On algebraic independence of algebraic powers of algebraic numbers

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 $\mathbf Q$.
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