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Gröbner Bases in Orders of Algebraic Number Fields

✍ Scribed by David Andrew Smith

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
281 KB
Volume
33
Category
Article
ISSN
0747-7171

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✦ Synopsis

We prove that any order O of any algebraic number field K is a reduction ring. Rather than showing the axioms for a reduction ring hold, we start from scratch by well-ordering O, defining a division algorithm, and demonstrating how to use it in a Buchberger algorithm which computes a Gröbner basis given a finite generating set for an ideal. It is shown that our theory of Gröbner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Gröbner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper.

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Gröbner Bases in Clifford and Grassmann
Gröbner Bases in Clifford and Grassmann algebras
✍ David Hartley; Philip Tuckey 📂 Article 📅 1995 🏛 Elsevier Science 🌐 English ⚖ 287 KB

We give an account of the theory of Gröbner bases for Clifford and Grassmann algebras, both important in physical applications. We describe a characterization criterion tailored to these algebras which is significantly simpler than those given earlier or for more general non-commuting algebras. Our