Advanced Modern Algebra Third Edition, Part 1 by Joseph J. Rotman
| Modern Algebra , Number Theory | |
| Language | English |
| File Type | |
| 723 | |
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| File Size & Downloads | Size 40.0 MiB Downloads 589 |
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| Advanced Modern Algebra Third Edition Contents | |
| Preface to Third Edition: Part 1 | xi |
| Acknowledgments | xiv |
| Part A. Course I | 3 |
| Chapter A-1. Classical Formulas | 4 |
| Cubics | 6 |
| Quartics | 9 |
| Chapter A-2. Classical Number Theory | 9 |
| Divisibility | 16 |
| Euclidean Algorithms | 19 |
| Congruence | 29 |
| Chapter A-3. Commutative Rings | 41 |
| Polynomials | 47 |
| Homomorphisms | 55 |
| Quotient Rings | 62 |
| From Arithmetic to Polynomials | 74 |
| Maximal Ideals and Prime Ideals | 83 |
| Finite Fields | 89 |
| Irreducibility | 97 |
| Euclidean Rings and Principal Ideal Domains | 104 |
| Unique Factorization Domains | |
| Chapter A-4. Groups | 115 |
| Permutations | 116 |
| Even and Odd | 123 |
| Groups | 127 |
| Lagrange’s Theorem | 139 |
| Homomorphisms | 150 |
| Quotient Groups | 159 |
| Simple Groups | 173 |
| Chapter A-5. Galois Theory | 179 |
| Insolvability of the Quintic | 179 |
| Classical Formulas and Solvability by Radicals | 187 |
| Translation into Group Theory | 190 |
| Fundamental Theorem of Galois Theory | 200 |
| Calculations of Galois Groups | 223 |
| Chapter A-6. Appendix: Set Theory | 235 |
| Equivalence Relations | 243 |
| Chapter A-7. Appendix: Linear Algebra | 247 |
| Vector Spaces | 247 |
| Linear Transformations and Matrices | 259 |
| Part B. Course II | |
| Chapter B-1. Modules | 273 |
| Noncommutative Rings | 273 |
| Chain Conditions on Rings | 282 |
| Left and Right Modules | 288 |
| Chain Conditions on Modules | 300 |
| Exact Sequences | 305 |
| Chapter B-2. Zorn’s Lemma | 313 |
| Zorn, Choice, and Well-Ordering | 313 |
| Zorn and Linear Algebra | 319 |
| Zorn and Free Abelian Groups | 323 |
| Semisimple Modules and Rings | 334 |
| Algebraic Closure | 339 |
| Transcendence | 345 |
| Liiroth’s Theorem | 353 |
| Chapter B-3. Advanced Linear Algebra | 359 |
| Torsion and Torsion-free | 359 |
| Basis Theorem | 362 |
| Fundamental Theorem | 371 |
| Elementary Divisors | 371 |
| Invariant Factors | 374 |
| From Abelian Groups to Modules | 378 |
| Rational Canonical Forms | 383 |
| Eigenvalues | 388 |
| Jordan Canonical Forms | 395 |
| Smith Normal Forms | 402 |
| Inner Product Spaces | 417 |
| Orthogonal and Symplectic Groups | 429 |
| Hermitian Forms and Unitary Groups | 436 |
| Chapter B-4. Categories of Modules | 441 |
| Categories | 441 |
| Functors | 461 |
| Galois Theory for Infinite Extensions | 475 |
| Free and Projective Modules | 481 |
| Injective Modules | 492 |
| Divisible Abelian Groups | 501 |
| Tensor Products | 509 |
| Adjoint Isomorphisms | 522 |
| Flat Modules | 529 |
| Chapter B-5. Multilinear Algebra | 543 |
| Algebras and Graded Algebras | 543 |
| Tensor Algebra | 552 |
| Exterior Algebra | 561 |
| Grassmann Algebras | 566 |
| Exterior Algebra and Differential Forms | 573 |
| Determinants | 575 |
| Chapter B-6. Commutative Algebra II | 591 |
| Old-Fashioned Algebraic Geometry | 591 |
| Affine Varieties and Ideals | 593 |
| Nullstellensatz | 599 |
| Nullstellensatz Redux | 604 |
| Irreducible Varieties | 614 |
| Affine Morphisms | 623 |
| Algorithms in k[x1, … , Xn] | 628 |
| Monomial Orders | 629 |
| Division Algorithm | 636 |
| Grabner Bases | 639 |
| Chapter B-7. Appendix: Categorical Limits | 651 |
| Inverse Limits | 651 |
| Direct Limits | 657 |
| Directed Index Sets | 659 |
| Adjoint Functors | 666 |
| Chapter B-8. Appendix: Topological Spaces | 673 |
| Topological Groups | 678 |
| Bibliography | 681 |
| Special Notation | 687 |
| Index | 693 |
Advanced Modern Algebra Third Edition PDF
