| Preface | |
| How to Use This Book | |
| List of Algorithms | |
| Basics | p. 1 |
| Commutative Rings with Unity | p. 15 |
| Why Abstract Algebra? | p. 15 |
| Groups | p. 16 |
| Rings | p. 19 |
| Subrings and Homomorphisms | p. 22 |
| Ideals and Residue Class Rings | p. 25 |
| The Homomorphism Theorem | p. 32 |
| Gcd's, Lcm's, and Principal Ideal Domains | p. 38 |
| Maximal and Prime Ideals | p. 47 |
| Prime Rings and Characteristic | p. 49 |
| Adjunction, Products, and Quotient Rings | p. 51 |
| Polynomial Rings | p. 61 |
| Definitions | p. 61 |
| Euclidean Domains | p. 78 |
| Unique Factorization Domains | p. 87 |
| The Gaussian Lemma | p. 92 |
| Polynomial Gcd's | p. 97 |
| Squarefree Decomposition of Polynomials | p. 99 |
| Factorization of Polynomials | p. 109 |
| The Chinese Remainder Theorem | p. 114 |
| Vector Spaces and Modules | p. 119 |
| Vector Spaces | p. 119 |
| Independent Sets and Dimension | p. 129 |
| Modules | p. 133 |
| Orders and Abstract Reduction Relations | p. 141 |
| The Axiom of Choice and Some Consequences in Algebra | p. 141 |
| Relations | p. 149 |
| Foundedness Properties | p. 155 |
| Some Special Orders | p. 167 |
| Reduction Relations | p. 173 |
| Computing in Algebraic Structures | p. 178 |
| Grobner Bases | p. 187 |
| Term Orders and Polynomial Reductions | p. 188 |
| Grobner Bases - Existence and Uniqueness | p. 204 |
| Grobner Bases - Construction | p. 210 |
| Standard Representations | p. 218 |
| Improved Grobner Basis Algorithms | p. 222 |
| The Extended Grobner Basis Algorithm | p. 237 |
| First Applications of Grobner Bases | p. 243 |
| Computation of Syzygies | p. 243 |
| Basic Algorithms in Ideal Theory | p. 256 |
| Dimension of Ideals | p. 270 |
| Uniform Word Problems | p. 276 |
| Field Extensions and the Hilbert Nullstellensatz | p. 293 |
| Field Extensions | p. 293 |
| The Algebraic Closure of a Field | p. 306 |
| Separable Polynomials and Perfect Fields | p. 309 |
| The Hilbert Nullstellensatz | p. 312 |
| Height and Depth of Prime Ideals | p. 323 |
| Implicitization of Rational Parametrizations | p. 327 |
| Invertibility of Polynomial Maps | p. 330 |
| Decomposition, Radical, and Zeroes of Ideals | p. 335 |
| Preliminaries | p. 335 |
| The Radical of a Zero-Dimensional Ideal | p. 340 |
| The Number of Zeroes of an Ideal | p. 346 |
| Primary Ideals | p. 350 |
| Primary Decomposition in Noetherian Rings | p. 358 |
| Primary Decomposition of Zero-Dimensional Ideals | p. 366 |
| Radical and Decomposition in Higher Dimensions | p. 388 |
| Computing Real Zeroes of Polynomial Systems | p. 399 |
| Linear Algebra in Residue Class Rings | p. 423 |
| Grobner Bases and Reduced Terms | p. 423 |
| Computing in Finitely Generated Algebras | p. 433 |
| Dimensions and the Hilbert Function | p. 441 |
| Variations on Grobner Bases | p. 453 |
| Grobner Bases over PID's and Euclidean Domains | p. 453 |
| Homogeneous Grobner Bases | p. 466 |
| Homogenization | p. 477 |
| Grobner Bases for Polynomial Modules | p. 485 |
| Systems of Linear Equations | p. 488 |
| Standard Bases and the Tangent Cone | p. 496 |
| Symmetric Functions | p. 501 |
| Appendix: Outlook on Advanced and Related Topics | p. 511 |
| Complexity of Grobner Basis Constructions | p. 511 |
| Term Orders and Universal Grobner Bases | p. 514 |
| Comprehensive Grobner Bases | p. 515 |
| Grobner Bases and Automatic Theorem Proving | p. 518 |
| Characteristic Sets and Wu-Ritt Reduction | p. 520 |
| Term Rewriting | p. 523 |
| Standard Bases in Power Series Rings | p. 525 |
| Non-Commutative Grobner Bases | p. 526 |
| Grobner Bases and Differential Algebra | p. 528 |
| Selected Bibliography | p. 531 |
| List of Symbols | p. 561 |
| Index | p. 567 |
| Table of Contents provided by Blackwell. All Rights Reserved. |
