Algebraic Number Theory CHAPMAN AND HALL MATHEMATICS SERIES Edited by Professor R. Brown Head of the Department of Pure Mathematics, University College of North Wales, Bangor and Dr M. A. H. Dempster, University Lecturer in Industrial Mathematics and Fellow of Balliol College, Oxford A Preliminary Course in Analysis R. M. F. Moss and G. T. Roberts Elementary Differential Equations R. 1. E. Schwarzenberger A First Course on Complex Functions G. J. O. Jameson Rings, Modules and Linear Algebra B. Hartley and T. O. Hawkes Regular Algebra and Finite Machines J. H. Conway Complex Numbers w. H. Cockcroft Galois Theory Ian Stewart Topology and Normed Spaces G. J. O. Jameson Introduction to Optimization Methods P. R. Adby and M. A. H. Dempster Graphs, Surfaces and Homology P. J. Giblin Linear Estimation and Stochastic Control M. H. A. Davis Mathematical Programming and Control Theory B. D. Craven Algebraic Number Theory IAN STEWART DAVID TALL Mathematics Institute, University of Warwick, Coventry LONDON CHAPMAN AND HALL A Halsted Press Book John Wiley & Sons, New York First published 1979 by Chapman and Hall Ltd 11 New Fetter Lane London EC4P 4EE © 19791 N. Stewart and D. O. Tall Typeset by the Alden Press, Oxford, London & Northampton and printed in Great Britain at the University Press, Cambridge ISBN 978-0-412-13840-9 ISBN 978-1-4615-6412-6 (eBook) DOl 10.1007/978-1-4615-6412-6 This title is available in both hardbound and paperback editions. The paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. All rights reserved. No part of this book may be reprinted, or reproduced or utilized in any form or by an electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the Publisher. Distributed in the U.S.A. by Halsted Press, a Division of John Wiley & Sons, Inc., New York Library of Congress Cataloging in Publication Data Stewart, Ian. Algebraic number theory. (Chapman and Hall mathematics series) Bibliography: p. Includes index. 1. Algebraic number theory. I. Tall, David Orme, joint author. II. Title. QA247.S76 512'.74 78-31625 TO RONNIE BROWN WHOSE BRAINCHILD IT WAS Contents Preface page x Readers' guide xiv Index of notation xvi The origins of algebraic number theory PART I ALGEBRAIC METHODS 1 Algebraic background 9 1.1 Rings and fields 10 1.2 Factorization of polynomials 14 1.3 Field extensions 22 1.4 Symmetric polynomials 24 1.5 Modules 27 1.6 Free abelian groups 28 2 Algebraic numbers 38 2.1 Algebraic numbers 39 2.2 Conjugates and discriminants 41 2.3 Algebraic integers 45 2.4 Integral bases 50 2.5 Norms and traces 54 vii viii CONTENTS 3 Quadratic and cyclotomic fields 59 3.1 Quadratic fields 59 3.2 Cyclotomic fields 62 4 Factorization into irreducibles 71 4.1 Historical background 72 4.2 Trivial factorizations 74 4.3 Factorization into irreducibles 78 4.4 Examples of non-unique factorization into irreducibles 82 4.5 Prime factorization 87 4.6 Euclidean domains 90 4.7 Euclidean quadratic fields 92 4.8 Consequences of unique factorization 97 4.9 The Ramanujan-Nagell theorem 99 5 Ideals 105 5.1 Historical background 106 5.2 Prime factorization of ideals 109 5.3 The norm of an ideal 120 PART II GEOMETRIC METHODS 6 Lattices 135 6.1 Lattices 135 6.2 The quotient torus 138 7 Minkowski's theorem 145 7.1 Minkowski's theorem 145 7.2 The two-squares theorem 148 7.3 The four-squares theorem 149 8 Geometric representation of algebraic numbers 152 8.1 The space Lst 152 9 Class-group and class-number 158 9.1 The class-group 158 9.2 An existence theorem 160 9.3 Finiteness of the class-group 165 9.4 How to make an ideal principal .166 9.5 Unique factorization of elements in an extension ring 170 CONTENTS ~ PART III NUMBER·THEORETIC APPLICA TIONS 10 Computational methods 179 10.1 Factorization of a rational prime 179 10.2 Minkowski's constants 183 10.3 Some class-number calculations 187 10.4 Tables 190 11 Fennat's Last Theorem 192 11.1 Some history 192 11.2 Elementary considerations 195 11.3 Kummer's lemma 198 11.4 Kummer's Theorem 202 11.5 Regular primes 206 12 Dirichlet's Units Theorem 211 12.1 Introduction 211 12.2 Logarithmic space 212 12.3 Embedding the unit group in logarithmic space 213 12.4 Dirichlet's theorem 215 Appendix 1 Quadratic Residues 223 A.1 Quadratic equations in Zm 224 A.2 The units of Zm 226 A.3 Quadratic Residues 232 Appendix 2 Valuations 245 References 250 Index 253 Preface The title of this book may be read in two ways. One is 'algebraic number-theory', that is, the theory of numbers viewed algebraically; the other, 'algebraic-number theory', the study of algebraic numbers. Both readings are compatible with our aims, and both are perhaps misleading. Misleading, because a proper coverage of either topic would require more space than is available, and demand more of the reader than we wish to; compatible, because our aim is to illustrate how some of the basic notions of the theory of algebraic numbers may be applied to problems in number theory. Algebra is an easy subject to compartmentalize, with topics such as 'groups', 'rings' or 'modules' being taught in comparative isolation. Many students view it this way. While it would be easy to exaggerate this tendency, it is not an especially desirable one. The leading mathematicians of the nineteenth and early twentieth centuries developed and used most of the basic results and techniques of linear algebra for perhaps a hundred years, without ever defining an abstract vector space: nor is there anything to suggest that they suf fered thereby. This historical fact may indicate that abstrac tion is not always as necessary as one commonly imagines; on the other hand the axiomatization of mathematics has led to enormous organizational and conceptual gains. x PREFACE xi Algebraic number theory illustrates both of these tend encies, and the 'creative tension' engendered by their overt opposition (and covert collaboration); and we hope that a study of it will encourage an awareness in the student that it is possible to use abstract algebra to prove theorems about something else. A particular target here is a partial proof of Fermat's Last Theorem, chosen for its historical importance and cultural notoriety, which add a certain piquancy to its excellence as a motive for the introduction of a number of key ideas. Fermat stated that the equation + xn yn = zn has no non-zero integer solutions for x, y, and z, when n is an integer greater than or equal to 3. We shall prove this, follow ing the original ideas of Kummer, for the case where n is a 'regular' prime; but under the simplifying assumption that n does not divide any of x, y, or z. Our algebra will not be as abstract as it might be. Pro fessionals may be aghast to learn that Galois group theory is not used, that valuations are given only perfunctory treat ment in an appendix, and that generalizations such as Dedekind domains have been suppressed. Instead, we make use of such 'classical' (a less polite word would be 'old fashioned') devices as symmetric polynomials and deter minants. The reason is that the latter are accessible to a much wider readership. For the same reason we have devoted more space to some aspects of the subject-matter (notably factor ization into primes) than current importance would warrant: these topics provide ample opportunity for concrete com putations which help familiarize the student with the under lying concepts. (In a lecture course some of this material could of course be omitted.) In mathematics it is important to 'get one's hands dirty', and the elegance of polished theories does not always provide suitable material. Prime factorization in specific number fields also displays the tendency of mathematical objects to take on a life of their own: sometimes something works, sometimes it does not,