Algebra (2nd Edition)

Algebra, Second Edition, by Michael Artin, provides comprehensive coverage at the level of an honors-undergraduate or introductory-graduate course. The second edition of this classic text incorporates twenty years of feedback plus the author’s own teaching experience. This book discusses concrete topics of algebra in greater detail than others, preparing readers for the more abstract concepts; linear algebra is tightly integrated throughout. Contents......Page 4 Preface......Page 10 1.1 The Basic Operations......Page 16 1.2 Row Reduction......Page 25 1.3 The Matrix Transpose......Page 32 1.4 Determinants......Page 33 1.5 Permutations......Page 39 1.6 Other Formulas for the Determinant......Page 42 Exercises......Page 46 2.1 Laws of Composition......Page 52 2.2 Groups and Subgroups......Page 55 2.3 Subgroups of the Additive Group of Integers......Page 58 2.4 Cyclic Groups......Page 61 2.5 Homomorphisms......Page 62 2.6 Isomorphisms......Page 66 2.7 Equivalence Relations and Partitions......Page 67 2.8 Cosets......Page 71 2.9 Modular Arithmetic......Page 75 2.10 The Correspondence Theorem......Page 76 2.11 Product Groups......Page 79 2.12 Quotient Groups......Page 81 Exercises......Page 84 3.1 Subspaces of Rn......Page 93 3.2 Fields......Page 95 3.3 Vector Spaces......Page 99 3.4 Bases and Dimension......Page 101 3.5 Computing with Bases......Page 106 3.6 Direct Sums......Page 110 3.7 Infinite-Dimensional Spaces......Page 111 Exercises......Page 113 4.1 The Dimension Formula......Page 117 4.2 The Matrix of a Linear Transformation......Page 119 4.3 Linear Operators......Page 123 4.4 Eigenvectors......Page 125 4.5 The Characteristic Polynomial......Page 128 4.6 Triangular and Diagonal Forms......Page 131 4.7 Jordan Form......Page 135 Exercises......Page 140 5.1 Orthogonal Matrices and Rotations......Page 147 5.2 Using Continuity......Page 153 5.3 Systems of Differential Equations......Page 156 5.4 The Matrix Exponential......Page 160 Exercises......Page 165 6.1 Symmetry of Plane Figures......Page 169 6.2 Isometries......Page 171 6.3 Isometries of the Plane......Page 174 6.4 Finite Groups of Orthogonal Operators on the Plane......Page 178 6.5 Discrete Groups of Isometries......Page 182 6.6 Plane Crystallographic Groups......Page 187 6.7 Abstract Symmetry: Group Operations......Page 191 6.8 The Operation on Cosets......Page 193 6.9 The Counting Formula......Page 195 6.11 Permutation Representations......Page 196 6.12 Finite Subgroups of the Rotation Group......Page 198 Exercises......Page 203 7.2 The Class Equation......Page 210 7.3 p-Groups......Page 212 7.4 The Class Equation of the Icosahedral Group......Page 213 7.5 Conjugation in the Symmetric Group......Page 215 7.7 The Sylow Theorems......Page 218 7.8 Groups of Order 12......Page 223 7.9 The Free Group......Page 225 7.10 Generators and Relations......Page 227 7.11 The Todd-Coxeter Algorithm......Page 231 Exercises......Page 236 8.1 Bilinear Forms......Page 244 8.2 Symmetric Forms......Page 246 8.3 Hermitian Forms......Page 247 8.4 Orthogonality......Page 250 8.5 Euclidean Spaces and Hermitian Spaces......Page 256 8.6 The Spectral Theorem......Page 257 8.7 Conics and Quadrics......Page 260 8.8 Skew-Symmetric Forms......Page 264 8.9 Summary......Page 267 Exercises......Page 269 9.1 The Classical Groups......Page 276 9.2 Interlude: Spheres......Page 278 9.3 The Special Unitary Group SU2......Page 281 9.4 The Rotation Group SO3......Page 284 9.5 One-Parameter Groups......Page 287 9.6 The Lie Algebra......Page 290 9.7 Translation in a Group......Page 292 9.8 Normal Subgroups of SL2......Page 295 Exercises......Page 298 10.1 Definitions......Page 305 10.2 Irreducible