On free products of an absolute Galois group
type: Item
instance of: scholarly article
title: On free products of an absolute Galois group
zbMATH Open document ID: 0935.12005
DOI: 10.1007/BF02674295
author: Oleg Melnikov
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: The author gives a partial answer to the question by \textit{M.~Jarden} [Infinite Galois theory, Handbook of algebra. Vol. 1, 271-319 (1996; Zbl 0864.12003)] whether the class of profinite groups that are absolute Galois groups of fields is closed under free profinite products. Denote by \(G(K)\) an absolute Galois group of a field \(K\). Recall that a separable group is a topological group that has a countable base of neighborhoods about identity. The author proves: Theorem A. Let the absolute Galois groups \(\mathfrak G(K_i)=G_i\) of fields \(K_i\) be separable groups, \(i=1,2,\dots,n\). Let \(G=G_1*G_2*\dots *G_n\) be the free profinite product. Then, for some field \(L\), the group \(G\) is an absolute Galois group, \(G=\mathfrak G(L)\). Besides, the author proves: Theorem B. A profinite group \(G\) is isomorphic to an absolute Galois group of some field, \(G=\mathfrak G(K)\), if and only if it is a subgroup of an absolute Galois group of some field of rational functions \(\Phi_p(X)\) over a prime field \(\Phi_p\) of \(p\) elements for some set of variables \(X\).
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 1432167
zbMATH Keywords: profinite group
zbMATH Keywords: free profinite product
zbMATH Keywords: absolute Galois group
MaRDI profile type: Publication
cites work: Q4893201
cites work: Q3682579
cites work: Galois groups of intersections of local fields
cites work: Projective Profinite Groups
cites work: Ein Darstellungssatz für kompakte, total unzusammenhängende Gruppen
cites work: Maximal abelian subgroups of free profinite groups
cites work: Q3824581
cites work: Free pro-\b C-groups
full work available at URL: https://doi.org/10.1007/bf02674295
OpenAlex ID: W2328780364
description: scientific article; zbMATH DE number 1432167