Cover image for Algebra and number theory : an integrated approach
Title:
Algebra and number theory : an integrated approach
Publication Information:
Hoboken, N.J. : Wiley, c2010
Physical Description:
xi, 523 p. ; 25 cm.
ISBN:
9780470496367
General Note:
Includes index
Subject Term:

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30000010263482 QA241 D59 2010 Open Access Book Book
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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

Prefacep. ix
Chapter 1 Setsp. 1
1.1 Operations on Setsp. 1
Exercise Set 1.1p. 6
1.2 Set Mappingsp. 8
Exercise Set 1.2p. 19
1.3 Products of Mappingsp. 20
Exercise Set 1.3p. 26
1.4 Some Properties of Integersp. 28
Exercise Set 1.4p. 39
Chapter 2 Matrices and Determinantsp. 41
2.1 Operations on Matricesp. 41
Exercise Set 2.1p. 52
2.2 Permutations of Finite Setsp. 54
Exercise Set 2.2p. 64
2.3 Determinants of Matricesp. 66
Exercise Set 2.3p. 77
2.4 Computing Determinantsp. 79
Exercise Set 2.4p. 91
2.5 Properties of the Product of Matricesp. 93
Exercise Set 2.5p. 103
Chapter 3 Fieldsp. 105
3.1 Binary Algebraic Operationsp. 105
Exercise Set 3.1p. 118
3.2 Basic Properties of Fieldsp. 119
Exercise Set 3.2p. 129
3.3 The Field of Complex Numbersp. 130
Exercise Set 3.3p. 144
Chapter 4 Vector Spacesp. 145
4.1 Vector Spacesp. 146
Exercise Set 4.1p. 158
4.2 Dimensionp. 159
Exercise Set 4.2p. 172
4.3 The Rank of a Matrixp. 174
Exercise Set 4.3p. 181
4.4 Quotient Spacesp. 182
Exercise Set 4.4p. 186
Chapter 5 Linear Mappingsp. 187
5.1 Linear Mappingsp. 187
Exercise Set 5.1p. 199
5.2 Matrices of Linear Mappingsp. 200
Exercise Set 5.2p. 207
5.3 Systems of Linear Equationsp. 209
Exercise Set 5.3p. 215
5.4 Eigenvectors and Eigenvaluesp. 217
Exercise Set 5.4p. 223
Chapter 6 Bilinear Formsp. 226
6.1 Bilinear Formsp. 226
Exercise Set 6.1p. 234
6.2 Classical Formsp. 235
Exercise Set 6.2p. 247
6.3 Symmetric Forms over Rp. 250
Exercise Set 6.3p. 257
6.4 Euclidean Spacesp. 259
Exercise Set 6.4p. 269
Chapter 7 Ringsp. 272
7.1 Rings, Subrings, and Examplesp. 272
Exercise Set 7.1p. 287
7.2 Equivalence Relationsp. 288
Exercise Set 7.2p. 295
7.3 Ideals and Quotient Ringsp. 297
Exercise Set 7.3p. 303
7.4 Homomorphisms of Ringsp. 303
Exercise Set 7.4p. 313
7.5 Rings of Polynomials and Formal Power Seriesp. 315
Exercise Set 7.5p. 327
7.6 Rings of Multivariable Polynomialsp. 328
Exercise Set 7.6p. 336
Chapter 8 Groupsp. 338
8.1 Groups and Subgroupsp. 338
Exercise Set 8.1p. 348
8.2 Examples of Groups and Subgroupsp. 349
Exercise Set 8.2p. 358
8.3 Cosetsp. 359
Exercise Set 8.3p. 364
8.4 Normal Subgroups and Factor Groupsp. 365
Exercise Set 8.4p. 374
8.5 Homomorphisms of Groupsp. 375
Exercise Set 8.5p. 382
Chapter 9 Arithmetic Properties of Ringsp. 384
9.1 Extending Arithmetic to Commutative Ringsp. 384
Exercise Set 9.1p. 399
9.2 Euclidean Ringsp. 400
Exercise Set 9.2p. 404
9.3 Irreducible Polynomialsp. 406
Exercise Set 9.3p. 415
9.4 Arithmetic Functionsp. 416
Exercise Set 9.4p. 429
9.5 Congruencesp. 430
Exercise Set 9.5p. 446
Chapter 10 The Real Number Systemp. 448
10.1 The Natural Numbersp. 448
10.2 The Integersp. 458
10.3 The Rationalsp. 468
10.4 The Real Numbersp. 477
Answers to Selected Exercisesp. 489
Indexp. 513