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Kenneth Ireland (deceased) J.H. Ewing Editorial Board Department of M athematics Indiana University Bloomington, IN 47405 USA With 1 illustration. of Mathematics Michael Rosen Department Brown University Providence, USA RI02912 F. W. Gehring Department of P.R. Halmos Department of Mathematics Mathematics of Michigan University Ann Arbor, MI 48109 USA Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classification: 11-01, 11-02 Library of Congress Cataloging-in-Publication Ireland, Kenneth F. A classical to modem number theory / Kenneth introduction Data Ireland, Michael Rosen.-2nd ed. p. cm.-(Graduate texts in mathematics; Includes bibliographical references 1. Number theory. I. Rosen, Michael I. II. Title. III. Series. 84) and index. QA241.I667 1990 512.7-dc20 90-9848 "A Classical "Elements of Number Theory" Introduction to Modem Number Theory" is a revised and expanded version of published in 1972 by Bogden and Quigley, Inc., Publishers. © 1972, 1982, 1990 Springer-Verlag New York, Inc. Use in connection (Springer-Verlag or copied in whole or in part without the analysis. adaptation, This work may not be translated of the publisher All rights reserved. written permission New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly electronic known or hereafter developed The use of general descriptive even if the fonner are not especially as understood freely by anyone. is forbidden. names, trade names, trademarks, is not to be taken as a sign that such names, be used computer software, or by similar or dissimilar by the Trade Marks and Merchandise with any fonn of information New York, Inc., 175 Fifth Avenue, Marks Act, may accordingly methodology now identified, storage and retrieval, etc., in this pubJication, ISBN 0-387-97329-X Springer-Verlag ISBN 3-540-97329-X New York Berlin Heidelberg Berlin Heidelberg Springer-Verlag New York

Preface to the Second Edition It is now 10 years since the first edition of this book appeared in 1980. The decade has seen tremendous advances take place in mathe intervening matics generally, able to treat some of these advances, and with the addition of two new we are able to cover some portion of this new material. chapters, As examples of important new work that we have not included, we and in number theory in particular. It would seem desir mention the following two results: (2) Let PI, P2, and P3 be three distinct (1) The first case of Fermat's last theorem is true for infinitely many many primes p, the prime exponents p. This means that, for infinitely equation xP + yP = zP has no solutions with p r in nonzero integers xyz. This was proved by L.M. Adelman and D.R. Heath-Brown and by E. Fouvry. An overview of the proof is given by independently Heath-Brown in the Mathematical Intelligencer (Vol. 7, No.6, 1985). primes. Then at least one of them is many primes q. R-ecall that E. Artin a primitive conjectured that, if a E Z. is not 0, 1, -1, or a square, then there are q. The infinitely theorem we have stated was proved in a weaker form by R. Gupta and M.R. Murty, and then strengthened by the combined efforts of R. Gupta, M.R. Murty, V.K. Murty, and D.R. Heath-Brown. tion of this result, M.R. Murty in the Mathematical Intelligencer (Vol. 10, No.4, 1988). An exposi as well as an analogue on elliptic curves, is given by many primes q such that a is a primitive root modulo root for infinitely The new material that we have added falls within the frame geometry. In Chapter 19 we give a complete proof of principally work of arithmetic L.J. Mordell's fundamental theorem, which asserts that the group of ra- v

VI Preface to the Second Edition curve, defined over the rational numbers, is In keeping with the spirit of the book, the proof (due in tional points on an elliptic finitely generated. essence to A. Weill is elementary. or any other advanced machinery. and a weak form of the Dirichlet unit theorem; the text. It makes no use of cohomology groups It does use finiteness of class number both results are proved in The second new chapter, Chapter 20, is an overview of G. Faltings's curves, especially and recent progress on the arithmetic of the work of B. Gross, V.A. Kolyvagin, K. to proof of the Mordell conjecture elliptic Rubin, and D. Zagier. Some of this work has surprising applications other areas of number theory. We discuss one application theorem, due to G. Frey, J.P. Serre, and K. Ribet. Another important application class numbers of imaginary quadratic number fields. This comes about by combining the work of B. Gross and D. Zagier with a result of D. Gold feld. This chapter contains few proofs. Its main purpose is to give an informative background necessary to a better understanding important new developments. is the solution of an old problem due to K.F. Gauss about survey in the hope that the reader will be inspired and appreciation to Fermat's last to learn the of these The rest of the book is essentially_ unchanged. An attempt has been In an effort to keep confusion to a made to correct errors and misprints. minimum, we have not changed the bibliography New references found at the end of those chapters. and others for submitting thank Linda Guthrie for typing portions of the final chapters. Chapters 19 and 20, will be for the two new chapters, from the first edition. a list of misprints We would like at the end of the book. to thank TofU Nakahara Also, we We have both been very pleased with the warm reception that the first It is our hope that the new edition will edition of this book received. continue to entice readers to delve deeper into the mysteries of this an cient, beautiful, February 1990 and still vital subject. Kenneth Ireland Michael Rosen Second Corrected Addendum to Second Edition, The.second tions and the addition of a few clarifyin g comments. K. Conrad, M. Jastrzebski, F. Lemmermeyer trouble to send us detail printing of mispr of the second edition is unchanged Printing except ed lists ints. and others who took the I would like to thank for correc November 1992 Michael Rosen

