Contemporary Abstract Algebra

Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students. Front Cover......Page 1 Title Page......Page 2 Copyright......Page 3 Contents......Page 4 Preface......Page 12 PART 1: Integers and Equivalence Relations......Page 16 Properties of Integers......Page 18 Modular Arithmetic......Page 22 Mathematical Induction......Page 27 Equivalence Relations......Page 30 Functions (Mappings)......Page 33 Exercises......Page 36 Computer Exercises......Page 40 PART 2: Groups......Page 42 Symmetries of a Square......Page 44 The Dihedral Groups......Page 47 Exercises......Page 50 Biography of Niels Abel......Page 54 Definition and Examples of Groups......Page 55 Elementary Properties of Groups......Page 63 Historical Note......Page 66 Exercises......Page 67 Computer Exercises......Page 70 Terminology and Notation......Page 72 Subgroup Tests......Page 73 Examples of Subgroups......Page 76 Exercises......Page 79 Computer Exercises......Page 85 Properties of Cyclic Groups......Page 87 Classification of Subgroups of Cyclic Groups......Page 92 Exercises......Page 96 Computer Exercises......Page 101 Biography of J. J. Sylvester......Page 104 Supplementary Exercises for Chapters 1–4......Page 106 Definition and Notation......Page 110 Cycle Notation......Page 113 Properties of Permutations......Page 115 Exercises......Page 128 Computer Exercises......Page 133 Biography of Augustin Cauchy......Page 136 Definition and Examples......Page 137 Cayley’s Theorem......Page 141 Automorphisms......Page 144 Exercises......Page 148 Computer Exercise......Page 151 Biography of Arthur Cayley......Page 152 Properties of Cosets......Page 153 Lagrange’s Theorem and Consequences......Page 156 An Application of Cosets to Permutation Groups......Page 160 Exercises......Page 164 Computer Exercise......Page 168 Biography of Joseph Lagrange......Page 169 Definition and Examples......Page 170 Properties of External Direct Products......Page 171 The Group of Units Modulo n as an External Direct Product......Page 174 Applications......Page 176 Exercises......Page 182 Computer Exercises......Page 185 Biography of Leonard Adleman......Page 188 Supplementary Exercises for Chapters 5–8......Page 189 Normal Subgroups......Page 193 Factor Groups......Page 195 Applications of Factor Groups......Page 200 Internal Direct Products......Page 203 Exercises......Page 208 Biography of Évariste Galois......Page 214 Definition and Examples......Page 215 Properties of Homomorphisms......Page 217 The First Isomorphism Theorem......Page 221 Exercises......Page 226 Computer Exercise......Page 231 Biography of Camille Jordan......Page 232 The Isomorphism Classes of Abelian Groups......Page 233 Proof of the Fundamental Theorem......Page 238 Exercises......Page 241 Computer Exercises......Page 243 Supplementary Exercises for Chapters 9–11......Page 245 PART 3: Rings......Page 250 Motivation and Definition......Page 252 Examples of Rings......Page 253 Properties of Rings......Page 254 Subrings......Page 255 Exercises......Page 257 Computer Exercises......Page 260 Biography of I. N. Herstein......Page 263 Definition and Examples......Page 264 Fields......Page 265 Characteristic of a Ring......Page 267 Exercises......Page 270 Computer Exercises......Page 274 Biography of Nathan Jacobson......Page 276 Ideals......Page 277 Factor Rings......Page 278 Prime Ideals and Maximal Ideals......Page 282 Exercises......Page 284 Computer Exercises......Page 288 Biography of Richard Dedekind......Page 289 Biography of Emmy Noether......Page 290 Supplementary Exercises for Chapters 12–14......Page 291 Definition and Examples......Page 295 Properties of Ring Homomorphisms......Page 298 The Field of Quotients......Page 300 Exercises......Page 302 Notation and Terminology......Page 308 The Division Algorithm and Consequences......Page 311 Exercises......Page 315 Biography of Saunders Mac Lane......Page 319 Reducibility Tests......Page 320 Irreducibility Tests......Page 323 Unique Factorization in Z[x]......Page 328 Factorization......Page 329 Exercises......Page 331 Computer Exercises......Page 334 Biography of Serge Lang......Page 336 