From Fermat To Minkowski: Lectures On The Theory Of Numbers And Its Historical Development (undergraduate Texts In Mathematics)

This book arose from a course of lectures given by the first author during the winter term 1977/1978 at the University of Münster (West Germany). The course was primarily addressed to future high school teachers of mathematics; it was not meant as a systematic introduction to number theory but rather as a historically motivated invitation to the subject, designed to interest the audience in number-theoretical questions and developments. This is also the objective of this book, which is certainly not meant to replace any of the existing excellent texts in number theory. Our selection of topics and examples tries to show how, in the historical development, the investigation of obvious or natural questions has led to more and more comprehensive and profound theories, how again and again, surprising connections between seemingly unrelated problems were discovered, and how the introduction of new methods and concepts led to the solution of hitherto unassailable questions. All this means that we do not present the student with polished proofs (which in turn are the fruit of a long historical development); rather, we try to show how these theorems are the necessary consequences of natural questions. Two examples might illustrate our objectives. Title ......Page 3 Copyright ......Page 4 Preface ......Page 5 Contents ......Page 7 Literature ......Page 10 CHAPTER 1 The Beginnings ......Page 13 CHAPTER 2 Fermat ......Page 17 CHAPTER 3 Euler ......Page 26 CHAPTER 4 Lagrange ......Page 44 CHAPTER 5 Legendre ......Page 69 CHAPTER 6 Gauss ......Page 76 CHAPTER 7 Fourier ......Page 114 CHAPTER 8 Dirichlet ......Page 121 CHAPTER 9 From Hermite to Minkowski ......Page 163 CHAPTER 10 A Preview of Reduction Theory ......Page 181 APPENDIX English Translation of Gauss's Letter to Dirichlet, November 1838 ......Page 190 Index ......Page 193 The book provides a self-contained introduction to classical Number Theory. All the proofs of the individual theorems and the solutions of the exercises are being presented step by step. Some historical remarks are also presented. The book will be directed to advanced undergraduate, beginning graduate students as well as to students who prepare for mathematical competitions (ex. Mathematical Olympiads and Putnam Mathematical competition) Tracing the story from its earliest sources, this book celebrates the lives and work of pioneers of modern mathematics: Fermat, Euler, Lagrange, Legendre, Gauss, Fourier, Dirichlet and more. Includes an English translation of Gauss's 1838 letter to Dirichlet. Winfried Scharlau, Hans Opolka ; [translators, Walter K. Bühler, Gary Cornell]. Translation Of: Von Fermat Bis Minkowski. Includes Bibliographies And Index.