Algebra and Number Theory : An Integrated Approach

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  • Format: Hardcover
  • Copyright: 2010-09-27
  • Publisher: Wiley

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This book successfully blends algebra and number theory as an integrated discipline and consists of seven parts: Part 1 discusses the elements of set theory; Parts 2 and 3 address number systems; Parts 4 and 5 cover the main topics of linear algebra; Part 6 develops the main ideas of algebraic structures; and Part 7 demonstrates the applications of algebraic ideas to number theory. Based on the experience of the authors, this book was developed for one course that integrates three disciplines - linear algebra, abstract algebra, and number theory - in an effort to use time more efficiently. Many theorems in number theory have very simple proofs using algebraic tools, and most importantly, the book's integrated approach helps to build a deeper understanding of the subject for readers as well as improve their retention of knowledge. Applications are provided at the end of each chapter to further explain the results found in the book, and exercises are also ample throughout. While the book is mathematically self-contained, readers should be comfortable with mathematical formalism and have some experience in reading and writing mathematical proofs.

Author Biography

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.

Table of Contents

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