Geodesic
A~criterion for algebraic dependence of transcendental numbers
Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 553-562

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We obtain a theorem concerning the algebraic dependence of the $p+1$ numbers $\theta,\theta_1,\dots,\theta_p$ subject to the condition that the numbers $\theta_1,\dots,\theta_p$ are algebraically independent and possess a “sufficiently good” estimate of measure of algebraic independence. 
@article{MZM_1974_16_4_a6,
     author = {A. A. Shmelev},
     title = {A~criterion for algebraic dependence of transcendental numbers},
     journal = {Matemati\v{c}eskie zametki},
     pages = {553--562},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a6/}
}
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A. A. Shmelev. A~criterion for algebraic dependence of transcendental numbers. Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 553-562. http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a6/