Irreducibles, Primes......Page 337 Historical Discussion of Fermat’s Last Theorem......Page 340 Unique Factorization Domains......Page 343 Euclidean Domains......Page 346 Exercises......Page 350 Computer Exercise......Page 352 Biography of Sophie Germain......Page 354 Biography of Andrew Wiles......Page 355 Supplementary Exercises for Chapters 15–18......Page 356 PART 4: Fields......Page 358 Definition and Examples......Page 360 Subspaces......Page 361 Linear Independence......Page 362 Exercises......Page 364 Biography of Emil Artin......Page 367 Biography of Olga Taussky-Todd......Page 368 The Fundamental Theorem of Field Theory......Page 369 Splitting Fields......Page 371 Zeros of an Irreducible Polynomial......Page 377 Exercises......Page 381 Biography of Leopold Kronecker......Page 384 Characterization of Extensions......Page 385 Finite Extensions......Page 387 Properties of Algebraic Extensions......Page 391 Exercises......Page 393 Biography of Irving Kaplansky......Page 396 Classification of Finite Fields......Page 397 Structure of Finite Fields......Page 398 Subfields of a Finite Field......Page 402 Exercises......Page 404 Computer Exercises......Page 406 Biography of L. E. Dickson......Page 407 Historical Discussion of Geometric Constructions......Page 408 Constructible Numbers......Page 409 Exercises......Page 411 Supplementary Exercises for Chapters 19–23......Page 414 PART 5: Special Topics......Page 416 Conjugacy Classes......Page 418 The Class Equation......Page 419 The Probability That Two Elements Commute......Page 420 The Sylow Theorems......Page 421 Applications of Sylow Theorems......Page 426 Computer Exercise......Page 433 Biography of Ludwig Sylow......Page 434 Historical Background......Page 435 Nonsimplicity Tests......Page 440 The Simplicity of A5......Page 444 The Cole Prize......Page 445 Exercises......Page 446 Computer Exercises......Page 447 Biography of Michael Aschbacher......Page 449 Biography of Daniel Gorenstein......Page 450 Biography of John Thompson......Page 451 Motivation......Page 452 Definitions and Notation......Page 453 Free Group......Page 454 Generators and Relations......Page 455 Classification of Groups of Order Up to 15......Page 459 Characterization of Dihedral Groups......Page 461 Realizing the Dihedral Groups with Mirrors......Page 462 Exercises......Page 464 Biography of Marshall Hall, Jr.......Page 467 Isometries......Page 468 Classification of Finite Plane Symmetry Groups......Page 470 Classification of Finite Groups of Rotations in R3......Page 471 Exercises......Page 473 The Frieze Groups......Page 476 The Crystallographic Groups......Page 482 Identification of Plane Periodic Patterns......Page 488 Exercises......Page 494 Biography of M. C. Escher......Page 499 Biography of George Pólya......Page 500 Biography of John H. Conway......Page 501 Motivation......Page 502 Burnside’s Theorem......Page 503 Applications......Page 505 Group Action......Page 508 Exercises......Page 509 Biography of William Burnside......Page 512 The Cayley Digraph of a Group......Page 513 Hamiltonian Circuits and Paths......Page 517 Some Applications......Page 523 Exercises......Page 526 Biography of William Rowan Hamilton......Page 531 Biography of Paul Erdös......Page 532 Motivation......Page 533 Linear Codes......Page 538 Parity-Check Matrix Decoding......Page 543 Coset Decoding......Page 546 Historical Note: The Ubiquitous Reed-Solomon Codes......Page 550 Exercises......Page 552 Biography of Richard W. Hamming......Page 557 Biography of Jessie MacWilliams......Page 558 Biography of Vera Pless......Page 559 Fundamental Theorem of Galois Theory......Page 560 Solvability of Polynomials by Radicals......Page 567 Insolvability of a Quintic......Page 571 Exercises......Page 572 Biography of Philip Hall......Page 575 Motivation......Page 576 Cyclotomic Polynomials......Page 577 The Constructible Regular n-gons......Page 581 Exercises......Page 583 Computer Exercise......Page 584 Biography of Carl Friedrich Gauss......Page 585 Supplementary Exercises for Chapters 24–33......Page 586 Selected Answers......Page 590 Text Credits......Page 629 Photo Credits......Page 631 Index of