Diophantine Equations and Power Integral Bases

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2 Istvan Gaal Diophantine Equations and Power Integral Bases New Computational Methods Springer Science+Business Media, LLC

3 Istvan Gaâl University of Debrecen Institute of Mathematics and Informatics Debrecen Pf. 12 H-4010 Hungary Library of Congress Cataloging-in-Publication Data Gaal, Istvan, Diophantine equations and power integral bases : new computational methods I Istvan Gaal. p. cm. lnc1udes bibliographical references and index. ISBN ISBN (ebook) DOI / Diophantine equations. 2. Algebraic fields. 3. Bases (Linear topological spaces) 1. Title. QA242.G '.72-dc CIP AMS Subject Classifications: Primary: lly50; Secondary: IID57, lld59, llr33 Printed on acid-free paper 2002 Springer Science+BusÎness Media New York Originally published by Birkhăuser Boston În 2002 AlI rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dis similar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especialiy identified, is not to be taken as a sign that such names, as understood by the SPIN ISBN Reformatted from author's files by TEXniques, Inc., Cambridge, MA

4 To Gabi, Zsuzsi and Szilvi

5 Contents Preface Acknowledgements 1 Introduction 1.1 Basic concepts 1.2 Related results. 2 Auxiliary Results, Tools 2.1 Baker's method, effective finiteness theorems 2.2 Reduction Davenport lemma The general case 2.3 Enumeration methods. 2.4 Software, hardware 3 Auxiliary Equations 3.1 Thue equations Elementary estimates Thue's theorem Fast algorithm for finding "small" solutions Effective methods The method ofbilu and Hanrot 3.2 Inhomogeneous Thue equations... xi xv

6 viii Contents Elementary estimates. Baker's method Reduction, test An analogue of the Bilu-Hanrot method. 3.3 Relative Thue equations Baker's method, reduction Enumeration An example The resolution of norm form equations Preliminaries Solving the unit equation Ca1culating the solutions of the norm form equation Examples Index Form Equations in General 4.1 The structure of the index form. 4.2 Using resolvents Factorizing the index form when proper subfields exist 4.4 Composite fields Coprime discriminants Non-coprime discriminants Cubic Fields Arbitrary cubic fields Simplest cubic fields 54 6 Quartic Fields Algorithm for arbitrary quartic fields The resolvent equation The quartic Thue equations Proof of the theorem on the quartic Thue equations Examples Simplest quartic fields An interesting application to mixed dihedral quartic fields Totally complex quartic fields Parametrie farnilies of totally complex quartic fields Bicyclic biquadratic number fields Integral basis, index form The totally real case The totally complex case The field index of bicyclic biquadratic number fields 75 7 Quintic Fields Algorithm for arbitrary quintic fields Preliminaries... 80

7 Contents IX Baker's method, reduction Enumeration Examples Lehmer's quintics Integer basis, unit group The index form The index form equation The exceptional case 8 Sextic Fields 8.1 Sextic fields with a quadratic subfield Real quadratic subfield Totally real sextic fields with a quadratic and a cubic subfield Imaginary quadratic subfield Sextic fields with an imaginary quadratic and a real cubic subfield Parametric families of sextic fields with imaginary quadratic and real cubic subfields. 8.2 Sextic fields with a cubic subfield 8.3 Sextic fields as composite fields A cyc1ic sextic field A non-cyc1ic sextic field The parametric family of simplest sextic fields Relative Power Integral Bases Basic concepts Relative cubic extensions Example 1. Cubic extension of a quintic field Example 2. Cubic extension of a sextic field Computational experiences Relative quartic extensions Preliminaries The cubic relative Thue equation Representing the variables as binary quadratic forms The quartic relative Thue equations An example for computing relative power integral bases in a field of degree 12 with a cubic subfield Some Higher Degree Fields 10.1 Octic fields with a quadratic subfield Preliminaries The unit equation The inhomogeneous Thue equation Sieving

8 x Contents An example for computing power integral bases in an octic field with a quadratic subfield Nonic fields with cubic subfields The relative Thue equations The unit equation over the normal closure The common variables Examples Some more fields ofhigher degree Power integral bases in imaginary quadratic extensions of totally real cyclic fields of prime degree Power integral bases in imaginary quadratic extensions of Lehmer's quintics One more composite field Tables Cubic fields Totally real cubic fields Complex cubic fields Quartic fields The distribution of the minimal indices The average behavior of the minimal indices Totally real cyclic quartic fields Monogenie mixed dihedral extensions of real quadratic fie1ds Totally real bicyclic biquadratic number fields Totally complex bicyclic biquadratic number fields Some more quartic fields Sextic fields Totally real cyclic sextic fields Sextic fields with imaginary quadratic subfields 168 References 171 Author Index 181 Subject Index 183

9 Preface One of the classical problems in algebraie number theory, going back, among others, to K. Hensel [He08] and H. Hasse [Ha63], is to deeide if an algebraie number field K of degree n has a power integral basis, that is, an integral basis oftype {I, Ci,, Ci n - 1}. This is equivalent to 7l,K being monogenie, that is ofthe form 7l,[ Ci]. The main purpose of this book is to deseribe algorithms for determining generators Ci of power integral bases. This problem is equivalent to solving the eorresponding index form equations. It is important to emphasize that in addition to providing the reader with some efficient algorithms for eomputing generators of power integral bases, the other goal in this work is to show the development of constructive (algorithmic) methods for solving diophantine equations, whieh has eome about as a eonsequenee of a systematic study of index form equations. This has a signifieant impact on our investigations of power integral bases. Many of these methods ean also be applied to solving other types of diophantine equations. The most straightforward benefit of a power integral basis is to have an easy way of performing arithmetie ealeulations in 7l,K, especially an easy way of multiplieation. It also helps to improve minimal polynornials of generating elements of number fields in numerieal tables. But it has many other, more or less independent applieations; to mention only one quite eharaeteristie applieation, let us reeall the result of B. Kovaes and A. Peth6 [KP91], who proved that in 7l,K there exists a generalized number system if and only if 7l,K is monogenie; moreover the generators of power integral bases are needed to eonstruet these number systems. This was one of the reasons it beeame interesting not only to deeide the monogenity of 7l, K but also to determine all possible generators of power integral bases. The algorithm for determining power integral bases in eubie fields was one of the

