Explore the main algebraic structures and number systems that play a central role across the field of mathematicsAlgebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines—linear algebra, abstract algebra, and number theory—into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers' retention of the main concepts.The book begins with an introduction to the elements of set theory. Next, the authors discuss matrices, determinants, and elements of field theory, including preliminary information related to integers and complex numbers. Subsequent chapters explore key ideas relating to linear algebra such as vector spaces, linear mapping, and bilinear forms. The book explores the development of the main ideas of algebraic structures and concludes with applications of algebraic ideas to number theory.Interesting applications are provided throughout to demonstrate the relevance of the discussed concepts. In addition, chapter exercises allow readers to test their comprehension of the presented material.Algebra and Number Theory is an excellent book for courses on linear algebra, abstract algebra, and number theory at the upper-undergraduate level. It is also a valuable reference for researchers working in different fields of mathematics, computer science, and engineering as well as for individuals preparing for a career in mathematics education.
Author(s): Martyn Dixon, Leonid Kurdachenko, Igor Subbotin
Edition: 1
Publisher: Wiley
Year: 2010
Language: English
Pages: 535
Tags: Математика;Общая алгебра;
CONTENTS......Page 6
PREFACE......Page 10
1.1 Operations on Sets ......Page 13
Exercise Set 1.1 ......Page 18
1.2 Set Mappings ......Page 20
Exercise Set 1.2 ......Page 31
1.3 Products of Mappings ......Page 32
Exercise Set 1.3 ......Page 38
1.4 Some Properties of Integers ......Page 40
Exercise Set 1.4 ......Page 51
2.1 Operations on Matrices ......Page 53
Exercise Set 2.1 ......Page 64
2.2 Permutations of Finite Sets ......Page 66
Exercise Set 2.2 ......Page 76
2.3 Determinants of Matrices ......Page 78
Exercise Set 2.3 ......Page 89
2.4 Computing Determinants ......Page 91
Exercise Set 2.4 ......Page 103
2.5 Properties of the Product of Matrices ......Page 105
Exercise Set 2.5 ......Page 115
3.1 Binary Algebraic Operations ......Page 117
Exercise Set 3 .1 ......Page 130
3.2 Basic Properties of Fields ......Page 131
Exercise Set 3.2 ......Page 141
3.3 The Field of Complex Numbers ......Page 142
Exercise Set 3.3 ......Page 156
CHAPTER 4 VECTOR SPACES ......Page 157
4.1 Vector Spaces ......Page 158
Exercise Set 4.1 ......Page 170
4.2 Dimension ......Page 171
Exercise Set 4.2 ......Page 184
4.3 The Rank of a Matrix ......Page 186
Exercise Set 4.3 ......Page 193
4.4 Quotient Spaces ......Page 194
Exercise Set 4.4 ......Page 198
5.1 Linear Mappings ......Page 199
Exercise Set 5 .1 ......Page 211
5.2 Matrices of Linear Mappings ......Page 212
Exercise Set 5.2 ......Page 219
5.3 Systems of Linear Equations ......Page 221
Exercise Set 5.3 ......Page 227
5.4 Eigenvectors and Eigenvalues ......Page 229
Exercise Set 5.4 ......Page 235
6.1 Bilinear Forms ......Page 238
Exercise Set 6.1 ......Page 246
6.2 Classical Forms ......Page 247
Exercise Set 6.2 ......Page 259
6.3 Symmetric Forms over ffi. ......Page 262
Exercise Set 6.3 ......Page 269
6.4 Euclidean Spaces ......Page 271
Exercise Set 6.4 ......Page 281
CHAPTER 7 RINGS ......Page 284
7.1 Rings ......Page 0
Exercise Set 7.1 ......Page 299
7.2 Equivalence Relations ......Page 300
Exercise Set 7.2 ......Page 307
7.3 Ideals and Quotient Rings ......Page 309
7.4 Homomorphisms of Rings ......Page 315
Exercise Set 7.4 ......Page 325
7.5 Rings of Polynomials and Formal Power Series ......Page 327
Exercise Set 7.5 ......Page 339
7.6 Rings of Multivariable Polynomials ......Page 340
Exercise Set 7.6 ......Page 348
8.1 Groups and Subgroups ......Page 350
Exercise Set 8.1 ......Page 360
8.2 Examples of Groups and Subgroups ......Page 361
Exercise Set 8.2 ......Page 370
8.3 Cosets ......Page 371
Exercise Set 8.3 ......Page 376
8.4 Normal Subgroups and Factor Groups ......Page 377
Exercise Set 8.4 ......Page 386
8.5 Homomorphisms of Groups ......Page 387
Exercise Set 8.5 ......Page 394
9.1 Extending Arithmetic to Commutative Rings ......Page 396
Exercise Set 9.1 ......Page 411
9.2 Euclidean Rings ......Page 412
Exercise Set 9.2 ......Page 416
9.3 Irreducible Polynomials ......Page 418
Exercise Set 9.3 ......Page 427
9.4 Arithmetic Functions ......Page 428
Exercise Set 9.4 ......Page 441
9.5 Congruences ......Page 442
Exercise Set 9.5 ......Page 458
10.1 The Natural Numbers ......Page 460
10.2 The Integers ......Page 470
10.3 The Rationals ......Page 480
10.4 The Real Numbers ......Page 489
ANSWERS TO SELECTED EXERCISES ......Page 501
INDEX ......Page 525
