0 Basics. - 0. 1 Natural Numbers and Integers. - 0. 2 Maps. - 0. 3 Mathematical Algorithms. - Notes. - 1 Commutative Rings with Unity. - 1. 1 Why Abstract Algebra?. - 1. 2 Groups. - 1. 3 Rings. - 1. 4 Subrings and Homomorphisms. - 1. 5 Ideals and Residue Class Rings. - 1. 6 The Homomorphism Theorem. - 1. 7 Gcd's Lcm's and Principal Ideal Domains. - 1. 8 Maximal and Prime Ideals. - 1. 9 Prime Rings and Characteristic. - 1. 10 Adjunction Products and Quotient Rings. - Notes. - 2 Polynomial Rings. - 2. 1 Definitions. - 2. 2 Euclidean Domains. - 2. 3 Unique Factorization Domains. - 2. 4 The Gaussian Lemma. - 2. 5 Polynomial Gcd's. - 2. 6 Squarefree Decomposition of Polynomials. - 2. 7 Factorization of Polynomials. - 2. 8 The Chinese Remainder Theorem. - Notes. - 3 Vector Spaces and Modules. - 3. 1 Vector Spaces. - 3. 2 Independent Sets and Dimension. - 3. 3 Modules. - Notes. - 4 Orders and Abstract Reduction Relations. - 4. 1 The Axiom of Choice and Some Consequences in Algebra. - 4. 2 Relations. - 4. 3 Foundedness Properties. - 4. 4 Some Special Orders. - 4. 5 Reduction Relations. - 4. 6 Computing in Algebraic Structures. - Notes. - 5 Gröbner Bases. - 5. 1 Term Orders and Polynomial Reductions. - 5. 2 Gröbner BasesExistence and Uniqueness. - 5. 3 Gröbner BasesConstruction. - 5. 4 Standard Representations. - 5. 5 Improved Gröbner Basis Algorithms. - 5. 6 The Extended Gröbner Basis Algorithm. - Notes. - 6 First Applications of Gröbner Bases. - 6. 1 Computation of Syzygies. - 6. 2 Basic Algorithms in Ideal Theory. - 6. 3 Dimension of Ideals. - 6. 4 Uniform Word Problems. - Notes. - 7 Field Extensions and the Hilbert Nullstellensatz. - 7. 1 Field Extensions. - 7. 2 The Algebraic Closure of a Field. - 7. 3 Separable Polynomials and Perfect Fields. - 7. 4 The Hilbert Nullstellensatz. - 7. 5 Height and Depth of Prime Ideals. - 7. 6 Implicitization of RationalParametrizations. - 7. 7 Invertibility of Polynomial Maps. - Notes. - 8 Decomposition Radical and Zeroes of Ideals. - 8. 1 Preliminaries. - 8. 2 The Radical of a Zero-Dimensional Ideal. - 8. 3 The Number of Zeroes of an Ideal. - 8. 4 Primary Ideals. - 8. 5 Primary Decomposition in Noetherian Rings. - 8. 6 Primary Decomposition of Zero-Dimensional Ideals. - 8. 7 Radical and Decomposition in Higher Dimensions. - 8. 8 Computing Real Zeroes of Polynomial Systems. - Notes. - 9 Linear Algebra in Residue Class Rings. - 9. 1 Gröbner Bases and Reduced Terms. - 9. 2 Computing in Finitely Generated Algebras. - 9. 3 Dimensions and the Hilbert Function. - Notes. - 10 Variations on Gröbner Bases. - 10. 1 Gröbner Bases over PID's and Euclidean Domains. - 10. 2 Homogeneous Gröbner Bases. - 10. 3 Homogenization. - 10. 4 Gröbner Bases for Polynomial Modules. - 10. 5 Systems of Linear Equations. - 10. 6 Standard Bases and the Tangent Cone. - 10. 7 Symmetric Functions. - Notes. - Appendix: Outlook on Advanced and Related Topics. - Complexity of Gröbner Basis Constructions. - Term Orders and Universal Gröbner Bases. - Comprehensive Gröbner Bases. - Gröbner Bases and Automatic Theorem Proving. - Characteristic Sets and Wu-Ritt Reduction. - Term Rewriting. - Standard Bases in Power Series Rings. - Non-Commutative Gröbner Bases. - Gröbner Bases and Differential Algebra. - Selected Bibliography. - Conference Proceedings. - Books and Monographs. - Articles. - List of Symbols. Language: English