10 xii Preface first "real" applications of the methods for solving Thue equations. The simple paper [GS89) on this cubic case is one of the most frequently cited papers of the author, which shows what kind of novelty such computations had at the end of the 1980s. It has been a great challenge to try to extend the algorithms for computing all generators of power integral bases to higher degree number fields; this was finally successful at least up to degree 5 in general, and for many special higher degree fields up to degree about 9, where we reached the limits of capability ofthe present methods and the capacity of the present computing machinery. Imagine that for a number field of degree n, the index form equation has n - 1 variables and degree n(n - 1)/2. This means that the index form equation is mostly (already in quartic fields) a very complicated equation that does not even fit onto one page. As mentioned above, the algorithms we developed for solving index form equations are clearly applicable and fruitful for also solving other types of decomposable form equations, e.g., the algorithm for solving norm form equations. On the other hand, in special types of fields, in order to make our computations easier, we investigated the structure of index forms and detected their correspondence with simpler types of equations. This enables us to reduce the index form equation to other important types of diophantine equations such as Thue equations, inhomogeneous Thue equations, relative Thue equations, which are hopefully easier to solve. Since in the past no efficient algorithms were known to solve inhomogeneous Thue equations and relative Thue equations, we developed methods for solving these equations as well. This is another important contribution to the constructive theory of diophantine equations. More than ten years have passed since we began to study constructive methods for determining power integral bases in algebraic number fields. The material of this book is an outgrowth of severallectures given by the author at various conferences, cf. [Ga91), [Ga96b), [Ga98b), [Ga99), [GaOOa). Also, some parts have appeared in special university courses held by the author. The book is organized in the following way. In Chapter 1 we fix our notation, describe the basic concepts, and summarize those important results on power integral bases which (being non-algorithmic) do not fit into subsequent chapters of the book. In Chapter 2 we collected the main tools for solving our equations. U sing Baker's method (Section 2.1) we obtain huge upper bounds for the unknown exponents of the corresponding unit equations. These bounds must be reduced (Section 2.2) using numerical diophantine approximation techniques based on the LLL basis reduction algorithm. Finally, we enumerate the possible small values of the unknown exponents lying under the reduced bounds (Section 2.3). The statements of these Sections are refined versions of formerly used lemmas, formulated in a suitable way for our applications. In Chapter 3 we survey algorithms for solving simpler types of equations, such as Thue equations (Section 3.1), inhomogenous Thue equations (Section 3.2), and relative Thue equations (Section 3.3). The index form equations in special types of number fields can often be reduced to these types of equations. It is useful to have

11 Preface xiii a unifonn discussion of these classical types of equations, containing several very recent results, such as the algorithm for relative Thue equations. In this chapter we also include a (recent) algorithm for solving norm form equations (Section 3.4), since they are very close to the above types of equations and use the same tools for their resolution. Chapter 4 gives a general overview of the structure of indexforms. Especially, if the field K is a composite of its subfields (Section 4.4), we obtain results that make the resolution of the index fonn equations much easier. In Chapters 5-8 we describe those algorithms that can be used for solving index form equations in cubic, quartic, quintic, sextic number fields, respectively. In addition to several efficient methods that work in special types of fields, we have general algorithms up to degree five. For degree six we can detennine the generators of power integral bases only if there are subfields. We also include several infinite parametric familes of fields and solve the corresponding index fonn equations in a parametric fonn. In Chapter 9 we consider the problem of relative power integral bases in cubic and quartic relative extensions of fields. This is used in Chapter 10 to consider power integral bases in some types of higher degree number fields, among others, in octic fields with a quadratic subfield, which are considered as relative quartic extensions of the quadratic subfields. Finally, in Chapter 11 we provide the results of ouf computations: tables of generators of power integral bases in cubic, quartic, sextic fields. These tables might have many applications. The reader should have a basic knowledge of algebraic numbers and algebraic number fields. Almost all that is needed can be found in W. Narkiewicz's book [Nark74]. For a more sophisticated algorithmic approach, the reader is referred to the book of M. Pohst and H. Zassenhaus [PZ89]. Debrecen, Hungary March, 2002 Istvan Gaal

12 Acknow ledgements The author is very grateful to Professor George Anastassiou, consultant editor of Birkhäuser Boston, who encouraged the publication of this book. The author is indepted to Professor KaIman Gy6ry who introduced hirn to the theory of diophantine equations. The influence of Professor Attila PethO led hirn in the direction of constructive methods. The author enjoyed the hospitality of Professor Michael Pohst and the joint work with hirn as a fellow of the Alexander von Humboldt Foundation as weh as several other times when the author could escape from the everyday duties of his horne university and devote a couple of days to joint research in Berlin. This book could not have been written without the support of Ann Kostant, editor of Birkhäuser Boston, who helped the author to organize the material to become a book. The author also thanks the referees for ideas that improved the presentation and for pointing out several misprints in former versions of the book.

13 Diophantine Equations and Power Integral Bases New Computational Methods