Dixon / Kurdachenko / Subbotin Algebra and Number Theory

An Integrated Approach
1. Auflage 2011
ISBN: 978-0-470-64053-1
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 544 Seiten, E-Book

Explore the main algebraic structures and number systems thatplay a central role across the field of mathematics
Algebra and number theory are two powerful branches of modernmathematics at the forefront of current mathematical research, andeach plays an increasingly significant role in different branchesof mathematics, from geometry and topology to computing andcommunications. Based on the authors' extensive experience withinthe field, Algebra and Number Theory has an innovativeapproach that integrates three disciplines--linear algebra,abstract algebra, and number theory--into one comprehensiveand fluid presentation, facilitating a deeper understanding of thetopic and improving readers' retention of the main concepts.
The book begins with an introduction to the elements of settheory. Next, the authors discuss matrices, determinants, andelements of field theory, including preliminary information relatedto integers and complex numbers. Subsequent chapters explore keyideas relating to linear algebra such as vector spaces, linearmapping, and bilinear forms. The book explores the development ofthe main ideas of algebraic structures and concludes withapplications of algebraic ideas to number theory.
Interesting applications are provided throughout to demonstratethe relevance of the discussed concepts. In addition, chapterexercises allow readers to test their comprehension of thepresented material.
Algebra and Number Theory is an excellent book forcourses on linear algebra, abstract algebra, and number theory atthe upper-undergraduate level. It is also a valuable reference forresearchers working in different fields of mathematics, computerscience, and engineering as well as for individuals preparing for acareer in mathematics education.

Autoren/Hrsg.

Dixon, Martyn R.

Kurdachenko, Leonid A.

Subbotin, Igor Ya

Inhaltsverzeichnis

PREFACE ix
CHAPTER 1 SETS 1
1.1 Operations on Sets 1
Exercise Set 1.1 6
1.2 Set Mappings 8
Exercise Set 1.2 19
1.3 Products of Mappings 20
Exercise Set 1.3 26
1.4 Some Properties of Integers 28
Exercise Set 1.4 39
CHAPTER 2 MATRICES AND DETERMINANTS 41
2.1 Operations on Matrices 41
Exercise Set 2.1 52
2.2 Permutations of Finite Sets 54
Exercise Set 2.2 64
2.3 Determinants of Matrices 66
Exercise Set 2.3 77
2.4 Computing Determinants 79
Exercise Set 2.4 91
2.5 Properties of the Product of Matrices 93
Exercise Set 2.5 103
CHAPTER 3 FIELDS 105
3.1 Binary Algebraic Operations 105
Exercise Set 3.1 118
3.2 Basic Properties of Fields 119
Exercise Set 3.2 129
3.3 The Field of Complex Numbers 130
Exercise Set 3.3 144
CHAPTER 4 VECTOR SPACES 145
4.1 Vector Spaces 146
Exercise Set 4.1 158
4.2 Dimension 159
Exercise Set 4.2 172
4.3 The Rank of a Matrix 174
Exercise Set 4.3 181
4.4 Quotient Spaces 182
Exercise Set 4.4 186
CHAPTER 5 LINEAR MAPPINGS 187
5.1 Linear Mappings 187
Exercise Set 5.1 199
5.2 Matrices of Linear Mappings 200
Exercise Set 5.2 207
5.3 Systems of Linear Equations 209
Exercise Set 5.3 215
5.4 Eigenvectors and Eigenvalues 217
Exercise Set 5.4 223
CHAPTER 6 BILINEAR FORMS 226
6.1 Bilinear Forms 226
Exercise Set 6.1 234
6.2 Classical Forms 235
Exercise Set 6.2 247
6.3 Symmetric Forms over R 250
Exercise Set 6.3 257
6.4 Euclidean Spaces 259
Exercise Set 6.4 269
CHAPTER 7 RINGS 272
7.1 Rings, Subrings, and Examples 272
Exercise Set 7.1 287
7.2 Equivalence Relations 288
Exercise Set 7.2 295
7.3 Ideals and Quotient Rings 297
Exercise Set 7.3 303
7.4 Homomorphisms of Rings 303
Exercise Set 7.4 313
7.5 Rings of Polynomials and Formal Power
Series 315
Exercise Set 7.5 327
7.6 Rings of Multivariable Polynomials 328
Exercise Set 7.6 336
CHAPTER 8 GROUPS 338
8.1 Groups and Subgroups 338
Exercise Set 8.1 348
8.2 Examples of Groups and Subgroups 349
Exercise Set 8.2 358
8.3 Cosets 359
Exercise Set 8.3 364
8.4 Normal Subgroups and Factor Groups 365
Exercise Set 8.4 374
8.5 Homomorphisms of Groups 375
Exercise Set 8.5 382
CHAPTER 9 ARITHMETIC PROPERTIES OF RINGS 384
9.1 Extending Arithmetic to Commutative Rings 384
Exercise Set 9.1 399
9.2 Euclidean Rings 400
Exercise Set 9.2 404
9.3 Irreducible Polynomials 406
Exercise Set 9.3 415
9.4 Arithmetic Functions 416
Exercise Set 9.4 429
9.5 Congruences 430
Exercise Set 9.5 446
CHAPTER 10 THE REAL NUMBER SYSTEM 448
10.1 The Natural Numbers 448
10.2 The Integers 458
10.3 The Rationals 468
10.4 The Real Numbers 477
ANSWERS TO SELECTED EXERCISES 489
INDEX 513

Über Autor(innen)

MARTYN R. DIXON, PhD, is Professor in the Department ofMathematics at the University of Alabama, Tuscaloosa. He hasauthored more than sixty published journal articles on infinitegroup theory, formation theory and Fitting classes, wreathproducts, and automorphism groups.
LEONID A. KURDACHENKO, PhD, is Distinguished Professorand Chair of the Department of Algebra at the DnepropetrovskNational University (Ukraine). Dr. Kurdachenko has authored morethan 150 journal articles on the topics of infinite-dimensionallinear groups, infinite groups, and module theory.
IGOR YA. SUBBOTIN, PhD, is Professor in the Department ofMathematics and Natural Sciences at National University(California). Dr. Subbotin is the author of more than 100 publishedjournal articles on group theory, cybernetics, and mathematicseducation.