Transcendental Number Theory

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Alan Baker
Cambridge University Press, Sep 28, 1990 - Mathematics - 165 pages
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First 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
Index
162

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About the author (1990)

Alan Baker was born on August 19, 1939 in London, England. He is a British mathematician. He is known for his work in Number Theory. He attended the University of Cambridge and is a fellow of the American Mathematical Society. His thesis in 1964 was entitled, Some Aspects of Diophantine Approximation. At age 31, he was awarded the Fields Medal and in 1972 we won the Adams Prize.

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