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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…mehr
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.
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.
Inhaltsangabe
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
Preface. Chapter 1. Sets. 1.1. Operations on Sets. Exercise Set 1.1. 1.2. Set mappings. Exercise Set 1.2. 1.3. Products of Mappings. Exercise Set 1.3. 1.4. Some properties of integers. Exercise Set 1.4. Chapter 2. Matrices and Determinants. 2.1. Operations on matrices. Exercise Set 2.1. 2.2. Permutations of finite sets. Exercise Set 2.2. 2.3. Determinants of matrices. Exercise Set 2.3. 2.4. Computing Determinants. Exercise Set 2.4. 2.5. Properties of the product of matrices. Exercise Set 2.5. Chapter 3. Fields. 3.1. Binary Algebraic Operations. Exercise Set 3.1. 3.2. Basic Properties of Fields. Exercise Set 3.2. 3.3. The Field of Complex Numbers. Exercise Set 3.3. Chapter 4. Vector Spaces. 4.1. Vector Spaces. Exercise Set 4.1. 4.2. Dimension. Exercise Set 4.2. 4.3. The Rank of a Matrix. Exercise Set 4.3. 4.4. Quotient Spaces. Exercise Set 4.4. Chapter 5. Linear Mappings. 5.1. Linear Mappings. Exercise Set 5.1. 5.2. Matrices of Linear Mappings. Exercise Set 5.2. 5.3. Systems of Linear Equations. Exercise Set 5.3. 5.4. Eigenvectors and eigenvalues. Exercise Set 5.4. Chapter 6. Bilinear Forms. 6.1. Bilinear Forms. Exercise Set 6.1. 6.2. Classical Forms. Exercise Set 6.2. 6.3. Symmetric forms over R. Exercise Set 6.3. 6.4. Euclidean Spaces. Exercise Set 6.4. Chapter 7. Rings. 7.1. Rings, Subrings and Examples. Exercise Set 7.1. 7.2. Equivalence Relations. Exercise Set 7.2. 7.3. Ideals and Quotient Rings. Exercise Set 7.3. 7.4. Homomorphisms of rings. Exercise Set 7.4. 7.5. Rings of polynomials and formal power series. Exercise Set 7.5. 7.6. Rings of multivariable polynomials. Exercise Set 7.6. Chapter 8. Groups. 8.1. Groups and Subgroups. Exercise Set 8.1. 8.2. Examples of Groups and Subgroups. Exercise Set 8.2. 8.3. Cosets. Exercise Set 8.3. 8.4. Normal subgroups and Factor groups. Exercise Set 8.4. 8.5. Homomorphisms of Groups. Exercise Set 8.5. Chapter 9. Arithmetic Properties of Rings. 9.1. Extending Arithmetic to Commutative Rings. Exercise Set 9.1. 9.2. Euclidean Rings. Exercise Set 9.2. 9.3. Irreducible Polynomials. Exercise Set 9.3. 9.4. Arithmetic Functions. Exercise Set 9.4. 9.5. Congruences. Exercise Set 9.5. Chapter 10. The Real Number System. 10.1. The Natural Numbers. 10.2. The Integers. 10.3. The Rationals. 10.4. The Real Numbers. Answers to selected exercises. Index.
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
Preface. Chapter 1. Sets. 1.1. Operations on Sets. Exercise Set 1.1. 1.2. Set mappings. Exercise Set 1.2. 1.3. Products of Mappings. Exercise Set 1.3. 1.4. Some properties of integers. Exercise Set 1.4. Chapter 2. Matrices and Determinants. 2.1. Operations on matrices. Exercise Set 2.1. 2.2. Permutations of finite sets. Exercise Set 2.2. 2.3. Determinants of matrices. Exercise Set 2.3. 2.4. Computing Determinants. Exercise Set 2.4. 2.5. Properties of the product of matrices. Exercise Set 2.5. Chapter 3. Fields. 3.1. Binary Algebraic Operations. Exercise Set 3.1. 3.2. Basic Properties of Fields. Exercise Set 3.2. 3.3. The Field of Complex Numbers. Exercise Set 3.3. Chapter 4. Vector Spaces. 4.1. Vector Spaces. Exercise Set 4.1. 4.2. Dimension. Exercise Set 4.2. 4.3. The Rank of a Matrix. Exercise Set 4.3. 4.4. Quotient Spaces. Exercise Set 4.4. Chapter 5. Linear Mappings. 5.1. Linear Mappings. Exercise Set 5.1. 5.2. Matrices of Linear Mappings. Exercise Set 5.2. 5.3. Systems of Linear Equations. Exercise Set 5.3. 5.4. Eigenvectors and eigenvalues. Exercise Set 5.4. Chapter 6. Bilinear Forms. 6.1. Bilinear Forms. Exercise Set 6.1. 6.2. Classical Forms. Exercise Set 6.2. 6.3. Symmetric forms over R. Exercise Set 6.3. 6.4. Euclidean Spaces. Exercise Set 6.4. Chapter 7. Rings. 7.1. Rings, Subrings and Examples. Exercise Set 7.1. 7.2. Equivalence Relations. Exercise Set 7.2. 7.3. Ideals and Quotient Rings. Exercise Set 7.3. 7.4. Homomorphisms of rings. Exercise Set 7.4. 7.5. Rings of polynomials and formal power series. Exercise Set 7.5. 7.6. Rings of multivariable polynomials. Exercise Set 7.6. Chapter 8. Groups. 8.1. Groups and Subgroups. Exercise Set 8.1. 8.2. Examples of Groups and Subgroups. Exercise Set 8.2. 8.3. Cosets. Exercise Set 8.3. 8.4. Normal subgroups and Factor groups. Exercise Set 8.4. 8.5. Homomorphisms of Groups. Exercise Set 8.5. Chapter 9. Arithmetic Properties of Rings. 9.1. Extending Arithmetic to Commutative Rings. Exercise Set 9.1. 9.2. Euclidean Rings. Exercise Set 9.2. 9.3. Irreducible Polynomials. Exercise Set 9.3. 9.4. Arithmetic Functions. Exercise Set 9.4. 9.5. Congruences. Exercise Set 9.5. Chapter 10. The Real Number System. 10.1. The Natural Numbers. 10.2. The Integers. 10.3. The Rationals. 10.4. The Real Numbers. Answers to selected exercises. Index.
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