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Cubic Fields with Geometry

  • Book
  • © 2018

Overview

Authors:
  1. Samuel A. Hambleton

    1. School of Mathematics and Physics, The University of Queensland, St. Lucia, Brisbane, Australia

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    You can also search for this author in PubMed   Google Scholar

  2. Hugh C. Williams

    1. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

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    You can also search for this author in PubMed   Google Scholar

  • Provides an up-to-date compendium of results
  • Helps the reader to envision what is explained in the text
  • Introduces the reader to several tools and disciplines which are applicable in the study of cubic fields

Part of the book series: CMS Books in Mathematics (CMSBM)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xix
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  2. Cubic Fields

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 1-61

  3. Cubic Ideals and Lattices

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 63-116

  4. Binary Cubic Forms

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 117-172

  5. Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler)

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 173-203

  6. Cubic Pell Equations

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 205-245

  7. The Minima of Forms and Units by Approximation

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 247-276

  8. Voronoi’s Theory of Continued Fractions

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 277-321

  9. Relative Minima Adjacent to 1 in a Reduced Lattice

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 323-425

  10. Parametrization of Norm 1 Elements of \(\mathbb{K}\)

    • Samuel A. Hambleton, Hugh C. Williams
    Pages 427-464

  11. Back Matter

    Pages 465-493
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Keywords

About this book

The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics.  The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory.

The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics. 

Authors and Affiliations

  • School of Mathematics and Physics, The University of Queensland, St. Lucia, Brisbane, Australia

    Samuel A. Hambleton

  • Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

    Hugh C. Williams

Bibliographic Information

  • Book Title: Cubic Fields with Geometry

  • Authors: Samuel A. Hambleton, Hugh C. Williams

  • Series Title: CMS Books in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-01404-9

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2018

  • Hardcover ISBN: 978-3-030-01402-5Published: 19 November 2018

  • eBook ISBN: 978-3-030-01404-9Published: 07 November 2018

  • Series ISSN: 1613-5237

  • Series E-ISSN: 2197-4152

  • Edition Number: 1

  • Number of Pages: XIX, 493

  • Number of Illustrations: 26 b/w illustrations, 27 illustrations in colour

  • Topics: Algebraic Geometry, Number Theory, Algorithms

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