Completion of r.t. extensions of local fields. I
type: Item
instance of: scholarly article
title: Completion of r.t. extensions of local fields. I
zbMATH Open document ID: 0852.12003
DOI: 10.1007/PL00004526
author: V. Alexandru
author: Angel Popescu
author: N. Popescu
published in: Mathematische Zeitschrift
publication date: 1996-11-07T00:00:00Z
full work available at URL: https://eudml.org/doc/174873
review text: Let \((K,\nu)\) be a local field and \((K(X),w_0)\) the so called Gauss residual transcendental extension of it; so, if \(P(X)=a_0+a_1 X+\dots+a_k X^k\) is a polynomial in \(K[ X]\) we have \(w_0 (P(X))=\inf_{0\leq i\leq k} \{\nu (a_i)\}\). We denote by \(K\{\{ X\}\}\) the local field of all formal Laurent series \(\alpha=\sum_{n\in \mathbb{Z}} a_n X^n\) such that \(\{\nu (a_n) \}_n\) is lower bounded and \(\nu(a_{-n} )\to \infty\) if \(n\to \infty\), with the valuation \(u\) where \(u(\alpha)=\inf_{n\in \mathbb{Z}} \{\nu (a_n)\}\). We consider the embedding \(\omega\) of the valued field \((K (X), w_0)\) in the local field \((K\{\{ X\}\},u)\) such that \[ \omega (P(X))=P(X), \quad \text{for any polynomial} \quad P(X)\in K[ X]; \tag{1} \] if \(P(X)=\prod^s_{i=1} (X- a_i)\), \(a_i\in \overline {K}\), the algebraic closure of \(K\), then \[ \omega (1/ P(X))=\omega \Biggl( \prod^s_{i=1} 1/(X- a_i) \Biggr)=(1/X^s) \prod^s_{i=1} \sum^\infty_{n=0} (a_i/ X)^n \tag{2} \] if \(\nu (a_i )>0\), for any \(i=1,\dots, s\), and \[ \omega \Biggl( \prod^s_{i=1} 1/ (X- a_i) \Biggr)=\prod^s_{i=1} (-1/ a_i) \sum^\infty_{n=0} (X/ a_i)^n \quad \text{if } \nu (a_i) \leq 0, \quad \text{for any }i=1, \dots, s. \tag{3} \] In this way we look at \((\widetilde {K(X)}, \widetilde {\omega}_0)\) as a valued subfield of \((K\{\{ X\}\}, \mu)\) where \(\widetilde {K(X)}\) is the completion of \(K(X)\) with respect to \(\omega_0\). A series \(\alpha=\sum_{n\geq 0} a_n X^n\) is called almost periodic if it is a limit in \((K \{\{ X\}\}, u)\) of elements from \(K(X)\). For any series \(\alpha=\sum_{n\geq 0} a_n X^n\) and for any two natural numbers \(m\), \(l\) we denote by \(\widetilde {H}^l_m (\alpha)\) the set of all determinants of order \(l+1\) formed by the first \(l+1\) columns and any \(l+1\) rows of the Hankel matrix \(H_m (\alpha)\) [see for instance \textit{Y. Amice}, Les nombres \(p\)-adiques, Paris (1975; Zbl 0313.12104)]. The fundamental result in this paper is Theorem 2.3: Let \(\alpha=\…
reviewed by: Nicolae Pop
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 923403
zbMATH Keywords: Gauss residual transcendental extension
zbMATH Keywords: local field of all formal Laurent series
zbMATH Keywords: valued field
zbMATH Keywords: almost periodic series
MaRDI profile type: Publication
OpenAlex ID: W4235262864
description: scientific article; zbMATH DE number 923403