Mathematicians......Page 632 Index of Terms......Page 634 Front Cover 1 Title Page 2 Copyright 3 Contents 4 Preface 12 PART 1: Integers and Equivalence Relations 16 0 Preliminaries 18 Properties of Integers 18 Modular Arithmetic 22 Mathematical Induction 27 Equivalence Relations 30 Functions (Mappings) 33 Exercises 36 Computer Exercises 40 PART 2: Groups 42 1 Introduction to Groups 44 Symmetries of a Square 44 The Dihedral Groups 47 Exercises 50 Biography of Niels Abel 54 2 Groups 55 Definition and Examples of Groups 55 Elementary Properties of Groups 63 Historical Note 66 Exercises 67 Computer Exercises 70 3 Finite Groups; Subgroups 72 Terminology and Notation 72 Subgroup Tests 73 Examples of Subgroups 76 Exercises 79 Computer Exercises 85 4 Cyclic Groups 87 Properties of Cyclic Groups 87 Classification of Subgroups of Cyclic Groups 92 Exercises 96 Computer Exercises 101 Biography of J. J. Sylvester 104 Supplementary Exercises for Chapters 1–4 106 5 Permutation Groups 110 Definition and Notation 110 Cycle Notation 113 Properties of Permutations 115 Exercises 128 Computer Exercises 133 Biography of Augustin Cauchy 136 6 Isomorphisms 137 Motivation 137 Definition and Examples 137 Cayley’s Theorem 141 Automorphisms 144 Exercises 148 Computer Exercise 151 Biography of Arthur Cayley 152 7 Cosets and Lagrange’s Theorem 153 Properties of Cosets 153 Lagrange’s Theorem and Consequences 156 An Application of Cosets to Permutation Groups 160 Exercises 164 Computer Exercise 168 Biography of Joseph Lagrange 169 8 External Direct Products 170 Definition and Examples 170 Properties of External Direct Products 171 The Group of Units Modulo n as an External Direct Product 174 Applications 176 Exercises 182 Computer Exercises 185 Biography of Leonard Adleman 188 Supplementary Exercises for Chapters 5–8 189 9 Normal Subgroups and Factor Groups 193 Normal Subgroups 193 Factor Groups 195 Applications of Factor Groups 200 Internal Direct Products 203 Exercises 208 Biography of Évariste Galois 214 10 Group Homomorphisms 215 Definition and Examples 215 Properties of Homomorphisms 217 The First Isomorphism Theorem 221 Exercises 226 Computer Exercise 231 Biography of Camille Jordan 232 11 Fundamental Theorem of Finite Abelian Groups 233 The Fundamental Theorem 233 The Isomorphism Classes of Abelian Groups 233 Proof of the Fundamental Theorem 238 Exercises 241 Computer Exercises 243 Supplementary Exercises for Chapters 9–11 245 PART 3: Rings 250 12 Introduction to Rings 237 252 Motivation and Definition 252 Examples of Rings 253 Properties of Rings 254 Subrings 255 Exercises 257 Computer Exercises 260 Biography of I. N. Herstein 263 13 Integral Domains 264 Definition and Examples 264 Fields 265 Characteristic of a Ring 267 Exercises 270 Computer Exercises 274 Biography of Nathan Jacobson 276 14 Ideals and Factor Rings 277 Ideals 277 Factor Rings 278 Prime Ideals and Maximal Ideals 282 Exercises 284 Computer Exercises 288 Biography of Richard Dedekind 289 Biography of Emmy Noether 290 Supplementary Exercises for Chapters 12–14 291 15 Ring Homomorphisms 295 Definition and Examples 295 Properties of Ring Homomorphisms 298 The Field of Quotients 300 Exercises 302 16 Polynomial Rings 308 Notation and Terminology 308 The Division Algorithm and Consequences 311 Exercises 315 Biography of Saunders Mac Lane 319 17 Factorization of Polynomials 320 Reducibility Tests 320 Irreducibility Tests 323 Unique Factorization in Z[x] 328 Factorization 329 Exercises 331 Computer Exercises 334 Biography of Serge Lang 336 18 Divisibility in Integral Domains 337 Irreducibles, Primes 337 Historical Discussion of Fermat’s Last Theorem 340 Unique Factorization Domains 343 Euclidean Domains 346 Exercises 350 Computer Exercise 352 Biography of Sophie Germain 354 Biography of Andrew Wiles 355 Supplementary Exercises for Chapters 15–18 356 PART 4: Fields 358 19 Vector Spaces 360 Definition and Examples 360 Subspaces 361 Linear Independence 362 Exercises 364 Biography of Emil Artin 367 Biography of Olga Taussky-Todd 368 20 Extension Fields 369 The Fundamental Theorem of Field Theory 369 Splitting Fields 371 Zeros of an Irreducible Polynomial 377 