{"product_id":"algebra-and-number-theory-isbn-9780470496367","title":"Algebra and Number Theory","description":"\u003cb\u003eExplore the main algebraic structures and number systems that play a central role across the field of mathematics\u003c\/b\u003e  \u003cp\u003eAlgebra 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, \u003ci\u003eAlgebra and Number Theory\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003eThe 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.\u003c\/p\u003e \u003cp\u003eInteresting 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eAlgebra and Number Theory\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003ePreface ix\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1 Sets 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Operations on Sets 1\u003c\/p\u003e \u003cp\u003eExercise Set 1.1 6\u003c\/p\u003e \u003cp\u003e1.2 Set Mappings 8\u003c\/p\u003e \u003cp\u003eExercise Set 1.2 19\u003c\/p\u003e \u003cp\u003e1.3 Products of Mappings 20\u003c\/p\u003e \u003cp\u003eExercise Set 1.3 26\u003c\/p\u003e \u003cp\u003e1.4 Some Properties of Integers 28\u003c\/p\u003e \u003cp\u003eExercise Set 1.4 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2 Matrices and Determinants 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Operations on Matrices 41\u003c\/p\u003e \u003cp\u003eExercise Set 2.1 52\u003c\/p\u003e \u003cp\u003e2.2 Permutations of Finite Sets 54\u003c\/p\u003e \u003cp\u003eExercise Set 2.2 64\u003c\/p\u003e \u003cp\u003e2.3 Determinants of Matrices 66\u003c\/p\u003e \u003cp\u003eExercise Set 2.3 77\u003c\/p\u003e \u003cp\u003e2.4 Computing Determinants 79\u003c\/p\u003e \u003cp\u003eExercise Set 2.4 91\u003c\/p\u003e \u003cp\u003e2.5 Properties of the Product of Matrices 93\u003c\/p\u003e \u003cp\u003eExercise Set 2.5 103\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3 Fields 105\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Binary Algebraic Operations 105\u003c\/p\u003e \u003cp\u003eExercise Set 3.1 118\u003c\/p\u003e \u003cp\u003e3.2 Basic Properties of Fields 119\u003c\/p\u003e \u003cp\u003eExercise Set 3.2 129\u003c\/p\u003e \u003cp\u003e3.3 The Field of Complex Numbers 130\u003c\/p\u003e \u003cp\u003eExercise Set 3.3 144\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4 Vector Spaces 145\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Vector Spaces 146\u003c\/p\u003e \u003cp\u003eExercise Set 4.1 158\u003c\/p\u003e \u003cp\u003e4.2 Dimension 159\u003c\/p\u003e \u003cp\u003eExercise Set 4.2 172\u003c\/p\u003e \u003cp\u003e4.3 The Rank of a Matrix 174\u003c\/p\u003e \u003cp\u003eExercise Set 4.3 181\u003c\/p\u003e \u003cp\u003e4.4 Quotient Spaces 182\u003c\/p\u003e \u003cp\u003eExercise Set 4.4 186\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5 Linear Mappings 187\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Linear Mappings 187\u003c\/p\u003e \u003cp\u003eExercise Set 5.1 199\u003c\/p\u003e \u003cp\u003e5.2 Matrices of Linear Mappings 200\u003c\/p\u003e \u003cp\u003eExercise Set 5.2 207\u003c\/p\u003e \u003cp\u003e5.3 Systems of Linear Equations 209\u003c\/p\u003e \u003cp\u003eExercise Set 5.3 215\u003c\/p\u003e \u003cp\u003e5.4 Eigenvectors and Eigenvalues 217\u003c\/p\u003e \u003cp\u003eExercise Set 5.4 223\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6 Bilinear Forms 226\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Bilinear Forms 226\u003c\/p\u003e \u003cp\u003eExercise Set 6.1 234\u003c\/p\u003e \u003cp\u003e6.2 Classical Forms 235\u003c\/p\u003e \u003cp\u003eExercise Set 6.2 247\u003c\/p\u003e \u003cp\u003e6.3 Symmetric Forms over R 250\u003c\/p\u003e \u003cp\u003eExercise Set 6.3 257\u003c\/p\u003e \u003cp\u003e6.4 Euclidean Spaces 259\u003c\/p\u003e \u003cp\u003eExercise Set 6.4 269\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 7 Rings 272\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Rings, Subrings, and Examples 272\u003c\/p\u003e \u003cp\u003eExercise Set 7.1 287\u003c\/p\u003e \u003cp\u003e7.2 Equivalence Relations 288\u003c\/p\u003e \u003cp\u003eExercise Set 7.2 295\u003c\/p\u003e \u003cp\u003e7.3 Ideals and