Arithmetic lifting of dihedral extensions
type: Item
instance of: scholarly article
title: Arithmetic lifting of dihedral extensions
zbMATH Open document ID: 0924.12003
author: Elena V. Black
published in: Journal of Algebra
publication date: 1998-11-18T00:00:00Z
review text: The standard and usual approach to inverse Galois problems relies on the construction of Galois branched coverings of the projective line and on their specialization using the Hilbert Irreducibility Theorem. Therefore it is fundamental to know whether every Galois extension of number fields is the specialization of a branched covering of the projective line with the same Galois group. The author proves that this holds true for every Galois extension of number fields whose Galois group is a dihedral group of odd degree (\(D_n\), with \(n\) odd). The same positive result was obtained by \textit{S. Beckmann} when the Galois group is an abelian finite group [J. Algebra 165, 430-451 (1994; Zbl 0802.12003)]. The author reinterprets the work of Beckmann in terms of étale cohomology and then generalizes to the case of dihedral groups of odd degree.
reviewed by: Leonardo Cangelmi
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 1203903
zbMATH Keywords: inverse Galois theory
zbMATH Keywords: Galois branched coverings
zbMATH Keywords: arithmetic lifting
zbMATH Keywords: dihedral extensions
zbMATH Keywords: étale cohomology
MaRDI profile type: Publication
full work available at URL: jabr.1998.7466
OpenAlex ID: W2043832520
cites work: Is every extension of \(\mathbb{Q}\) the specialization of a branched covering?
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cites work: Generic Galois extensions and problems in field theory
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DOI: 10.1006/JABR.1998.7466
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description: scientific article; zbMATH DE number 1203903