Solitary Galois extensions of algebraic number fields
type: Item
instance of: scholarly article
title: Solitary Galois extensions of algebraic number fields
zbMATH Open document ID: 0822.11075
author: Leonid Stern
author: Robert Guralnick
published in: Journal of Number Theory
publication date: 1995-02-28T00:00:00Z
review text: Let \(k\) be an algebraic number field. For any finite Galois extensions \(K/k\) and \(L/k\), \textit{L. Stern} [J. Number Theory 32, 203-219 (1989; Zbl 0687.12006)] has proved that the norm groups \(N_{K/k} K^ \times\) and \(N_{L/k} L^ \times\) are equal iff \(K=L\). A finite extension \(K\) of \(k\) is called ``\(k\)-solitary'' if for any finite extension \(L\) of \(k\) the equality \(N_{K/k} K^ \times= N_{L/k} L^ \times\) implies that \(K\) and \(L\) are conjugate over \(k\) (i.e. \(k\)-isomorphic). The classification problem of \(k\)-solitary extensions seems to be difficult to solve in general. In the present paper, the authors describe finite Galois extensions \(K\) of \(k\) which are \(k\)-solitary by first proving that \(K/k\) is necessarily a 2-extension. They state that \(k\) itself and every quadratic extension \(K/k\) are \(k\)-solitary. They provide a necessary condition for a finite Galois 2-extension \(K/k\) with \([K: k]\geq 4\) to be \(k\)-solitary. Afterwards, they investigate covering subgroups and show that this necessary condition is also sufficient for extensions of degree 4 or 8. In the last section, a dihedral Galois extension \(E/\mathbb{Q}\) of degree 8, with \(\mathbb{Q}\)-solitary field \(E\), is constructed.
reviewed by: Richard Massy
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 714107
zbMATH Keywords: solitary extensions
zbMATH Keywords: norm groups
zbMATH Keywords: quadratic extension
zbMATH Keywords: dihedral Galois extension
MaRDI profile type: Publication
full work available at URL: jnth.1995.1001
OpenAlex ID: W2089374410
DOI: 10.1006/JNTH.1995.1001
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description: scientific article; zbMATH DE number 714107