Gröbner Bases

1. Front Matter

   Pages i-xxii

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2. Basics

   • Thomas Becker, Volker Weispfenning

   Pages 1-13

3. Commutative Rings with Unity

   • Thomas Becker, Volker Weispfenning

   Pages 15-59

4. Polynomial Rings

   • Thomas Becker, Volker Weispfenning

   Pages 61-118

5. Vector Spaces and Modules

   • Thomas Becker, Volker Weispfenning

   Pages 119-139

6. Orders and Abstract Reduction Relations

   • Thomas Becker, Volker Weispfenning

   Pages 141-185

7. Gröbner Bases

   • Thomas Becker, Volker Weispfenning

   Pages 187-242

8. First Applications of Gröbner Bases

   • Thomas Becker, Volker Weispfenning

   Pages 243-292

9. Field Extensions and the Hilbert Nullstellensatz

   • Thomas Becker, Volker Weispfenning

   Pages 293-333

10. Decomposition, Radical, and Zeroes of Ideals

    • Thomas Becker, Volker Weispfenning

    Pages 335-421

11. Linear Algebra in Residue Class Rings

    • Thomas Becker, Volker Weispfenning

    Pages 423-452

12. Variations on Gröbner Bases

    • Thomas Becker, Volker Weispfenning

    Pages 453-509

13. Back Matter

    Pages 511-576

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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 ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbek­ 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 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.

• Abstract algebra
• Algebra
• Algebraic structure
• Finite
• Morphism
• Vector space
• axiom of choice
• equation
• function
• mathematics
• theorem

• Fakultät für Mathematik und Informatik Universität Passau, Passau, Germany

  Thomas Becker, Volker Weispfenning

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