Advanced Modern Algebra Third Edition, Part 1 by Joseph J. Rotman
Modern Algebra , Number Theory | |
Language | English |
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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