Galois parallelograms
type: Item
instance of: scholarly article
title: Galois parallelograms
zbMATH Open document ID: 0973.12002
author: Richard Massy
author: Sylvie Monier-Derviaux
published in: Journal of Algebra
publication date: 2001-12-04T00:00:00Z
review text: Let \(K/J\) be an algebraic extension of fields and \(N/K\) be a Galois extension. A natural question is, does there exist a field \(L\), \(J\subseteq L \subseteq N\) such that \(L/J\) is a Galois extension with \(L\cap K = J\) and \(N = KL\)? In the case that such \(L\) exists, it is said that \(N/K\) descends to \(J\) and that \(L/J\) is a descent of \(N/K\). We have \(\text{Gal}(N/K) \cong \text{Gal}(L/J)\). The authors generalize the notion of Galois extension by defining Galois parallelograms as above and considering \(K/J\) also a Galois extension. A Galois extension is just a ``flat'' parallelogram, that is, when \(K=J\) and \(N=L\). We may consider the theory of Galois parallelograms as two dimensional Galois theory. After stating the principal general and arithmetic properties of Galois parallelograms, generalizing some of their previous results [J. Théor. Nombres Bordx. 11, No. 1, 161-172 (1999; Zbl 0944.12002)], the authors define the Galois group of a Galois parallelogram and prove the usual properties of classical Galois theory. The last section is dedicated to prove the general properties of Galois parallelograms stated at the beginning of the paper.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 1340611
zbMATH Keywords: Galois descent
zbMATH Keywords: Galois parallelogram
zbMATH Keywords: two-dimensional Galois theory
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: jabr.1998.7769
OpenAlex ID: W4212850202
cites work: On $p$-groups as Galois groups.
cites work: Subgroups of \(\text{Hol} Q_ 8\) as Galois groups
cites work: Construction de p-extensions galoisiennes d'un corps de caractéristique différente de p. (Construction of Galois p-extensions of fields with characteristic different from p)
cites work: Descent and Galois parallelograms
cites work: Descente de $p$-extensions galoisiennes kummeriennes.
cites work: Q3856856
cites work: Central \(p\)-extensions of \((p, p, \dots, p)\)-type Galois groups
DOI: 10.1006/JABR.1998.7769
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description: scientific article; zbMATH DE number 1340611