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Number Theory I

Fundamental Problems, Ideas and Theories

  • A. N. Parshin
  • I. R. Shafarevich
Book

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 49)

Table of contents

  1. Front Matter
    Pages I-7
  2. Problems and Tricks

    1. A. N. Parshin, I. R. Shafarevich
      Pages 8-48
    2. A. N. Parshin, I. R. Shafarevich
      Pages 49-76
  3. Ideas and Theories

    1. A. N. Parshin, I. R. Shafarevich
      Pages 77-93
    2. A. N. Parshin, I. R. Shafarevich
      Pages 94-161
    3. A. N. Parshin, I. R. Shafarevich
      Pages 162-208
    4. A. N. Parshin, I. R. Shafarevich
      Pages 208-274
  4. Back Matter
    Pages 275-306

About this book

Introduction

This book surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems (including some modern areas such as cryptography, factorization and primality testing), the central ideas of modern theories are exposed: algebraic number theory, calculations and properties of Galois groups, non-Abelian generalizations of class field theory, recursive computability and links with Diophantine equations, the arithmetic of algebraic varieties, connections with modular forms, zeta- and L-functions. The authors have tried to present the most significant results and methods of modern time. An overview of the major conjectures is also given in order to illustrate current thinking in number theory. Most of these conjectures are based on analogies between functions and numbers, and on connections with other branches of mathematics such as algebraic topology, analysis, representation theory and geometry

Keywords

Arakelov geometry Arithmetic der algebraischen Zahlen Elementare Zahlentheorie Elementary number theory Langlands program Langlands-Programm Modular forms Non-commutative geometry arithmetic of algebraic numbers diophantine equations diophantische Gleichungen elliptic curves elliptische Kurven logic public public key Verschlüsselungssysteme public key cryptosystems zeta-functions

Editors and affiliations

  • A. N. Parshin
    • 1
  • I. R. Shafarevich
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-08005-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1995
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-662-08007-8
  • Online ISBN 978-3-662-08005-4
  • Series Print ISSN 0938-0396
  • Buy this book on publisher's site