Algebraic independence of elements from \(\mathbb C_p\) over \(\mathbb Q_p\). I
type: Item
instance of: scholarly article
title: Algebraic independence of elements from \(\mathbb C_p\) over \(\mathbb Q_p\). I
zbMATH Open document ID: 1027.11050
DOI: 10.1007/PL00012456
author: Peter Bundschuh
author: V. G. Chirskii
published in: Archiv der Mathematik
publication date: 2003-03-25T00:00:00Z
review text: 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 \(b_j\).
Mathematics Subject Classification ID: 11J85
Mathematics Subject Classification ID: 11J61
zbMATH DE Number: 1887596
zbMATH Keywords: p-adic field
zbMATH Keywords: algebraic independence
zbMATH Keywords: p-adic numbers
reviewed by: Daniel Bertrand
MaRDI profile type: Publication
full work available at URL: pl00012456
OpenAlex ID: W2337122012
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description: scientific article; zbMATH DE number 1887596