Representations......Page 309 10.3 Unitary Representations......Page 311 10.4 Characters......Page 313 10.5 One-Dimensional Characters......Page 318 10.6 The Regular Representation......Page 319 10.7 Schur's Lemma......Page 322 10.8 Proof of the Orthogonality Relations......Page 324 10.9 Representations of SU2......Page 326 Exercises......Page 329 11.1 Definition of a Ring......Page 338 11.2 Polynomial Rings......Page 340 11.3 Homomorphisms and Ideals......Page 343 11.4 Quotient Rings......Page 349 11.5 Adjoining Elements......Page 353 11.6 Product Rings......Page 356 11.7 Fractions......Page 357 11.8 Maximal Ideals......Page 359 11.9 Algebraic Geometry......Page 362 Exercises......Page 369 12.1 Factoring Integers......Page 374 11.2 Unique Factorization Domains......Page 375 12.3 Gauss's Lemma......Page 382 12.4 Factoring Integer Polynomials......Page 386 12.5 Gauss Primes......Page 391 Exercises......Page 393 13.1 Algebraic Integers......Page 398 13.2 Factoring Algebraic Integers......Page 400 13.3 Ideals in Z[√-5]......Page 402 13.4 Ideal Multiplication......Page 404 13.5 Factoring Ideals......Page 407 13.6 Prime Ideals and Prime Integers......Page 409 13.7 Ideal Classes......Page 411 13.8 Computing the Class Group......Page 414 13.9 Real Quadratic Fields......Page 417 13.10 About Lattices......Page 420 Exercises......Page 423 14.1 Modules......Page 427 14.2 Free Modules......Page 429 14.3 Identities......Page 432 14.4 Diagonalizing Integer Matrices......Page 433 14.5 Generators and Relations......Page 438 14.6 Noetherian Rings......Page 441 14.7 Structure of Abelian Groups......Page 444 14.8 Application to Linear Operators......Page 447 14.9 Polynomial Rings in Several Variables......Page 451 Exercises......Page 452 15.1 Examples of Fields......Page 457 15.2 Algebraic and Transcendental Fields......Page 458 15.3 The Degree of a Field Extension......Page 461 15.4 Finding the Irreducible Polynomial......Page 464 15.5 Ruler and Compass Constructions......Page 465 15.6 Adjoining Roots......Page 471 15.7 Finite Fields......Page 474 15.8 Primitive Elements......Page 477 15.9 Function Fields......Page 478 15.10 The Fundamental Theorem of Algebra......Page 486 Exercises......Page 487 16.1 Symmetric Functions......Page 492 16.2 The Discriminant......Page 496 16.3 Splitting Fields......Page 498 16.4 Isomorphisms of Field Extensions......Page 499 16.5 Fixed Fields......Page 501 16.6 Galois Extensions......Page 503 16.7 The Main Theorem......Page 504 16.8 Cubic Equations......Page 507 16.9 Quartic Equations......Page 508 16.10 Roots of Unity......Page 512 16.11 Kummer Extensions......Page 515 16.12 Quintic Equations......Page 517 Exercises......Page 520 A.1 About Proofs......Page 528 A.2 The Integers......Page 531 A.3 Zorn's Lemma......Page 533 A.4 The Implicit Function Theorem......Page 534 Bibliography......Page 538 Notation......Page 540 Index......Page 544 Algebra, Second Edition, By Michael Artin, Is Ideal For The Honors Undergraduate Or Introductory Graduate Course. The Second Edition Of This Classic Text Incorporates Twenty Years Of Feedback And The Author's Own Teaching Experience. The Text Discusses Concrete Topics Of Algebra In Greater Detail Than Most Texts, Preparing Students For The More Abstract Concepts; Linear Algebra Is Tightly Integrated Throughout. -- Publisher's Description. Matrices -- Groups -- Vector Spaces -- Linear Operators -- Applications Of Linear Operators -- Symmetry -- More Group Theory -- Bilinear Forms -- Linear Groups -- Group Representations -- Rings -- Factoring -- Quadratic Number Fields -- Linear Algebra In A Ring -- Fields -- Galois Theory. Michael Artin. Includes Bibliographical References (p. 523-524) And Index.