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Introduction to abstract algebra (4th Ed., 4th Edition)

Langue : Anglais

Auteur :

Couverture de l’ouvrage Introduction to abstract algebra

Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."*Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book"s unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.

The Fourth Edition features important concepts as well as specialized topics, including:

  • The treatment of nilpotent groups, including the Frattini and Fitting subgroups

  • Symmetric polynomials

  • The proof of the fundamental theorem of algebra using symmetric polynomials

  • The proof of Wedderburn"s theorem on finite division rings

  • The proof of the Wedderburn-Artin theorem

Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book"s exercises.

Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

Preface ix

Notations Used in the Text xix

A Sketch of the History of Algebra to 1929 xxii

0. Preliminaries 1

1

0.1 Proofs 1

0.2 Sets 5

0.3 Mappings 9

0.4 Equivalences 17

1. Integers and Permutations 23

1.1 Induction 23

1.2 Divisors and Prime Factorization 31

1.3 Integers modulo n 42

1.4 Permutations 53

1.5 An Application of Cryptography 65

2. Groups 67

2.1 Binary Operations 67

2.2 Groups 74

2.3 Subgroups 83

2.4 Cyclic Groups and the Order of an Element 88

2.5 Homomorphisms and Isomorphisms 97

2.6 Gosets and Lagrange's Theorem 107

2.7 Groups of Motions and Symmetries 116

2.8 Normal Subgroups 120

2.9 Factor Groups 129

2.10 The Isomorphism Theorem 135

2.11 An Application to Binary Linear Codes 141

3. Rings 157

3.1 Examples and Basic Properties 157

3.2 Integral Domains and Fields 168

3.3 Ideals and Factor Rings 177

3.4 Homomorphisms 186

3.5 Ordered Integral Domains 196

4. Polynomials 199

4.1 Polynomials 199

4.2 Factorization of Polynomials over a Field 211

4.3 Factor Rings of Polynomials over a Field 224

4.4 Partial Fractions 233

4.5 Symmetric Polynomials 246

4.6 Formal Construction of Polynomials 246

5. Factorization in Integral Domains 249

5.1 Irreducibles and Unique Factorization 250

5.2 Principal Ideal Domains 262

6. Field 273

6.1 Vector Spaces 274

6.2 Algebraic Extensions 282

6.3 Splitting Fields 290

6.4 Finite Fields 297

6.5 Geometric Constructions 303

6.6 The Fundamental Theorem of Algebra 308

6.7 An Application to Cyclic and BCH Codes 310

7. Modules over Principal Ideal Domains 323

7.1 Modules 323

7.2 Modules over a PID 334

8. p-Groups and the Sylow Theorems 349

8.1 Factors and Products 349

8.2 Cauchy's Theorem and p-Groups 357

8.3 Group Actions 364

8.4 The Sylow Theorems 372

8.5 Semidirect Products 379

8.6 An Application to Combinatorics 383

9. Series of Subgroups 389

9.1 The Jordan-Hölder Theorem 390

9.2 Solvable Groups 395

9.3 Nilpotent Groups 402

10. Galois Theory 413

10.1 Galois Groups and Separability 414

10.2 The Main Theorem of Galois Theory 423

10.3 Insolvability of Polynomials 435

10.4 Cyclotomic Polynomials and Wedderburn's Theorem 443

11. Finiteness Conditions for Rings and Modules 449

11.1 Wedderburn's Theorem 449

11.2 The Wedderburn-Artin Theorem 459

Appendices

App. A Complex Numbers 473

App. B Matrix Algebra 480

App. C Zorn's Lemma 488

App. D Proof of the Recursion Theorem 492

Bibliography 595

Selected Answers 597

Index 499

Date de parution :

Ouvrage de 560 p.

18.2x26 cm

Disponible chez l'éditeur (délai d'approvisionnement : 12 jours).

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