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### Summary

### Summary

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.

### Author Notes

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.

### Reviews 1

### Choice Review

The title of this work may be misleading; it is designed to integrate topics in algebra, linear algebra, and elementary number theory. As such, its appeal is the integration. There are dedicated books in each area that are better, but there are virtually none that integrate the three. Dixon (Univ. of Alabama, Tuscaloosa), Kurdachenko (Dnepropetrovsk National Univ., Ukraine), and Subbotin (National Univ., California) write well and include ample, appropriate examples and exercises. Viewing this book as an integrated resource, it would be difficult to pick and choose topics for a course. The first six chapters are devoted mainly to linear algebra. The remaining chapters are "Rings," "Groups," "Arithmetic Properties of Rings" (elementary number theory in a commutative ring setting), and "The Real Number System." The book is aimed at students bound for graduate work in mathematics, so the intended audience is a talented group. It would be an ambitious one-year course for "normal" undergraduates; falling short of completion, it misses some key topics--an inherent risk of an integrated program. However, instructors contemplating such a unified approach should give this book serious consideration. Summing Up; Recommended. Upper-division undergraduates through researchers/faculty. J. R. Burke Gonzaga University

### Table of Contents

Preface | p. ix |

Chapter 1 Sets | p. 1 |

1.1 Operations on Sets | p. 1 |

Exercise Set 1.1 | p. 6 |

1.2 Set Mappings | p. 8 |

Exercise Set 1.2 | p. 19 |

1.3 Products of Mappings | p. 20 |

Exercise Set 1.3 | p. 26 |

1.4 Some Properties of Integers | p. 28 |

Exercise Set 1.4 | p. 39 |

Chapter 2 Matrices and Determinants | p. 41 |

2.1 Operations on Matrices | p. 41 |

Exercise Set 2.1 | p. 52 |

2.2 Permutations of Finite Sets | p. 54 |

Exercise Set 2.2 | p. 64 |

2.3 Determinants of Matrices | p. 66 |

Exercise Set 2.3 | p. 77 |

2.4 Computing Determinants | p. 79 |

Exercise Set 2.4 | p. 91 |

2.5 Properties of the Product of Matrices | p. 93 |

Exercise Set 2.5 | p. 103 |

Chapter 3 Fields | p. 105 |

3.1 Binary Algebraic Operations | p. 105 |

Exercise Set 3.1 | p. 118 |

3.2 Basic Properties of Fields | p. 119 |

Exercise Set 3.2 | p. 129 |

3.3 The Field of Complex Numbers | p. 130 |

Exercise Set 3.3 | p. 144 |

Chapter 4 Vector Spaces | p. 145 |

4.1 Vector Spaces | p. 146 |

Exercise Set 4.1 | p. 158 |

4.2 Dimension | p. 159 |

Exercise Set 4.2 | p. 172 |

4.3 The Rank of a Matrix | p. 174 |

Exercise Set 4.3 | p. 181 |

4.4 Quotient Spaces | p. 182 |

Exercise Set 4.4 | p. 186 |

Chapter 5 Linear Mappings | p. 187 |

5.1 Linear Mappings | p. 187 |

Exercise Set 5.1 | p. 199 |

5.2 Matrices of Linear Mappings | p. 200 |

Exercise Set 5.2 | p. 207 |

5.3 Systems of Linear Equations | p. 209 |

Exercise Set 5.3 | p. 215 |

5.4 Eigenvectors and Eigenvalues | p. 217 |

Exercise Set 5.4 | p. 223 |

Chapter 6 Bilinear Forms | p. 226 |

6.1 Bilinear Forms | p. 226 |

Exercise Set 6.1 | p. 234 |

6.2 Classical Forms | p. 235 |

Exercise Set 6.2 | p. 247 |

6.3 Symmetric Forms over R | p. 250 |

Exercise Set 6.3 | p. 257 |

6.4 Euclidean Spaces | p. 259 |

Exercise Set 6.4 | p. 269 |

Chapter 7 Rings | p. 272 |

7.1 Rings, Subrings, and Examples | p. 272 |

Exercise Set 7.1 | p. 287 |

7.2 Equivalence Relations | p. 288 |

Exercise Set 7.2 | p. 295 |

7.3 Ideals and Quotient Rings | p. 297 |

Exercise Set 7.3 | p. 303 |

7.4 Homomorphisms of Rings | p. 303 |

Exercise Set 7.4 | p. 313 |

7.5 Rings of Polynomials and Formal Power Series | p. 315 |

Exercise Set 7.5 | p. 327 |

7.6 Rings of Multivariable Polynomials | p. 328 |

Exercise Set 7.6 | p. 336 |

Chapter 8 Groups | p. 338 |

8.1 Groups and Subgroups | p. 338 |

Exercise Set 8.1 | p. 348 |

8.2 Examples of Groups and Subgroups | p. 349 |

Exercise Set 8.2 | p. 358 |

8.3 Cosets | p. 359 |

Exercise Set 8.3 | p. 364 |

8.4 Normal Subgroups and Factor Groups | p. 365 |

Exercise Set 8.4 | p. 374 |

8.5 Homomorphisms of Groups | p. 375 |

Exercise Set 8.5 | p. 382 |

Chapter 9 Arithmetic Properties of Rings | p. 384 |

9.1 Extending Arithmetic to Commutative Rings | p. 384 |

Exercise Set 9.1 | p. 399 |

9.2 Euclidean Rings | p. 400 |

Exercise Set 9.2 | p. 404 |

9.3 Irreducible Polynomials | p. 406 |

Exercise Set 9.3 | p. 415 |

9.4 Arithmetic Functions | p. 416 |

Exercise Set 9.4 | p. 429 |

9.5 Congruences | p. 430 |

Exercise Set 9.5 | p. 446 |

Chapter 10 The Real Number System | p. 448 |

10.1 The Natural Numbers | p. 448 |

10.2 The Integers | p. 458 |

10.3 The Rationals | p. 468 |

10.4 The Real Numbers | p. 477 |

Answers to Selected Exercises | p. 489 |

Index | p. 513 |