Dewey Class 
512.2 
Title 
Gröbner Bases ([EBook] :) : A Computational Approach to Commutative Algebra / / by Thomas Becker, Volker Weispfenning. 
Author 
Becker, Thomas 
Added Personal Name 
Weispfenning, Volker author. 
Other name(s) 
SpringerLink (Online service) 
Publication 
New York, NY : : Springer New York : : Imprint: Springer, , 1993. 
Physical Details 
XXII, 576 p. : online resource. 
Series 
Graduate texts in mathematics 00725285 ; ; 141 
ISBN 
9781461209133 
Summary Note 
The origins of the mathematics in this book date back more than two thou 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 Eu 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 ninthcentury scientist Mohammed ibn Musa alKhowarizmi, who was born in what is now Uzbek istan and worked in Baghdad at the court of Harun alRashid's son. The word "algorithm" is actually a westernization of alKhowarizmi's name, while "algebra" derives from "aljabr," a term that appears in the title of his book Kitab aljabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and al 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.: 
Contents note 
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 Bases—Existence and Uniqueness  5.3 Gröbner Bases—Construction  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 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 ZeroDimensional 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 ZeroDimensional 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 WuRitt Reduction  Term Rewriting  Standard Bases in Power Series Rings  NonCommutative Gröbner Bases  Gröbner Bases and Differential Algebra  Selected Bibliography  Conference Proceedings  Books and Monographs  Articles  List of Symbols. 
System details note 
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users) 
Internet Site 
http://dx.doi.org/10.1007/9781461209133 
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