## Transcendental Number TheoryFirst published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979, however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references. |

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### Contents

The origins | 1 |

2 Transcendence of e | 3 |

3 Lindemanns theorem | 6 |

Linear forms in logarithms | 9 |

2 Corollaries | 11 |

3 Notation | 12 |

4 The auxiliary function | 13 |

5 Proof of main theorem | 20 |

7 Successive minima | 76 |

8 Comparison of minima | 79 |

9 Exterior algebra | 81 |

10 Proof of main theorem | 82 |

Mahlers classification | 85 |

2 Anumbers | 87 |

3 Algebraic dependence | 88 |

4 Heights of polynomials | 89 |

Lower Bounds for linear forms | 22 |

2 Preliminaries | 24 |

3 The auxiliary function | 28 |

4 Proof of main theorem | 34 |

Diophantine equations | 36 |

2 The Thue equation | 38 |

3 The hyperelliptic equation | 40 |

4 Curves of genus 1 | 43 |

5 Quantitative bounds | 44 |

Class numbers of imaginary quadratic fields | 47 |

2 Lfunctions | 48 |

3 Limit formula | 50 |

4 Class number 1 | 51 |

5 Class number 2 | 52 |

Elliptic functions | 55 |

2 Corollaries | 56 |

3 Linear equations | 58 |

5 Proof of main theorem | 60 |

6 Periods and quasiperiods | 61 |

Rational approximations to algebraic numbers | 66 |

2 Wronskians | 69 |

4 A combinatorial lemma | 73 |

5 Grids | 74 |

6 The auxiliary polynomial | 75 |

5 Snumbers | 90 |

7 Tnumbers | 92 |

Metrical theory | 95 |

2 Zeros of polynomials | 96 |

3 Null sets | 98 |

4 Intersections of intervals | 99 |

5 Proof of main theorem | 100 |

The exponential function | 103 |

2 Fundamental polynomials | 104 |

3 Proof of main theorem | 108 |

The SiegalShidlovsky theorems | 109 |

2 Basic construction | 111 |

3 Further lemmas | 114 |

4 Proof of main theorem | 115 |

Algebraic independence | 118 |

2 Exponential polynomials | 120 |

3 Heights | 122 |

4 Algebraic criterion | 124 |

5 Main arguments | 125 |

Bibliography | 129 |

Original papers | 130 |

Further publications | 145 |

New developments | 155 |

162 | |

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### Common terms and phrases

Abelian Abelian functions absolute values Acta Arith algebraic independence algebraic integers algebraic number field algébriques Amer applications Academic Press assume Australian Math auxiliary function Chapter class number clearly complex numbers Compositio Math conjugates defined denotes Diophantine approximation Diophantine equations E-functions elements elliptic functions estimates exist finitely follows forms in logarithms Further Gelfond given height Hence hypothesis ibid induction inequality integer coefficients leading coefficient Lemma linear forms linearly independent logarithms of algebraic London Math lower bound Mahler main theorem Mathematika nombres transcendants non-negative integers Number Th number theory numbers with degrees obtained p-adic parallelepiped plainly polynomial with degree positive integer problem Proc Proof of main proof of Theorem prove quadratic irrational Rational approximations rational integers result S-numbers satisfying Siegel similar titles solutions sufficiently large suppose Theorem 3.1 Thue Thue equation TIJDEMAN Trans Transcendence theory transcendental numbers U-number Uber variables whence Wronskian Zametki zeros