Algebraic independence of elements from \(\mathbb C_p\) over \(\mathbb Q_p\). I
The noncompleteness of its algebraic closure provides the \(p\)-adic field \(\mathbb Q_p\) with a richer transcendence theory than its classical counterpart. For previous work on this topic, see \textit{A. Escassut} [J. Number Theory 16, 395--402 (1983; Zbl 0519.10023), Correction 19, 451 (1984; Zbl 0562.10014)] and \textit{D. Lampert} [J. Number Theory 23, 279--284 (1986; Zbl 0586.12021)]. In this paper, the authors prove the algebraic independence over \(\mathbb Q_p\) of \(p\)-adic numbers of the type \(\sum_{i\geq 0}a_ip^{r_i}\), where the \(a_i\)'s are \(p\)-adic integers, and the exponents \(r_i\)'s are rational integers, with sufficiently independent denominators (in particular, the truncated sums lie in highly ramified extensions of \(\mathbb Q_p\)). The general criterion they obtain apply to such families of numbers as \(\alpha_j:= \sum_{k\geq 0}p^{r_j^k}\), with \(r_j\in \mathbb Q\), \(r_j>1\), \((\text{den}(r_j), \text{den}(r_{j'}))= 1\), or \(\beta_j= \sum_{k\geq 0}(b_j)_k p^{r^k}\), with distinct positive integers \(b_j\).
scientific article; zbMATH DE number 1887596