数论．5，超越数

Notation

Introduction

0.1 Preliminary Remarks

0.2 Irrationality of 2

0.3 The Number π

0.4 Transcendental Numbers

0.5 Approximation of Algebraic Numbers

0.6 Transcendence Questions and Other Branches of Number Theory

0.7 The Basic Problems Studied in Transcendental Number Theory

0.8 Different Ways of Giving the Numbers

0.9 Methods

Chapter 1 Approximation of Algebraic Numbers

1 Preliminaries

1.1 Parameters for Algebraic Numbers and Polynomials

1.2 Statement of the Problem

1.3 Approximation of Rational Numbers

1.4 Continued Fractions

1.5 Quadratic Irrationalities

1.6 Liouvilles Theorem and Liouville Numbers

1.7 Generalization of Liouvilles Theorem

2 Approximations of Algebraic Numbers and Thues Equation

2.1 Thues Equation

2.2 The Case n = 2

2.3 The Case n > 3

3 Strengthening Liouvilles Theorem First Version of Thues Method

3.1 A Way to Bound qθ-ρ

3.2 Construction of Rational Approximations for

3.3 Thues First Result

3.4 Effectiveness

3.5 Effective Analogues of Theorem 1.6

3.6 The First Effective Inequalities of Baker

3.7 Effective Bounds on Linear Forms in Algebraic Numbers

4 Stronger and More General Versions of Liouvilles Theorem and Thues Theorem

4.1 The Dirichlet Pigeonhole Principle

4.2 Thues Method in the General Case

4.3 Thues Theorem on Approximation of Algebraic Numbers

4.4 The Non-effectiveness of Thues Theorems

5 Further Development of Thues Method

5.1 Siegels Theorem

5.2 The Theorems of Dyson and Gelfond

5.3 Dysons Lemma

5.4 Bombieris Theorem

6 Multidimensional Variants of the Thue-Siegel Method

6.1 Preliminary Remarks

6.2 Siegels Theorem

6.3 The Theorems of Schneider and Mahler

7 Roths Theorem

7.1 Statement of the Theorem

7.2 The Index of a Polynomial

7.3 Outline of the Proof of Roths Theorem

7.4 Approximation of Algebraic Numbers by Algebraic Numbers

7.5 The Number k in Roths Theorem

7.6 Approximation by Numbers of a Special Type

7.7 Transcendence of Certain Numbers

7.8 The Number of Solutions to the Inequality (62) and Certain Diophantine Equations

8 Linear Forms in Algebraic Numbers and Schmidts Theorem

8.1 Elementary Estimates

8.2 Schmidts Theorem

8.3 Minkowskis Theorem on Linear Forms

8.4 Schmidts Subspace Theorem

Chapter 2 Effective Constructions in Transcendental Number Theory

Chapter 3 Hillberts Seventh Problem

Chapter 4 Multidimensional Generalization of Hillberts Seventh Problem

Chapter 5 Values of Analytic Functions That Satisgy Linear Differential Equations

Chapter 6 Algebraic Independence of the Values of Analytic Functions That Have an Additaon La

《数论4：超越数(影印版)》is a survey of the most important directions of research in transcendental number theory. The central topics in this theory include proofs of irrationality and transcendence of various numbers，especially those，that arise as the values of special functions. Questions of this sort go back to ancient times. An example is the old problem of squaring the circle，which Lindemann showed to bc impossible in 1882，when hc proved that Pi is a trandental number. Eulers conjecture that the logarithm of an algebraic number to an algebraic base is transcendental was included in Hilberts famous list of open problems; this conjecture was proved by Gelfond and Schneider in 1934. A more recent result was Anervs surprising proof of the irrationality of ξ(3)in 1979.
　　The quantitative aspects of the theory have important applications to the study of Diophantine equations and other areas of number theory. For a reader interested in different branches of number theory，this monograph provides both an overview of the central ideas and techniques of transcendental number theory，and also a guide to the most important results and references.

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