Answer key for Contemporary abstract algebra

Contemporary Abstract Algebra 7e, written by Joseph Gallian, a well-known active researcher and award-winning teacher upholds the text's reputation for providing students a solid introduction to traditional abstract algebra topics. The text includes concepts and methodologies used by working mathematicians, computer scientists, physicists and chemists. Front cover......Page 1 Title page......Page 3 Date-line......Page 4 Acknowledgments......Page 5 Preface to the 2nd edition......Page 9 Preface to the 1st edition......Page 11 Contents......Page 15 Integers and Equivalence Relations......Page 27 Properties of Integers......Page 29 Modular Arithmetic......Page 33 Mathematical Induction......Page 36 Equivalence Relations......Page 39 Functions (Mappings)......Page 41 Exercises......Page 43 Groups......Page 47 Symmetries of a Square......Page 49 The Dihedral Groups......Page 52 Exercises......Page 54 Biography of Niels Abel......Page 58 Definition and Examples of Groups......Page 59 Elementary Properties of Groups......Page 66 Applications of Modular Arithmetic......Page 68 Historical Note......Page 71 Exercises......Page 72 Programming Exercises......Page 76 Terminology and Notation......Page 79 Subgroup Tests......Page 80 Examples of Subgroups......Page 82 Exercises......Page 86 Programming Exercises......Page 90 Properties of Cyclic Groups......Page 92 Classification of Subgroups of Cyclic Groups......Page 96 Exercises......Page 98 Programming Exercises......Page 102 Biography of J. J. Sylvester......Page 104 Supplementary Exercises for Chapters 1-4......Page 106 Definition and Notation......Page 109 Cycle Notation......Page 112 Properties of Permutations......Page 114 A Check-Digit Scheme Based on $D_5$......Page 118 Exercises......Page 120 Programming Exercise......Page 122 Biography of Augustin Cauchy......Page 124 Motivation......Page 125 Definition and Examples......Page 126 Cayley's Theorem......Page 129 Properties of Isomorphisms......Page 130 Automorphisms......Page 131 Exercises......Page 134 Biography of Arthur Cayley......Page 138 Definition and Examples......Page 139 Properties of External Direct Products......Page 140 Exercises......Page 142 Programming Exercises......Page 144 Definition and Examples......Page 145 The Group of Units Modulo $n$ As an Internal and External Direct Product......Page 148 Exercises......Page 150 Programming Exercises......Page 152 Supplementary Exercises for Chapters 5-8......Page 154 Properties of Cosets......Page 156 Lagrange's Theorem and Consequences......Page 158 An Application of Cosets to Permutation Groups......Page 161 The Rotation Group of a Cube and a Soccer Ball......Page 162 Exercises......Page 165 Biography of Joseph Lagrange......Page 170 Normal Subgroups......Page 171 Factor Groups......Page 172 Applications of Factor Groups......Page 177 Exercises......Page 179 Biography of Evariste Galois......Page 184 Definition and Examples......Page 185 Properties of Homomorphisms......Page 187 The First Isomorphism Theorem......Page 188 Exercises......Page 193 Biography of Camille Jordan......Page 197 The Fundamental Theorem......Page 198 The Isomorphism Classes of Abelian Groups......Page 199 Proof of the Fundamental Theorem......Page 203 Exercises......Page 205 Programming Exercises......Page 207 Supplementary Exercises for Chapters 9-12......Page 209 Rings......Page 211 Motivation and Definition......Page 213 Examples of Rings......Page 214 Exercises......Page 215 Programming Exercise......Page 217 Properties of Rings......Page 218 Subrings......Page 219 Exercises......Page 221 Programming Exercise......Page 223 Biography of I. N. Herstein......Page 224 Definition and Examples......Page 225 Fields......Page 227 Characteristic of a Ring......Page 228 Exercises......Page 230 Programming Exercises......Page 234 Biography of Nathan Jacobson......Page 235 Ideals......Page 236 Factor Rings......Page 237 Prime Ideals and Maximal Ideals......Page 239 Exercises......Page 241 Biography of Richard Dedekind......Page 244 Biography of Emmy Noether......Page 245 Supplementary Exercises for Chapters 13-16......Page 246 Definition and Examples......Page 248 Properties of Ring Homomorphisms......Page 