Geodesic
Algebraic independence of $p$-adic numbers
Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 565-579

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We prove lower bounds for the transcendence degree of fields generated by values of the $p$-adic exponential function. In particular, we estimate the transcendence degree of the field $\mathbb Q(e^{\alpha_1},\dots,e^{\alpha_d})$, where $\alpha_1,\dots,\alpha_d$ are algebraic (over the field of rational numbers) $p$-adic numbers that form a basis of a finite extension of $\mathbb Q$. 
@article{IM2_2008_72_3_a5,
     author = {Yu. V. Nesterenko},
     title = {Algebraic independence of $p$-adic numbers},
     journal = {Izvestiya. Mathematics },
     pages = {565--579},
     publisher = {mathdoc},
     volume = {72},
     number = {3},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a5/}
}
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Yu. V. Nesterenko. Algebraic independence of $p$-adic numbers. Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 565-579. http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a5/