Algebra and Number Theory An Integrated Approach

by Dixon, Martyn R.; Kurdachenko, Leonid A.; Subbotin, Igor Ya

• ISBN13: 9780470496367
• ISBN10: 0470496363
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-09-27
• Publisher: Wiley

Summary

This book successfully blends algebra and number theory as an integrated discipline and consists of seven parts: Part 1 discusses the elements of set theory; Parts 2 and 3 address number systems; Parts 4 and 5 cover the main topics of linear algebra; Part 6 develops the main ideas of algebraic structures; and Part 7 demonstrates the applications of algebraic ideas to number theory. Based on the experience of the authors, this book was developed for one course that integrates three disciplines - linear algebra, abstract algebra, and number theory - in an effort to use time more efficiently. Many theorems in number theory have very simple proofs using algebraic tools, and most importantly, the book's integrated approach helps to build a deeper understanding of the subject for readers as well as improve their retention of knowledge. Applications are provided at the end of each chapter to further explain the results found in the book, and exercises are also ample throughout. While the book is mathematically self-contained, readers should be comfortable with mathematical formalism and have some experience in reading and writing mathematical proofs.

Author Biography

Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama, Tuscaloosa. He has authored more than sixty published journal articles on infinite group theory, formation theory and Fitting classes, wreath products, and automorphism groups. Leonid A. Kurdachenko, PhD, is Distinguished Professor and Chair of the Department of Algebra at the Dnepropetrovsk National University (Ukraine). Dr. Kurdachenko has authored more than 150 journal articles on the topics of infinite-dimensional linear groups, infinite groups, and module theory. Igor Ya. Subbotin, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University (California). Dr. Subbotin is the author of more than 100 published journal articles on group theory, cybernetics, and mathematics education.

Table of Contents

Preface	p. ix
Sets	p. 1
Operations on Sets	p. 1
Exercise Set 1.1	p. 6
Set Mappings	p. 8
Exercise Set 1.2	p. 19
Products of Mappings	p. 20
Exercise Set 1.3	p. 26
Some Properties of Integers	p. 28
Exercise Set 1.4	p. 39
Matrices and Determinants	p. 41
Operations on Matrices	p. 41
Exercise Set 2.1	p. 52
Permutations of Finite Sets	p. 54
Exercise Set 2.2	p. 64
Determinants of Matrices	p. 66
Exercise Set 2.3	p. 77
Computing Determinants	p. 79
Exercise Set 2.4	p. 91
Properties of the Product of Matrices	p. 93
Exercise Set 2.5	p. 103
Fields	p. 105
Binary Algebraic Operations	p. 105
Exercise Set 3.1	p. 118
Basic Properties of Fields	p. 119
Exercise Set 3.2	p. 129
The Field of Complex Numbers	p. 130
Exercise Set 3.3	p. 144
Vector Spaces	p. 145
Vector Spaces	p. 146
Exercise Set 4.1	p. 158
Dimension	p. 159
Exercise Set 4.2	p. 172
The Rank of a Matrix	p. 174
Exercise Set 4.3	p. 181
Quotient Spaces	p. 182
Exercise Set 4.4	p. 186
Linear Mappings	p. 187
Linear Mappings	p. 187
Exercise Set 5.1	p. 199
Matrices of Linear Mappings	p. 200
Exercise Set 5.2	p. 207
Systems of Linear Equations	p. 209
Exercise Set 5.3	p. 215
Eigenvectors and Eigenvalues	p. 217
Exercise Set 5.4	p. 223
Bilinear Forms	p. 226
Bilinear Forms	p. 226
Exercise Set 6.1	p. 234
Classical Forms	p. 235
Exercise Set 6.2	p. 247
Symmetric Forms over R	p. 250
Exercise Set 6.3	p. 257
Euclidean Spaces	p. 259
Exercise Set 6.4	p. 269
Rings	p. 272
Rings, Subrings, and Examples	p. 272
Exercise Set 7.1	p. 287
Equivalence Relations	p. 288
Exercise Set 7.2	p. 295
Ideals and Quotient Rings	p. 297
Exercise Set 7.3	p. 303
Homomorphisms of Rings	p. 303
Exercise Set 7.4	p. 313
Rings of Polynomials and Formal Power Series	p. 315
Exercise Set 7.5	p. 327
Rings of Multivariable Polynomials	p. 328
Exercise Set 7.6	p. 336
Groups	p. 338
Groups and Subgroups	p. 338
Exercise Set 8.1	p. 348
Examples of Groups and Subgroups	p. 349
Exercise Set 8.2	p. 358
Cosets	p. 359
Exercise Set 8.3	p. 364
Normal Subgroups and Factor Groups	p. 365
Exercise Set 8.4	p. 374
Homomorphisms of Groups	p. 375
Exercise Set 8.5	p. 382
Arithmetic Properties of Rings	p. 384
Extending Arithmetic to Commutative Rings	p. 384
Exercise Set 9.1	p. 399
Euclidean Rings	p. 400
Exercise Set 9.2	p. 404
Irreducible Polynomials	p. 406
Exercise Set 9.3	p. 415
Arithmetic Functions	p. 416
Exercise Set 9.4	p. 429
Congruences	p. 430
Exercise Set 9.5	p. 446
The Real Number System	p. 448
The Natural Numbers	p. 448
The Integers	p. 458
The Rationals	p. 468
The Real Numbers	p. 477
Answers to Selected Exercises	p. 489
Index	p. 513
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