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CMS Books in Mathematics

Cubic Fields with Geometry

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

  • 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
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Buy this book

eBook $109.00
price for USA in USD (gross)
  • ISBN 978-3-030-01404-9
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.99
price for USA in USD
  • ISBN 978-3-030-01402-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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. 

Table of contents (9 chapters)

  • Cubic Fields

    Hambleton, Samuel A. (et al.)

    Pages 1-61

  • Cubic Ideals and Lattices

    Hambleton, Samuel A. (et al.)

    Pages 63-116

  • Binary Cubic Forms

    Hambleton, Samuel A. (et al.)

    Pages 117-172

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

    Hambleton, Samuel A. (et al.)

    Pages 173-203

  • Cubic Pell Equations

    Hambleton, Samuel A. (et al.)

    Pages 205-245

Buy this book

eBook $109.00
price for USA in USD (gross)
  • ISBN 978-3-030-01404-9
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.99
price for USA in USD
  • ISBN 978-3-030-01402-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Cubic Fields with Geometry
Authors
Series Title
CMS Books in Mathematics
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-030-01404-9
DOI
10.1007/978-3-030-01404-9
Hardcover ISBN
978-3-030-01402-5
Series ISSN
1613-5237
Edition Number
1
Number of Pages
XIX, 493
Number of Illustrations
26 b/w illustrations, 27 illustrations in colour
Topics