Geodesic
On algebraic independence of algebraic powers of algebraic numbers
Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 429-454

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It is proved that among the numbers $\alpha^\beta,\alpha^{\beta^2},\dots,\alpha^{\beta^{d-1}}$, where $\alpha$ is algebraic, $\alpha\ne0,1$ and $\beta$ is algebraic of degree $d\geqslant2$, there are no fewer than $[\log_2(d+1)]$ which are algebraically independent over $\mathbf Q$. Bibliography: 17 titles. 
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     author = {Yu. V. Nesterenko},
     title = {On algebraic independence of algebraic powers of algebraic numbers},
     journal = {Sbornik. Mathematics},
     pages = {429--454},
     publisher = {mathdoc},
     volume = {51},
     number = {2},
     year = {1985},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1985_51_2_a7/}
}
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Yu. V. Nesterenko. On algebraic independence of algebraic powers of algebraic numbers. Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 429-454. http://geodesic.mathdoc.fr/item/SM_1985_51_2_a7/