Advertisement

Table of contents

  1. Front Matter
    Pages i-xix
  2. Samuel A. Hambleton, Hugh C. Williams
    Pages 1-61
  3. Samuel A. Hambleton, Hugh C. Williams
    Pages 63-116
  4. Samuel A. Hambleton, Hugh C. Williams
    Pages 117-172
  5. Samuel A. Hambleton, Hugh C. Williams
    Pages 205-245
  6. Samuel A. Hambleton, Hugh C. Williams
    Pages 247-276
  7. Samuel A. Hambleton, Hugh C. Williams
    Pages 277-321
  8. Samuel A. Hambleton, Hugh C. Williams
    Pages 323-425
  9. Samuel A. Hambleton, Hugh C. Williams
    Pages 427-464
  10. Back Matter
    Pages 465-493

About this book

Introduction

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. 

Keywords

binary cubic forms cubic fields Voronoi's algorithm geometry of numbers continued fractions fundamental units norm equation cubic Pell equation parametrization simultaneous approximation arithmetic matrices linear algebraic groups cubic Thue equations

Authors and affiliations

  • Samuel A. Hambleton
    • 1
  • Hugh C. Williams
    • 2
  1. 1.School of Mathematics and PhysicsThe University of QueenslandSt. Lucia, BrisbaneAustralia
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-01404-9
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-01402-5
  • Online ISBN 978-3-030-01404-9
  • Series Print ISSN 1613-5237
  • Series Online ISSN 2197-4152
  • Buy this book on publisher's site