Hilbert's seventh problem: solutions and extensions (Q2796413)

The book deals with the Hilbert's seventh problem which says that for algebraic numbers \(\alpha\) and \(\beta\) with \(\beta\) irrational, the number \(\alpha^\beta\) is algebraic only in the case when \(\alpha\in\{ 0,1\}\). The content is split into 7 chapters. NEWLINENEWLINEThe first chapter is an introduction to the problem and includes the history during the last three centuries. This chapter deals with the several variations of the Hilbert's conjectures. Liouville's theorem, Euler's conjecture and Lindemann-Weierstrass theorem are included also. It starts with the Fourier's proof of the irrationality of the Napier's constant \(e\) and shows why Fourier's proof failed when we want to prove the transcendence of the number \(e\). NEWLINENEWLINEThe second chapter describes some variations of how to prove the transcendence of the numbers \(e\), \(\pi\) and \(e^{\sqrt 2}\). NEWLINENEWLINEThe third chapter deals with the Gelfond's proof that the number \(e^\pi\) is transcendental and Kuzmin's proof that the number \(2^{\sqrt 2}\) is transcendental. NEWLINENEWLINEThe topic of the fourth chapter is the Golfond's proof that if the ratio \(x=\frac {\log \alpha}{\log \beta}\) is an irrational number, where \(\alpha\) and \(\beta\) are algebraic numbers, then \(x\) is a transcendental number. NEWLINENEWLINESchneider's proof for the Hilbert's seventh problem including also the Siegel's lemma can be found in the fifth chapter. NEWLINENEWLINEThe sixth chapter deals with the extension of the Hilbert's seventh problem to transcendental functions. First the author shows the hint of the proof that if \(\{ x_1, x_2\}\) and \(\{ y_1, y_2, y_3\}\) are the \(\mathbb Q\)-linearly independent sets of complex numbers then at least one the six numbers \(e^{x_1y_1}\), \(e^{x_1y_2}\), \(e^{x_1y_3}\), \(e^{x_2y_1}\), \(e^{x_2y_2}\) and \(e^{x_2y_3}\) is transcendental. Then, the author looks for the analogue of Schneider-Lang theorem concerning the algebraic independence of meromorphic functions. Finally, the author presents the elliptic versions of the Hermite-Lindemann theorem and Gelfond-Schneider theorem. NEWLINENEWLINEThe last seventh chapter goes it a discussion of how to generalized Hilbert's seventh problem and the methods which would solve this problem. One way is an idea of Ricci. Second one is a method of Franklin. The development of Waldschmidt is also included.