Automorphism groups of fields
type: Item
instance of: scholarly article
title: Automorphism groups of fields
zbMATH Open document ID: 0824.12003
DOI: 10.1007/BF02568195
author: Manfred Dugas
author: Rüdiger Göbel
published in: Manuscripta Mathematica
publication date: 1995-11-12T00:00:00Z
full work available at URL: https://eudml.org/doc/156019
review text: This paper continues a long series of publications of Kuyk, Fried and Kollar, Geyer, Pröhle, Stichtenoth, Dugas and Göbel, Schmelzer and others. By two different constructions the following theorem is extended. To every group \(G\) and every field \(K\) there exists a field extension \(F\) with \(\Aut F\simeq G\) and \(K \subseteq \text{Fix} (\Aut F)\) where \(\text{Fix} (\Aut F)\) denotes the fixed field of \(\Aut F\). Especially \(K= \text{Fix} (\Aut F)\) holds, if \(G\) is infinite. 1) If \(K\) does not contain too many roots of unity then the cardinality of \(F\) can be estimated by \(| F|\leq \max \{\aleph_ 0, | G|, | K|\}\). 2) The theory of fields is ``group universal''. Let a group \(G\), a field \(K\) and a homomorphism \(\varphi: G\to \Aut K\) be given. Then there exists a field extension \(F\) with \(\Aut F\simeq G\) and \(\varphi (G)\) is the restriction of \(\Aut F\) on \(K\). If one restricts to right orderable groups the theory of formal real fields is group universal, too. The question for group universality was raised by Funk, Kegel and Strambach who showed this property for other algebraic structures with quite different methods. As in other papers the authors consider a transcendental base \(X\) on which the group \(G\) operates faithfully. The field \(K(X)\) can be extended algebraically to \(F\subseteq \widehat {K (X)}\) such that \(X\) is mapped onto itself by \(\Aut F\). Further \(\Aut F\) operates on \(X\) in the same way as \(G\). The elements of \(X\) are distinguished by root chains. To show group universality the authors have adjusted their construction from 1987 [Trans. Am. Math. Soc. 304, 355-384 (1987; Zbl 0644.12011)] which uses combinatorial tools from S. Shelah. From there they get a cardinal jump for \(F\) so that it remains open whether 1) and 2) hold simultaneously. As in their paper (loc. cit.) monoids with right cancellation law were considered instead of groups, that paper also presented an essential extension of the theorem q…
reviewed by: Dietmar Schweigert
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 20F29
zbMATH DE Number: 724655
zbMATH Keywords: field extension
zbMATH Keywords: group universality
MaRDI profile type: Publication
cites work: Endomorphism algebras of modules with distinguished partially ordered submodules over commutative rings
cites work: Model theoretic algebra. Selected topics
cites work: Right-ordered groups
cites work: Prescribing Endomorphism Algebras, a Unified Treatment
cites work: Every Countable Reduced Torsion-Free Ring is an Endomorphism Ring
cites work: Endomorphism algebras of large modules with distinguished submodules
cites work: Countable Butler groups and vector spaces with four distinguished subspaces
cites work: Prescribing endomorphism algebras. The cotorsion-free case
cites work: Q4742987
cites work: Q3966342
cites work: Automorphism groups of algebraic number fields
cites work: A Note on Automorphism Groups of Algebraic Number Fields
cites work: Q5609457
cites work: Gruppenuniversalität und Homogenisierbarkeit
cites work: Jede endliche Gruppe ist Automorphismengruppe einer endlichen Erweiterung K/Q
cites work: Four submodules suffice for realizing algebras over commutative rings
cites work: Independence in Completions and Endomorphism Algebras
cites work: Q5328221
cites work: Q4784966
cites work: Lectures on formally real fields
cites work: Does the Frobenius endomorphism always generate a direct summand in the endomorphism monoids of fields of prime characteristic?
cites work: Q3757954
cites work: Q4079762
cites work: Q4095071
cites work: A combinatorial principle and endomorphism rings. I: On \(p\)-groups
cites work: Q4093626
cites work: Q4271281
OpenAlex ID: W2051870651
description: scientific article; zbMATH DE number 724655