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 |

SUMMARY

The origins of the mathematics in this book date back more than two thouยญ sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Euยญ clid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbekยญ istan and worked in Baghdad at the court of Harun al-Rashid's son. The word "algorithm" is actually a westernization of al-Khowarizmi's name, while "algebra" derives from "al-jabr," a term that appears in the title of his book Kitab al-jabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and alยญ gorithms lasted roughly up to the beginning of this century; until then, the primary goal of algebra was the design of constructive methods for solving equations by means of symbolic transformations. During the second half of the nineteenth century, a new line of thought began to enter algebra from the realm of geometry, where it had been successful since Euclid's time, namely, the axiomatic method

CONTENT

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 Gcdโ{128}{153}s, Lcmโ{128}{153}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 Gcdโ{128}{153}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 Grรถbner Bases -- 5.1 Term Orders and Polynomial Reductions -- 5.2 Grรถbner Basesโ{128}{148}Existence and Uniqueness -- 5.3 Grรถbner Basesโ{128}{148}Construction -- 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 PIDโ{128}{153}s 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

Mathematics
Algebra
Group theory
Mathematics
Group Theory and Generalizations
Algebra