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Diophantine Equations and Power Integral Bases

Theory and Algorithms

  • Book
  • © 2019

Overview

Authors:
  1. István Gaál

    1. Institute of Mathematics, University of Debrecen, Debrecen, Hungary

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  • Provides a complete reference on index form equations and power integral bases

  • Describes algorithms and methods to efficiently solve several different types of classical Diophantine equations

  • Includes detailed numerical examples that demonstrate how the various methods of calculating index form equations can be applied

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xxii
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  2. Introduction

    • István Gaál
    Pages 1-11

  3. Auxiliary Results and Tools

    • István Gaál
    Pages 13-24

  4. Thue Equations

    • István Gaál
    Pages 25-37

  5. Inhomogeneous Thue Equations

    • István Gaál
    Pages 39-44

  6. Relative Thue Equations

    • István Gaál
    Pages 45-79

  7. The Resolution of Norm Form Equations

    • István Gaál
    Pages 81-92

  8. Index Form Equations in General

    • István Gaál
    Pages 93-103

  9. Cubic Fields

    • István Gaál
    Pages 105-111

  10. Quartic Fields

    • István Gaál
    Pages 113-150

  11. Quintic Fields

    • István Gaál
    Pages 151-168

  12. Sextic Fields

    • István Gaál
    Pages 169-195

  13. Pure Fields

    • István Gaál
    Pages 197-205

  14. Cubic Relative Extensions

    • István Gaál
    Pages 207-227

  15. Quartic Relative Extensions

    • István Gaál
    Pages 229-264

  16. Some Higher Degree Fields

    • István Gaál
    Pages 265-271

  17. Tables

    • István Gaál
    Pages 273-311

  18. Back Matter

    Pages 313-326
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Keywords

About this book

This monograph outlines the structure of index form equations, and makes clear their relationship to other classical types of Diophantine equations. In order to more efficiently determine generators of power integral bases, several algorithms and methods are presented to readers, many of which are new developments in the field. Additionally, readers are presented with various types of number fields to better facilitate their understanding of how index form equations can be solved. By introducing methods like Baker-type estimates, reduction methods, and enumeration algorithms, the material can be applied to a wide variety of Diophantine equations. This new edition provides new results, more topics, and an expanded perspective on algebraic number theory and Diophantine Analysis.


Notations, definitions, and tools are presented before moving on to applications to Thue equations and norm form equations. The structure of index forms is explained, which allows readers to approach several types of number fields with ease. Detailed numerical examples, particularly the tables of data calculated by the presented methods at the end of the book, will help readers see how the material can be applied.


Diophantine Equations and Power Integral Bases will be ideal for graduate students and researchers interested in the area. A basic understanding of number fields and algebraic methods to solve Diophantine equations is required.


Reviews

“The book is recommended to PhD students and researchers in the field of Diophantine equations. It can be used as a textbook for a specialized graduate course in Thue and index-form equations and as an additional reading for a general course in Diophantine equations.” (Andrej Dujella, zbMATH 1465.11090, 2021)

Authors and Affiliations

  • Institute of Mathematics, University of Debrecen, Debrecen, Hungary

    István Gaál

Bibliographic Information

  • Book Title: Diophantine Equations and Power Integral Bases

  • Book Subtitle: Theory and Algorithms

  • Authors: István Gaál

  • DOI: https://doi.org/10.1007/978-3-030-23865-0

  • Publisher: Birkhäuser Cham

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

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Hardcover ISBN: 978-3-030-23864-3Published: 19 September 2019

  • Softcover ISBN: 978-3-030-23867-4Published: 19 September 2020

  • eBook ISBN: 978-3-030-23865-0Published: 03 September 2019

  • Edition Number: 2

  • Number of Pages: XXII, 326

  • Number of Illustrations: 2 illustrations in colour

  • Topics: Number Theory

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