Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra 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.
Table of Contents:
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
