Algebraic independence of elements from $\mathbb{C}_p$ over $\mathbb{Q}_p$, I

Abstract.

Until now there are only very few contributions to the problem in the title, probably because it has no analogue in classical transcendence theory, $\mathbb{C}$ being a quadratic extension of $\mathbb{R}$. The main aim of this article is to prove a new criterion giving sufficient conditions for the algebraic independence of finitely many elements from $\mathbb{C}_p$ over $\mathbb{Q}_p$, if these elements are defined by certain p-adic series. Some applications of the criterion are also presented.