Algebra and Number Theory

An Integrated Approach

Inbunden, Engelska, 2010

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Beskrivning

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.

Produktinformation

Utgivningsdatum:2010-09-03
Mått:163 x 244 x 36 mm
Vikt:930 g
Format:Inbunden
Språk:Engelska
Antal sidor:544
Förlag:John Wiley & Sons Inc
ISBN:9780470496367

Utforska kategorier

Talteori inom Naturvetenskap och teknik
Algebra inom Naturvetenskap och teknik

Mer om författaren

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.

Recensioner i media

“The book is well-written and covers, with plenty of exercises, the material needed in the three aforementioned courses, albeit in a new order.” (Zentralblatt MATH, 1 December 2012)"However, instructors contemplating such a unified approach should give this book serious consideration. Recommended. Upper-division undergraduates through researchers/faulty." (Choice , 1 April 2011)

Innehållsförteckning

Preface ixChapter 1 Sets 11.1 Operations on Sets 1Exercise Set 1.1 61.2 Set Mappings 8Exercise Set 1.2 191.3 Products of Mappings 20Exercise Set 1.3 261.4 Some Properties of Integers 28Exercise Set 1.4 39Chapter 2 Matrices and Determinants 412.1 Operations on Matrices 41Exercise Set 2.1 522.2 Permutations of Finite Sets 54Exercise Set 2.2 642.3 Determinants of Matrices 66Exercise Set 2.3 772.4 Computing Determinants 79Exercise Set 2.4 912.5 Properties of the Product of Matrices 93Exercise Set 2.5 103Chapter 3 Fields 1053.1 Binary Algebraic Operations 105Exercise Set 3.1 1183.2 Basic Properties of Fields 119Exercise Set 3.2 1293.3 The Field of Complex Numbers 130Exercise Set 3.3 144Chapter 4 Vector Spaces 1454.1 Vector Spaces 146Exercise Set 4.1 1584.2 Dimension 159Exercise Set 4.2 1724.3 The Rank of a Matrix 174Exercise Set 4.3 1814.4 Quotient Spaces 182Exercise Set 4.4 186Chapter 5 Linear Mappings 1875.1 Linear Mappings 187Exercise Set 5.1 1995.2 Matrices of Linear Mappings 200Exercise Set 5.2 2075.3 Systems of Linear Equations 209Exercise Set 5.3 2155.4 Eigenvectors and Eigenvalues 217Exercise Set 5.4 223Chapter 6 Bilinear Forms 2266.1 Bilinear Forms 226Exercise Set 6.1 2346.2 Classical Forms 235Exercise Set 6.2 2476.3 Symmetric Forms over R 250Exercise Set 6.3 2576.4 Euclidean Spaces 259Exercise Set 6.4 269Chapter 7 Rings 2727.1 Rings, Subrings, and Examples 272Exercise Set 7.1 2877.2 Equivalence Relations 288Exercise Set 7.2 2957.3 Ideals and Quotient Rings 297Exercise Set 7.3 3037.4 Homomorphisms of Rings 303Exercise Set 7.4 3137.5 Rings of Polynomials and Formal Power Series 315Exercise Set 7.5 3277.6 Rings of Multivariable Polynomials 328Exercise Set 7.6 336Chapter 8 Groups 3388.1 Groups and Subgroups 338Exercise Set 8.1 3488.2 Examples of Groups and Subgroups 349Exercise Set 8.2 3588.3 Cosets 359Exercise Set 8.3 3648.4 Normal Subgroups and Factor Groups 365Exercise Set 8.4 3748.5 Homomorphisms of Groups 375Exercise Set 8.5 382Chapter 9 Arithmetic Properties of Rings 3849.1 Extending Arithmetic to Commutative Rings 384Exercise Set 9.1 3999.2 Euclidean Rings 400Exercise Set 9.2 4049.3 Irreducible Polynomials 406Exercise Set 9.3 4159.4 Arithmetic Functions 416Exercise Set 9.4 4299.5 Congruences 430Exercise Set 9.5 446Chapter 10 The Real Number System 44810.1 The Natural Numbers 44810.2 The Integers 45810.3 The Rationals 46810.4 The Real Numbers 477Answers to Selected Exercises 489Index 513