. 424
A.3 Decimal Representation of Rational Numbers	p. 427
A.4 Complex Numbers	p. 428
A.5 Polar Form of a Complex Number	p. 432
A.6 Exercises	p. 440
Appendix B Linear Algebra	p. 445
B.1 Vector Spaces	p. 445
B.2 Linear Transformations	p. 452
B.3 Inner Product Spaces	p. 462
B.4 Orthogonal Linear Transformations and Orthogonal Matrices	p. 468
B.5 Determinants	p. 471
B.6 Eigenvalues and Eigenvectors	p. 480
B.7 Exercise	p. 482
Index	p. 487

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