Exercises 381 Biography of Leopold Kronecker 384 21 Algebraic Extensions 385 Characterization of Extensions 385 Finite Extensions 387 Properties of Algebraic Extensions 391 Exercises 393 Biography of Irving Kaplansky 396 22 Finite Fields 397 Classification of Finite Fields 397 Structure of Finite Fields 398 Subfields of a Finite Field 402 Exercises 404 Computer Exercises 406 Biography of L. E. Dickson 407 23 Geometric Constructions 408 Historical Discussion of Geometric Constructions 408 Constructible Numbers 409 Angle-Trisectors and Circle-Squarers 411 Exercises 411 Supplementary Exercises for Chapters 19–23 414 PART 5: Special Topics 416 24 Sylow Theorems 418 Conjugacy Classes 418 The Class Equation 419 The Probability That Two Elements Commute 420 The Sylow Theorems 421 Applications of Sylow Theorems 426 Computer Exercise 433 Biography of Ludwig Sylow 434 25 Finite Simple Groups 435 Historical Background 435 Nonsimplicity Tests 440 The Simplicity of A5 444 The Fields Medal 445 The Cole Prize 445 Exercises 446 Computer Exercises 447 Biography of Michael Aschbacher 449 Biography of Daniel Gorenstein 450 Biography of John Thompson 451 26 Generators and Relations 452 Motivation 452 Definitions and Notation 453 Free Group 454 Generators and Relations 455 Classification of Groups of Order Up to 15 459 Characterization of Dihedral Groups 461 Realizing the Dihedral Groups with Mirrors 462 Exercises 464 Biography of Marshall Hall, Jr. 467 27 Symmetry Groups 468 Isometries 468 Classification of Finite Plane Symmetry Groups 470 Classification of Finite Groups of Rotations in R3 471 Exercises 473 28 Frieze Groups and Crystallographic Groups 476 The Frieze Groups 476 The Crystallographic Groups 482 Identification of Plane Periodic Patterns 488 Exercises 494 Biography of M. C. Escher 499 Biography of George Pólya 500 Biography of John H. Conway 501 29 Symmetry and Counting 502 Motivation 502 Burnside’s Theorem 503 Applications 505 Group Action 508 Exercises 509 Biography of William Burnside 512 30 Cayley Digraphs of Groups 513 Motivation 513 The Cayley Digraph of a Group 513 Hamiltonian Circuits and Paths 517 Some Applications 523 Exercises 526 Biography of William Rowan Hamilton 531 Biography of Paul Erdös 532 31 Introduction to Algebraic Coding Theory 533 Motivation 533 Linear Codes 538 Parity-Check Matrix Decoding 543 Coset Decoding 546 Historical Note: The Ubiquitous Reed-Solomon Codes 550 Exercises 552 Biography of Richard W. Hamming 557 Biography of Jessie MacWilliams 558 Biography of Vera Pless 559 32 An Introduction to Galois Theory 560 Fundamental Theorem of Galois Theory 560 Solvability of Polynomials by Radicals 567 Insolvability of a Quintic 571 Exercises 572 Biography of Philip Hall 575 33 Cyclotomic Extensions 576 Motivation 576 Cyclotomic Polynomials 577 The Constructible Regular n-gons 581 Exercises 583 Computer Exercise 584 Biography of Carl Friedrich Gauss 585 Biography of Manjul Bhargava 586 Supplementary Exercises for Chapters 24–33 586 Selected Answers 590 Text Credits 629 Photo Credits 631 Index of Mathematicians 632 Index of Terms 634 Part 1. Integers And Equivalence Relations. Preliminaries -- Part 2. Groups. Introduction To Groups -- Groups -- Finite Groups; Subgroups -- Cyclic Groups -- Permutation Groups -- Isomorphisms -- Cosets And Lagrange's Theorem -- External Direct Products -- Normal Subgroups And Factor Groups -- Group Homomorphisms -- Fundamental Theorem Of Finite Abelian Groups -- Part 3. Rings. Introduction To Rings -- Integral Domains -- Ideals And Factor Rings -- Ring Homomorphisms -- Polynomial Rings -- Factorization Of Polynomials -- Divisibility In Integral Domains -- Part 4. Fields -- Vector Spaces -- Extension Fields -- Algebraic Extensions -- Finite Fields -- Geometric Constructions -- Part 5. Special Topics -- Sylow Theorems -- Finite Simple Groups -- Generators And Relations -- Symmetry Groups -- Frieze Groups And Crystallographic Groups -- Symmetry And Counting -- Cayley Digraphs Of Groups -- Introduction To Algebraic Coding Theory -- An Introduction To Galois Theory -- Cyclotomic Extensions. Joseph A. Gallian. Includes Bibliographical References And Indexes.