250 The Field of Quotients......Page 253 Exercises......Page 254 Notation and Terminology......Page 259 The Division Algorithm and Consequences......Page 262 Exercises......Page 265 Reducibility Tests......Page 268 Irreducibility Tests......Page 270 Unique Factorization in $Z[x]$......Page 275 Weird Dice: An Application of Unique Factorization......Page 276 Exercises......Page 278 Programming Exercises......Page 281 Biography of Carl Friedrich Gauss......Page 283 Irreducibles, Primes......Page 285 Historical Discussion of Fermat's Last Theorem......Page 287 Unique Factorization Domains......Page 290 Euclidean Domains......Page 293 Exercises......Page 296 Biography of Ernst Eduard Kummer......Page 299 Biography of Sophie Germain......Page 300 Supplementary Exercises for Chapters 17-20......Page 301 Fields......Page 303 Definition and Examples......Page 305 Subspaces......Page 306 Linear Independence......Page 307 Exercises......Page 309 Biography of Emil Artin......Page 311 The Fundamental Theorem of Field Theory......Page 312 Splitting Fields......Page 314 Zeros of an Irreducible Polynomial......Page 319 Exercises......Page 321 Biography of Leopold Kronecker......Page 323 Characterization of Extensions......Page 324 Finite Extensions......Page 326 Properties of Algebraic Extensions......Page 330 Exercises......Page 331 Biography of Irving Kaplansky......Page 334 Classification of Finite Fields......Page 335 Structure of Finite Fields......Page 336 Subfields of a Finite Field......Page 339 Exercises......Page 341 Programming Exercises......Page 342 Biography ofL. E. Dickson......Page 343 Historical Discussion of Geometric Constructions......Page 344 Constructible Numbers......Page 345 Exercises......Page 347 Supplementary Exercises for Chapters 21-25......Page 350 Special Topics......Page 351 Conjugacy Classes......Page 353 The Class Equation......Page 354 The Probability That Two Elements Commute......Page 355 The Sylow Theorems......Page 356 Application of Sylow's Theorems......Page 359 Exercises......Page 364 Biography of Ludvig Sylow......Page 367 Historical Background......Page 368 Nonsimplicity Tests......Page 372 The Fields Medal......Page 376 Exercises......Page 377 Programming Exercises......Page 379 Biography of Michael Aschbacher......Page 381 Biography of Daniel Gorenstein......Page 382 Biography of John Thompson......Page 383 Motivation......Page 384 Definitions and Notation......Page 385 Free Group......Page 386 Generators and Relations......Page 387 Classification of Groups of Order up to 15......Page 390 Characterization of Dihedral Groups......Page 391 Realizing the Dihedral Groups with Mirrors......Page 392 Exercises......Page 394 Biography of William Burnside......Page 398 Isometries......Page 399 Classification of Finite Plane Symmetry Groups......Page 401 Classification of Finite Groups of Rotations in $\mathbb{R}^3$......Page 402 Exercises......Page 405 The Frieze Groups......Page 408 The Crystallographic Groups......Page 413 Identification of Plane Periodic Patterns......Page 415 Exercises......Page 424 Biography of M. C. Escher......Page 429 Motivation......Page 430 The Cayley Digraph of a Group......Page 431 Hamiltonian Circuits and Paths......Page 434 Some Applications......Page 441 Exercises......Page 446 Biography of William Rowan Hamilton......Page 450 Biography of Paul Erdos......Page 452 Motivation......Page 454 Linear Codes......Page 459 Parity-Check Matrix Decoding......Page 463 Coset Decoding......Page 465 Exercises......Page 468 Biography of Richard W. Hamming......Page 473 Fundamental Theorem of Galois Theory......Page 474 Solvability of Polynomials by Radicals......Page 480 Insolvability of a Quintic......Page 483 Exercises......Page 484 Biography of Philip Hall......Page 487 Motivation......Page 488 Definition and Properties......Page 490 The Algebra of Electric Circuits......Page 492 The Algebra of Logic......Page 495 Finite Boolean Algebras......Page 496 Exercises......Page 497 Biography of Claude E. Shannon......Page 500 Supplementary Exercises for Chapters 26-34......Page 501 Selected Answers......Page 503 Notations......Page 537 Index of Mathematicians......Page 541 Index of Terms......Page 543 Back cover......Page 550