Diophantine Equations and Power Integral Bases Theory and Algorithms
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
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Diophantine Equations and Power Integral Bases Theory and Algorithms Second Edition
Diophantine Equations and Power Integral Bases
István Gaál
Diophantine Equations and Power Integral Bases Theory and Algorithms Second Edition
István Gaál Institute of Mathematics University of Debrecen Debrecen, Debrecen, Hungary
ISBN 978-3-030-23864-3 ISBN 978-3-030-23865-0 (eBook) https://doi.org/10.1007/978-3-030-23865-0 Mathematics Subject Classification: 11D57, 11D59, 11D61, 11R04, 11Y50 1st edition: © Birkhäuser Boston 2002 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To Gabi, Zsuzsi, and Szilvi
Foreword
In the mid-1980s, the author started to investigate power integral bases in algebraic number fields and related Diophantine equations. To construct feasible methods requires both to develop the underlying theory of Diophantine equations and also to create algorithms. We described relations of index form equations and Thue equations and developed reduction and enumeration algorithms, involving Diophantine approximation tools and LLL algorithm. Starting with cubic number fields, within more than 10 years, we had feasible methods up to number fields of degree 5. The first edition of this book in 2002 (Diophantine Equations and Power Integral Bases, New Computational Methods, Birkhäuser Boston, 2002) was actually a thesis of the author. The book turned out to be useful in teaching university courses and in the research work of interested colleagues and PhD students. And the development of the methods did not stop. In the last 15 years, we developed several useful methods on higher degree number fields and relati
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