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

摘要

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.