Geodesic
The algebraic independence of certain transcendental numbers
Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 51-58 
A. A. Shmelev. The algebraic independence of certain transcendental numbers. Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 51-58. http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a6/
@article{MZM_1968_3_1_a6,
     author = {A. A. Shmelev},
     title = {The algebraic independence of certain transcendental numbers},
     journal = {Matemati\v{c}eskie zametki},
     pages = {51--58},
     year = {1968},
     volume = {3},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a6/}
}
TY  - JOUR
AU  - A. A. Shmelev
TI  - The algebraic independence of certain transcendental numbers
JO  - Matematičeskie zametki
PY  - 1968
SP  - 51
EP  - 58
VL  - 3
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a6/
LA  - ru
ID  - MZM_1968_3_1_a6
ER  - 
%0 Journal Article
%A A. A. Shmelev
%T The algebraic independence of certain transcendental numbers
%J Matematičeskie zametki
%D 1968
%P 51-58
%V 3
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a6/
%G ru
%F MZM_1968_3_1_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

Given the three numbers of, $a^\beta_1$, $a^\beta_2$, and $\frac{\ln a_2}{\ln a_1}$, where $a_1$ and $a_2$ are algebraic numbers whose logarithms are linearly independent in a rational field and $\beta$ is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.