Author | Becker, Thomas. author |
---|---|
Title | Grรถbner Bases [electronic resource] : A Computational Approach to Commutative Algebra / by Thomas Becker, Volker Weispfenning |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1993 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0913-3 |
Descript | XXII, 576 p. online resource |
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 Gcdโs, Lcmโs, 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 Gcdโs -- 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 Grรถbner Bases -- 5.1 Term Orders and Polynomial Reductions -- 5.2 Grรถbner BasesโExistence and Uniqueness -- 5.3 Grรถbner BasesโConstruction -- 5.4 Standard Representations -- 5.5 Improved Grรถbner Basis Algorithms -- 5.6 The Extended Grรถbner Basis Algorithm -- Notes -- 6 First Applications of Grรถbner 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 Rational Parametrizations -- 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 Grรถbner Bases and Reduced Terms -- 9.2 Computing in Finitely Generated Algebras -- 9.3 Dimensions and the Hilbert Function -- Notes -- 10 Variations on Grรถbner Bases -- 10.1 Grรถbner Bases over PIDโs and Euclidean Domains -- 10.2 Homogeneous Grรถbner Bases -- 10.3 Homogenization -- 10.4 Grรถbner 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 Grรถbner Basis Constructions -- Term Orders and Universal Grรถbner Bases -- Comprehensive Grรถbner Bases -- Grรถbner Bases and Automatic Theorem Proving -- Characteristic Sets and Wu-Ritt Reduction -- Term Rewriting -- Standard Bases in Power Series Rings -- Non-Commutative Grรถbner Bases -- Grรถbner Bases and Differential Algebra -- Selected Bibliography -- Conference Proceedings -- Books and Monographs -- Articles -- List of Symbols