On the prooves of Lindemann's theorem and Gelfond--Schneider's theorem

A. I. Galochkin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 4, pp. 1253-1260.

Résumé

The paper presents the new prooves of the Lindemann's theorem on the transcedence of the number $e^{\alpha}$ for non-zero algebraic $\alpha$ and the Gelfond–Schneider's theorem on the transcendence of the number $a^{\beta}$ for algebraic $a\ne0;1$ and algebraic irrational $\beta$. There is a difference from other prooves of Gelfond–Schneider's theorem. On the first step we construct the auxilary function with great order of zeroes at only the point $z=0$.