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Abstract Algebra: An Interactive Approach [Kõva köide]

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, (Arkansas State University, Jonesboro, USA)
  • Formaat: Hardback, 560 pages, kõrgus x laius: 235x156 mm, kaal: 930 g, Over 300; 104 Tables, black and white; 34 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 01-Jul-2009
  • Kirjastus: CRC Press Inc
  • ISBN-10: 1420094521
  • ISBN-13: 9781420094527
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Abstract Algebra: An Interactive ApproachSuurem pilt
  • Formaat: Hardback, 560 pages, kõrgus x laius: 235x156 mm, kaal: 930 g, Over 300; 104 Tables, black and white; 34 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 01-Jul-2009
  • Kirjastus: CRC Press Inc
  • ISBN-10: 1420094521
  • ISBN-13: 9781420094527
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781420094527.html
Märksõnad:
By integrating the use of GAP and Mathematica®, Abstract Algebra: An Interactive Approach presents a hands-on approach to learning about groups, rings, and fields. Each chapter includes both GAP and Mathematica commands, corresponding Mathematica notebooks, traditional exercises, and several interactive computer problems that utilize GAP and Mathematica to explore groups and rings.









Although the book gives the option to use technology in the classroom, it does not sacrifice mathematical rigor. It covers classical proofs, such as Abels theorem, as well as many graduate-level topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubiks Cube®-like puzzles, and Wedderburns theorem. He also incorporates problem sequences that allow students to delve into interesting topics in depth, including Fermats two square theorem.









This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area.

Arvustused

"The textbook gives an introduction to algebra. The course includes the explanation on how to use the computer algebra systems GAP and Mathematica The book can be used for an undergraduate-level course (chapter 1-4 and 9-12) or a second semester graduate-level course." Gerhard Pfister, Zentralblatt MATH 1173

List of Figures ix
List of Tables xi
Preface xiii
Acknowledgments xv
About the Author xvii
Symbol Description xix
Mathematica® vs. GAP xxiii
1 Understanding the Group Concept 1
1.1 Introduction to Groups
1
1.2 Modular Arithmetic
5
1.3 Prime Factorizations
10
1.4 The Definition of a Group
15
Problems for
Chapter 1
21
2 The Structure within a Group 27
2.1 Generators of Groups
27
2.2 Defining Finite Groups in Mathematica and GAP
31
2.3 Subgroups
38
Problems for
Chapter 2
48
3 Patterns within the Cosets of Groups 53
3.1 Left and Right Cosets
53
3.2 How to Write a Secret Message
58
3.3 Normal Subgroups
66
3.4 Quotient Groups
71
Problems for
Chapter 3
74
4 Mappings between Groups 79
4.1 Isomorphisms
79
4.2 Homomorphisms
86
4.3 The Three Isomorphism Theorems
93
Problems for
Chapter 4
103
5 Permutation Groups 107
5.1 Symmetric Groups
107
5.2 Cycles
111
5.3 Cayley's Theorem
121
5.4 Numbering the Permutations
127
Problems for
Chapter 5
130
6 Building Larger Groups from Smaller Groups 135
6.1 The Direct Product
135
6.2 The Fundamental Theorem of Finite Abelian Groups
141
6.3 Automorphisms
151
6.4 Semi-Direct Products
161
Problems for
Chapter 6
171
7 The Search for Normal Subgroups 175
7.1 The Center of a Group
175
7.2 The Normalizer and Normal Closure Subgroups
179
7.3 Conjugacy Classes and Simple Groups
183
7.4 The Class Equation and Sylow's Theorems
190
Problems for
Chapter 7
203
8 Solvable and Insoluble Groups 209
8.1 Subnormal Series and the Jordan-Holder Theorem
209
8.2 Derived Group Series
217
8.3 Polycyclic Groups
224
8.4 Solving the PyraminxTm
232
Problems for
Chapter 8
239
9 Introduction to Rings 245
9.1 Groups with an Additional Operation
245
9.2 The Definition of a Ring
252
9.3 Entering Finite Rings into GAP and Mathematica
256
9.4 Some Properties of Rings
264
Problems for
Chapter 9
269
10 The Structure within Rings 273
10.1 Subrings
273
10.2 Quotient Rings and Ideals
277
10.3 Ring Isomorphisms
284
10.4 Homomorphisms and Kernels
292
Problems for
Chapter 10
302
11 Integral Domains and Fields 309
11.1 Polynomial Rings
309
11.2 The Field of Quotients
318
11.3 Complex Numbers
324
11.4 Ordered Commutative Rings
338
Problems for
Chapter 11
345
12 Unique Factorization 351
12.1 Factorization of Polynomials
351
12.2 Unique Factorization Domains
362
12.3 Principal Ideal Domains
373
12.4 Euclidean Domains
379
Problems for
Chapter 12
385
13 Finite Division Rings 391
13.1 Entering Finite Fields in Mathematica or GAP
391
13.2 Properties of Finite Fields
396
13.3 Cyclotomic Polynomials
405
13.4 Finite Skew Fields
417
Problems for
Chapter 13
423
14 The Theory of Fields 429
14.1 Vector Spaces
429
14.2 Extension Fields
436
14.3 Splitting Fields
444
Problems for
Chapter 14
455
15 Galois Theory 459
15.1 The Galois Group of an Extension Field
459
15.2 The Galois Group of a Polynomial in Q
468
15.3 The Fundamental Theorem of Galois Theory
479
15.4 Solutions of Polynomial Equations Using Radicals
486
Problems for
Chapter 15
491
Answers to Odd-Numbered Problems 497
Bibliography 517
Index 519
William Paulsen is a Professor of Mathematics at Arkansas State University.