The embedding problem over a Hilbertian PAC-field
type: Item
instance of: scholarly article
title: The embedding problem over a Hilbertian PAC-field
zbMATH Open document ID: 0765.12002
DOI: 10.2307/2946573
author: Michael D. Fried
author: Helmut Völklein
published in: Annals of Mathematics. Second Series
publication date: 1993-01-16T00:00:00Z
review text: A field \(P\) is called pseudo-algebraically closed (a PAC-field) if every absolutely irreducible variety defined over \(P\) has a \(P\)-rational point. The algebraic closure of a field \(P\) is denoted by \(\overline{P}\), the absolute Galois group \(G(\overline{P}/P)\) of \(P\) by \(G_ P\). One says that all finite embedding problems over \(P\) are solvable if, for every surjection \(h: E\to C\), of finite groups and for every surjection \(\lambda: G_ P\to C\) there exists a surjection \(\varepsilon: G_ P\to E\) with \(h\circ \varepsilon=\lambda\). It is proved that every finite embedding problem over a Hilbertian PAC- field \(P\) is solvable. For countable fields this implies that the absolute Galois group of \(P\) is \(\omega\)-free that is \(G(\overline{P}/P)\) is a free profinite group of countably infinite rank. It is shown that a PAC-field \(P\) of characteristic 0 is Hilbertian if and only if all finite embedding problems over \(P\) are solvable. A field \(P\) is said to be regular Galois-Hilbertian if every regular (finite) Galois extension of \(P(x)\) can be specialized to a Galois extension of \(P\) with the same Galois group. It is proved that a PAC-field of char. 0 is RG-Hilbertian if and only if every finite group is a Galois group over \(P\).
Mathematics Subject Classification ID: 12E25
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 91439
zbMATH Keywords: pseudo-algebraically closed field
zbMATH Keywords: regular Galois-Hilbertian field
zbMATH Keywords: finite embedding problem
zbMATH Keywords: Hilbertian PAC-field
zbMATH Keywords: absolute Galois group
zbMATH Keywords: free profinite group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946573
OpenAlex ID: W1978839186
Wikidata QID: Q59664830
description: scientific article; zbMATH DE number 91439