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Transcendental Number Theory

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Cambridge University Press, Sep 28, 1990 - Mathematics - 176 pages
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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Review: Transcendental Number Theory

User Review  - Smcleish - Goodreads

Originally published on my blog here in June 1998. This book has been the standard survey of the theory of transcendental numbers for some time now - the first edition was published in the mid ... Read full review

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Selected pages

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

Common terms and phrases

Abelian Abelian functions absolute values Acta Arith algebraic independence algebraic integer algebraic number field algebriques Amer applications Academic Press approximations to algebraic arguments assume Australian Math auxiliary function Chapter class number clearly complex numbers conjugates defined denotes Diophantine approximation Diophantine equations elements elliptic functions estimate exist finitely forms in logarithms Further Gelfond given height Hence hypothesis ibid imaginary quadratic fields induction inequality integer coefficients leading coefficient Lemma linear forms linearly independent logarithms of algebraic London Math lower bound Mahler 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 respectively result S-numbers satisfying similar titles solutions sufficiently large suppose Theorem 3.1 Thue Thue equation TIJDEMAN Transcendence theory transcendental numbers vanish identically variables whence Wronskian Zametki zeros

References to this book

From other books

The Arithmetic of Elliptic Curves
The Arithmetic of Elliptic Curves
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Limited preview - 2000
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From Google Scholar

The 3x+ 1 Problem and Its Generalizations
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Two-Point Step Size Gradient Methods
JONATHAN BARZILAI, JONATHAN M BORWEIN - 1988 - IMA Journal of Numerical Analysis
Discrete Logarithms in GF (p) using the Number Field Sieve
Daniel M Gordon
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References from web pages

Alan Baker: Transcendental Number Theory
Transcendental Number Theory. Alan Baker. Review date: 26/6/98 Publisher: CUP Published: 1975. This book has been the standard survey of the theory of ...
www.geocities.com/ Athens/ Academy/ 6422/ rev0075.html

JSTOR: Transcendental Number Theory
Transcendental number theory, by Alan Baker. Pp. x, 147. ?4-90. 1975. SBN 0 521 20461 5 (Cambridge University Press) Within the space of a mere 130 pages ...
links.jstor.org/ sici?sici=0025-5572(197512)2%3A59%3A410%3C280%3ATNT%3E2.0.CO%3B2-G

Transcendental number theory, by Alan Baker, Cambridge Univ. Press ...
transcendental number theory. Baker's book is the book on transcendental numbers. In only 128 pp. he. covers a majority of those areas that have reached ...
projecteuclid.org/ DPubS/ Repository/ 1.0/ Disseminate?handle=euclid.bams/ 1183541479& view=body& content-type=pdf_1

11J: Diophantine approximation, transcendental number theory
We include material using transcendental number theory to solve Diophantine ... Baker, Alan: "Transcendental number theory", Cambridge University Press, ...
www.math.niu.edu/ ~rusin/ known-math/ index/ 11JXX.html

Transcendental number theory (Baker A., 1990)
Electronic literature gets more accessible and comfortable than the printed one: Dleex, the first Web 2.0 library
www.dleex.com/ details/ ?4176

4shared.com - online file sharing and storage - download ...
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Transcendental number - Wikipedia, the free encyclopedia
... Mathematische Annalen 43:216–219 (1893). Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975, ISBN 0-521-39791-X. ...
en.wikipedia.org/ wiki/ Transcendental_number

1939A.D.
Baker has also made contributions to research on transcendental number theory, including Hilbert's seventh problem, which asked whether or not aq was ...
faculty.oxy.edu/ jquinn/ home/ Math490/ Timeline/ 1939AD.html

Baker A. Трансцендентальная теория числа (1975) (ISBN 0521204615 ...
Please wait[ Download Baker A. Transcendental number theory (CUP, 1975)(K)(T)(ISBN 0521204615)(155s).djvu ]. © 2007 eknigu.
www.eknigu.com/ info/ M_Mathematics/ MT_Number%20theory/ Baker%20A.%20Transcendental%20number%20theory%20(CUP,%201975)(K...

Document sans-titre
BAKER A., Transcendental Number Theory, Camb. Math. Library Series, 1990. BLYTH ts and ROBERTSON ef, Basic Linear Algebra, Springer-Verlag, 1998. ...
www.cirs-tm.org/ books/ mathematics/ ALGEBRA.htm

About the author (1990)

Alan Baker has illustrated "The Odyssey" and "The Story of King Arthur," both published by Kingfisher. In addition, Alan created the popular and successful "Little Rabbits" series, which has sold over half-a-million copies in the U.S. Alan is a lecturer in illustration at the University of Brighton, England.

Bibliographic information

QR code for Transcendental Number Theory
TitleTranscendental Number Theory
Cambridge Mathematical Library
AuthorAlan Baker
Editionillustrated, reprint, reissue
PublisherCambridge University Press, 1990
ISBN052139791X, 9780521397919
Length176 pages
Subjects › 

Mathematics / Number Theory
  
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