Quotient Rings 297\u003c\/p\u003e \u003cp\u003eExercise Set 7.3 303\u003c\/p\u003e \u003cp\u003e7.4 Homomorphisms of Rings 303\u003c\/p\u003e \u003cp\u003eExercise Set 7.4 313\u003c\/p\u003e \u003cp\u003e7.5 Rings of Polynomials and Formal Power Series 315\u003c\/p\u003e \u003cp\u003eExercise Set 7.5 327\u003c\/p\u003e \u003cp\u003e7.6 Rings of Multivariable Polynomials 328\u003c\/p\u003e \u003cp\u003eExercise Set 7.6 336\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 8 Groups 338\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Groups and Subgroups 338\u003c\/p\u003e \u003cp\u003eExercise Set 8.1 348\u003c\/p\u003e \u003cp\u003e8.2 Examples of Groups and Subgroups 349\u003c\/p\u003e \u003cp\u003eExercise Set 8.2 358\u003c\/p\u003e \u003cp\u003e8.3 Cosets 359\u003c\/p\u003e \u003cp\u003eExercise Set 8.3 364\u003c\/p\u003e \u003cp\u003e8.4 Normal Subgroups and Factor Groups 365\u003c\/p\u003e \u003cp\u003eExercise Set 8.4 374\u003c\/p\u003e \u003cp\u003e8.5 Homomorphisms of Groups 375\u003c\/p\u003e \u003cp\u003eExercise Set 8.5 382\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 9 Arithmetic Properties of Rings 384\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Extending Arithmetic to Commutative Rings 384\u003c\/p\u003e \u003cp\u003eExercise Set 9.1 399\u003c\/p\u003e \u003cp\u003e9.2 Euclidean Rings 400\u003c\/p\u003e \u003cp\u003eExercise Set 9.2 404\u003c\/p\u003e \u003cp\u003e9.3 Irreducible Polynomials 406\u003c\/p\u003e \u003cp\u003eExercise Set 9.3 415\u003c\/p\u003e \u003cp\u003e9.4 Arithmetic Functions 416\u003c\/p\u003e \u003cp\u003eExercise Set 9.4 429\u003c\/p\u003e \u003cp\u003e9.5 Congruences 430\u003c\/p\u003e \u003cp\u003eExercise Set 9.5 446\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 10 The Real Number System 448\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 The Natural Numbers 448\u003c\/p\u003e \u003cp\u003e10.2 The Integers 458\u003c\/p\u003e \u003cp\u003e10.3 The Rationals 468\u003c\/p\u003e \u003cp\u003e10.4 The Real Numbers 477\u003c\/p\u003e \u003cp\u003eAnswers to Selected Exercises 489\u003c\/p\u003e \u003cp\u003eIndex 513\u003c\/p\u003e  \u003cp\u003e“The book is well-written and covers, with plenty of exercises, the material needed in the three aforementioned courses, albeit in a new order.”  (\u003ci\u003eZentralblatt MATH\u003c\/i\u003e, 1 December 2012)\u003c\/p\u003e \"However, instructors contemplating such a unified approach should give this book serious consideration. Recommended. Upper-division undergraduates through researchers\/faulty.\" (Choice , 1 April 2011)  \u003cp\u003e \u003c\/p\u003e  \u003cp\u003e\u003cb\u003eMARTYN R. DIXON, P\u003csmall\u003eH\u003c\/small\u003eD,\u003c\/b\u003e 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. \u003c\/p\u003e\u003cp\u003e\u003cb\u003eLEONID A. KURDACHENKO, P\u003csmall\u003eH\u003c\/small\u003eD,\u003c\/b\u003e 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. \u003c\/p\u003e\u003cp\u003e\u003cb\u003eIGOR YA. SUBBOTIN, P\u003csmall\u003eH\u003c\/small\u003eD,\u003c\/b\u003e 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.  \u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003e\u003cb\u003eExplore the main algebraic structures and number systems that play a central role across the field of mathematics\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003eAlgebra 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, \u003ci\u003eAlgebra and Number Theory\u003c\/i\u003e 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. \u003c\/p\u003e\u003cp\u003eThe 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. \u003c\/p\u003e\u003cp\u003eInteresting 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. \u003c\/p\u003e\u003cp\u003e\u003ci\u003eAlgebra and Number Theory\u003c\/i\u003e 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47988712243429,"sku":"NP9780470496367","price":179.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780470496367.jpg?v=1761781292","url":"https:\/\/k12savings.com\/products\/algebra-and-number-theory-isbn-9780470496367","provider":"K12savings","version":"1.0","type":"link"}