Fields with continuous local elementary properties. II
type: Item
instance of: scholarly article
title: Fields with continuous local elementary properties. II
zbMATH Open document ID: 0856.12005
DOI: 10.1007/BF02341870
published in: Algebra and Logic
publication date: 1997-02-17T00:00:00Z
full work available at URL: https://eudml.org/doc/187724
review text: The article is a continuation of Part I [Algebra Logic 33, No. 6, 351-365 (1994); translation from Algebra Logika 33, No. 6, 628-653 (1994; Zbl 0845.12005)]. The analogue of theorem 3 from that paper is proved in section 2 for the case of fields in \(RC^*_E\) (with absolutely unramified valuation rings). Such fields, in contrast to maximal fields which have ``the smallest'' possible absolute Galois group, possess a ``large'' Galois group. In section 1 some necessary group-theoretic notions are developed. Let \(G\) be a profinite group and let \(\text{Subg} G\) be the set of all closed subgroups of \(G\). \({\mathbf S} {\mathbf u} {\mathbf b} {\mathbf g} G\) endowed with the natural topology is a Boolean space. The group \(G\) acts on \({\mathbf S} {\mathbf u} {\mathbf b} {\mathbf g} G\) by conjugation. A pair \({\mathbf G} = \langle G,D_G \rangle\), where \(G\) is a profinite group and \(D_G \subseteq {\mathbf S} {\mathbf u} {\mathbf b} {\mathbf g} G\) is a closed subset such that \(\Gamma^g \in D_G\) for all \(\Gamma \in D_G\), \(g \in G\), is called a \(\Delta^*\)-group. If \({\mathbf G}_0 = \langle G_0, D_{G_0} \rangle\) and \({\mathbf G}_1 = \langle G_1, D_{G_1} \rangle\) are \(\Delta^*\)-groups, then any continuous homomorphism \(\varphi : G_0 \to G_1\) such that \(\varphi (\Gamma) \in D_{G_1}\) for every \(\Gamma \in D_{G_0}\) is called a homomorphism from \({\mathbf G}_0\) to \({\mathbf G}_1\). A homomorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) is called an epimorphism if \(\varphi : G_0 \to G_1\) is an epimorphism and \(D_{G_1} = \{\varphi (\Gamma) \mid \Gamma \in D_{G_0}\}\). An epimorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) is called flat if \(\Gamma \cap \text{Ker} \varphi = \{e\}\) for all \(\Gamma\) from \(D_{G_0}\). If \({\mathbf G}\) is a \(\Delta^*\)-group, \(X({\mathbf G})\) denotes the space \(\pi_G (D_G) \subseteq \text{Subg}^*G\), and each homomorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) of \(\Delta^*\)- groups indu…
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 907267
zbMATH Keywords: continuous local elementary properties
zbMATH Keywords: \(\Delta^*\)-groups
zbMATH Keywords: fields
zbMATH Keywords: absolutely unramified valuation rings
zbMATH Keywords: absolute Galois group
zbMATH Keywords: profinite group
author: Yuri L. Ershov
MaRDI profile type: Publication
cites work: Fields with continuous local elementary properties. I
cites work: The absolute Galois group of a pseudo p-adically closed field.
cites work: \(PC_ p\)-fields with universal Galois group
OpenAlex ID: W4252285471
description: scientific article; zbMATH DE number 907267