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Making transcendence transparent. An intuitive approach to classical transcendental number theory. (English) Zbl 1092.11031
New York, NY: Springer (ISBN 0-387-21444-5/hbk). viii, 263 p. \$ 39.95; EUR 39.95/net; £ 30.50; sFr 73.00 (2004).

One of the goals of the authors is to provide the reader with an intuitive framework in which the major classical results of transcendental number theory can be appreciated. In the title of the book, classical transcendental number theory refers to the most widely known results that were obtained in the nineteenth and early twentieth centuries. As pointed out by the authors, the underlying principles upon which the entire discipline is based are both straightforward and central to all of number theory. This book is an introduction to the subject which is supposed to enable the reader to pursue later his study with more modern results.

The traditional numbering with chapters is replaced here by a numbering with numbers, usually given by the first digits of the decimal expansion. Number 0 is $\sqrt{2}$. This chapter is a down to earth introduction to the basic concepts. Definitions of integers, rational numbers, real numbers, complex numbers are given with motivations. The two theorems of this section are as follows: the number $\sqrt{2}$ is not a rational number and the number $i$ is not the limit of a sequence of rational numbers (the proof of the latter is left as a challenge for the reader).

Number 1 is ${\sum }_{n=1}^{\infty }{10}^{-n!}$. This chapter is dedicated to Liouville’s numbers.

Number 2 is $e$. The corresponding chapter includes a proof of the transcendence of this number (Hermite’s Theorem) as well as a proof of the irrationality of $\pi$ (Lambert’s Theorem).

Number 3 is ${e}^{\sqrt{2}}$. This chapter deals with Hermite-Lindemann’s Theorem on the transcendence of ${e}^{\alpha }$ for $\alpha$ a non zero algebraic number, and with Lindemann-Weierstrass’ Theorem on the algebraic independence of such numbers for linearly independent $\alpha$’s.

Number 4 is ${e}^{\pi }$. The authors give a proof of the transcendence of this number (Gel’fond’s Theorem going back to 1929). They start by proving with some details ${e}^{\pi }\ne 23·25$, next they prove the irrationality, and finally the transcendence of this number.

Number 5 is ${2}^{\sqrt{2}}$. A solution of Hilbert’s seventh problem (Gel’fond-Schneider’s 1934 Theorem) is given.

Number 6 is $e+{\sum }_{n=1}^{\infty }{10}^{-n!}$. The transcendence of this number is deduced from the properties of Mahler’s classification of transcendental numbers.

Number 7 is ${\Gamma }{\left(1/4\right)}^{2}/\sqrt{\pi }$. Some of Schneider’s results on the transcendence of periods of elliptic curves are proved.

Number 8 is ${e}_{C}\left(1\right)$ where ${e}_{C}$ is the exponential function of Carlitz module over the field with two elements. This last chapter is an introduction to the theory of transcendence in fields of finite characteristic.

An appendix provides basic facts from complex analysis which are required for the proofs.

This book is aimed at beginners who like to have examples and detailed proofs with plenty of details and lengthy arguments. The reader who is looking for a concise treatment of the subject will prefer Alan Baker’s book [Transcendental number theory. Cambridge Mathematical Library. Cambridge etc.: Cambridge University Press (1975; Zbl 0715.11032) (1979; Zbl 0497.10023) (1990; Zbl 0297.10013)]. Those who either wish reliable information on the history or are interested with the state of the art of the subject will find them in the book by N. I. Fel’dman and Yu. V. Nesterenko [Transcendental numbers. Transl. from the Russian by Neal Koblitz. Encyclopaedia of Mathematical Sciences. 44. Berlin: Springer (1998; Zbl 0885.11004)].

##### MSC:
 11Jxx Diophantine approximation 11-01 Textbooks (number theory)