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Number Theory I

Fundamental Problems, Ideas and Theories

  • Book
  • © 1995

Overview

Editors:
  1. A. N. Parshin

    1. Steklov Mathematical Institute, Moscow, Russia

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  2. I. R. Shafarevich

    1. Steklov Mathematical Institute, Moscow, Russia

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  • Covers the most recent results around Fermat's Theorem (Andrew Wiles) and the Langlands Conjecture (Lafforgue)

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

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Table of contents (6 chapters)

  1. Front Matter

    Pages I-7
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  2. Problems and Tricks

    1. Elementary Number Theory

      • A. N. Parshin, I. R. Shafarevich
      Pages 8-48

    2. Some Modern Problems of Elementary Number Theory

      • A. N. Parshin, I. R. Shafarevich
      Pages 49-76

  3. Ideas and Theories

    1. Induction and Recursion

      • A. N. Parshin, I. R. Shafarevich
      Pages 77-93

    2. Arithmetic of Algebraic Numbers

      • A. N. Parshin, I. R. Shafarevich
      Pages 94-161

    3. Arithmetic of Algebraic Varieties

      • A. N. Parshin, I. R. Shafarevich
      Pages 162-208

    4. Zeta Functions and Modular Forms

      • A. N. Parshin, I. R. Shafarevich
      Pages 208-274

  4. Back Matter

    Pages 275-306
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Keywords

About this book

Preface Among the various branches of mathematics, number theory is characterized to a lesser degree by its primary subject ("integers") than by a psychologi­ cal attitude. Actually, number theory also deals with rational, algebraic, and transcendental numbers, with some very specific analytic functions (such as Dirichlet series and modular forms), and with some geometric objects (such as lattices and schemes over Z). The question whether a given article belongs to number theory is answered by its author's system of values. If arithmetic is not there, the paper will hardly be considered as number-theoretical, even if it deals exclusively with integers and congruences. On the other hand, any mathematical tool, say, homotopy theory or dynamical systems may become an important source of number-theoretical inspiration. For this reason, com­ binatorics and the theory of recursive functions are not usually associated with number theory, whereas modular functions are. In this report we interpret number theory broadly. There are compelling reasons to adopt this viewpoint. First of all, the integers constitute (together with geometric images) one of the primary subjects of mathematics in general. Because of this, the history of elementary number theory is as long as the history of all mathematics, and the history of modern mathematic began when "numbers" and "figures" were united by the concept of coordinates (which in the opinion of LR. Shafarevich also forms the basic idea of algebra).

Reviews

From the reviews of the second edition:

"Here is a welcome update to Number theory I. Introduction to number theory by the same authors … . the book now brings the reader up to date with some of the latest results in the field. … The book is generally well-written and should be of interest to both the general, non-specialist reader of Number Theory as well as established researchers who are seeking an overview of some of the latest developments in the field."

Philip Maynard, The Mathematical Gazette, Vol. 90 (519), 2006

[...] the first edition was a very good book; this edition is even better.

[...] Embedded in the text are a lot of interesting ideas, insights, and clues to how the authors think about the subject. [...]

Things get more interesting in Part II (by far the largest of the tree parts)[...] This part of the book covers such things as approaches through logic, algebraic number theory, arithmetic of algebraic varieties, zeta functions, and modular forms, followed by an extensive (50+ pages ) account of Wiles' proof of Fermat's Last Theorem. This is a valuable addition, new in this edition, and serves as a vivid example of the power of the "ideas and theories" that dominate this part of the book.

Also new and very interesting is Part III, entitled "Analogies and Visions,"

[...] The best surveys of mathematics are those written by deeply insightful mathematicians who are not afraid to infuse their ideas and insights  into their outline of subject. This is what we have here, and the result is an essential book. I only wish the price were lower so that I could encourage my students buy themselves a copy. Maybe I'll do that anyway.

Fernado Q. Gouvêa, on 09/10/2005

"This book is a revised and updated version of the first English translation. … Overall, the book is very well written, and has an impressive reference list. It is an excellent resource for thosewho are looking for both deep and wide understanding of number theory." (Alexander A. Borisov, Mathematical Reviews, Issue 2006 j)

"This edition feels altogether different from the earlier one … . There is much new and more in this edition than in the 1995 edition: namely, one hundred and fifty extra pages. … For my part, I come to praise this fine volume. This book is a highly instructive read with the usual reminder that there lots of facts one does not know … . the quality, knowledge, and expertise of the authors shines through. … The present volume is almost startlingly up-to-date … ." (Alf van der Poorten, Gazette of the Australian Mathematical Society, Vol. 34 (1), 2007)

Editors and Affiliations

  • Steklov Mathematical Institute, Moscow, Russia

    A. N. Parshin, I. R. Shafarevich

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