Preface book the primary audience of undergrad ch backgro of upper level and greatly expanded version of our book Elements published in 1972. As with the first This book is a revised N umber Theory uate mathematics majors and we envisage consists graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract portion of Chapters small amount of supplemen knowledge of Galois theory, and in Chapters 1 6 and 18 the theory 1-11 can be read even without su und with the aid of a assume some tary reading. The later algebra. A large Number theory is an ancient of complex variables is necessary. subject and , make a very limited selection from the array of possible topics. Our focus is on topics which point in the geometry. we have found it possible to exposit some material without requiring very much in the way of technical is classical in the sense that is was dis but it is also going on at the present ductory book must, of necessity fascinating direction of algebraic careful selection of subject matter rather background. Most of this material covered during the because time. nineteenth y related to important number theory and arithmetic century and earlier, an acquaintance with algebraic By a it is intimatel chapters advanced research modern its content is vast . Any intro In Chapters 1-5 we discuss prime numbers, unique factorizati and the law of quadratic reciprocity. und. Nevertheless it is remarkable how a metic functions, congruences, is demanded in the way of backgro modicum subject. to a natural questio Z/nZ? For example, of group and ring theory introduces unexpected order into the many scattered results n : What is the structur turn out to be parts e of the group of the ring units in of the answer on, arith Very little VB

Vllt Preface law, one of the major reciprocity, by formulating Artin reciprocity of algebraic number theory. We travel along the road beyond Reciprocity laws constitute a major theme in the later chapters. The law beautiful in itself, is the first of a series of reciprocity of quadratic laws which lead ultimately to the a�hievements quadratic reciprocity biquadratic reciprocity. In preparation for this many of the techniques algebra integers, investigation material the more advanced reciprocity cubic and of numbers and algebraic etc. Another important tool in this of Gauss and Jacobi sums. This we formulate and prove the Eisenstein of primes, theory generalization of these results, 6-9. Later in the book and proving the laws of are introduced; algebraic (and in others!) is the is covered in Chapters finite fields, splitting ic number theory partial law. A second major theme is that of diophant ine equations, at first over finite over the ra numbers. The discussion of polynomial over finite fields is begun in Chapters 8 and 10 and culminates in tional of a portion of the paper " Number of solutions development 11 with an exposition fields" by A. Weil. This paper, recent In Chapters fields and later equations Chapter of equations over finite been very influential in the and number theory. over the rational sums of squares to Fermat's developed earlier we are able point of view. Chapter 18 is about the arithmetic fers from the in that it is primarily definitions centrating something of the beauty of is being done and many mysteries remain. and statements of results on some important special 17 and 18 we cons 17 covers numbers. Chapter earlier chapters of elliptic curves. It dif an overview with many but few proofs. Nevertheless, by con to the reader cases we hope to convey the accomplishments in this area where much work published in 1948, has of both algebraic ider diophantine from many standard topics geometry equations Last Theorem. However, to treat a number of these topics because of material from a novel The third, and final, major theme is that of zeta functions. zeta function associated to varieties 11 we In Chapter defined over finite 16 we discuss the Riemann zeta function and the Dir ichlet discuss the congruence fields. In Chapter L-functions. In Chapter 18 we discuss algebrai ned over the rational Zeta functions single function analysis compress and make possibl to number theory. c curve defi the zeta function associat ed to an numbers and Hecke L-functions. a large amount of arithmetic information into a e the application of the powerful methods of l'hroughout the book we place considerab le emphasis on In the notes at the end of each chapter we give a brief the history of to the literature. The historical bibliography is extensi ve our subject. sketch and provide containing provide the reader with a wealth references many items both classical and modern. Our aim has been to of material for further study. There are many exercises, some routine, some challenging. Some of the by providing a step by step guide through the supplement the text exercises proofs of important from results been adapted results. In the later chapters which have appeared a number of exercises in the recent literatur have e. We

Preface IX hope that working through the exercises as instruction. will be a source of enjoyment as well In the writing of this book we have been helped immensely by the interest nce of many matherr�atical friends and acquaintances. We thank ar we would like to thank Henry Pohlmann who insisted conclusion, David Goss for allowing 16, and Oisin McGuiness logical themes to their some of his work into Chapter assistance patience and expertise in the prepara h, Janice Phillips, tion of Chapter cially large portions in typing and espe 1 8. We would Carol Ferreira, to express his support during his sabbati of the manuscript. gratitude to the Vaughn cal year in wishes Fund for financial In particul and assista them all. we follow certain us to incorporate for his invaluable like to thank Dale Cavanaug for their Finally, the second author Foundation Berkeley, California (1979/80). July 25, 1981 Kenneth Ireland Michael Rosen

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