A Galois correspondence for radical extensions of fields
type: Item
instance of: scholarly article
title: A Galois correspondence for radical extensions of fields
zbMATH Open document ID: 0607.12015
DOI: 10.1016/0022-4049(86)90067-8
author: Cornelius Greither
author: David K. Harrison
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(K/k\) be a finite extension of fields. \(\mathrm{Cog}(K/k)\) is the torsion subgroup of \(\dot K/\dot k\). With this start the authors develop a theory (dual to the usual Galois theory) for radical extensions of fields generalizing the work of \textit{M. Acosta de Oroczo} and \textit{W. Y. Vélez} [J. Number Theory 15, 388--405 (1982; Zbl 0493.12027)]. From the introduction: ``We call \(K/k\) cogalois with cogalois group \(\mathrm{Cog}(K/k)\) if: (1) (Conormality) card(Cog(K/k)) is finite and at most \([K: k]\); (2) (Coseparability) \(K = K_H\) with \(H = \mathrm{Cog}(K/k)\). In the first section, we prove: -- If \(K/k\) is cogalois and \(E\) is a field between \(K\) and \(k\), then \(K/E\) and \(E/k\) are cogalois. -- The maps \(\mathrm{Cog}(-/k)\) and \(K_{(-)}\) are inverse bijections between the lattices of intermediate fields of \(K/k\), and of subgroups of \(\mathrm{Cog}(K/k)\), respectively. - There are many examples; e.g. \(\mathbb Q \left(\root{n_1}\of {a_1}, \ldots, \root{n_t}\of {a_t}\right)\subset \mathbb R\) \((a_i > 0)\) is cogalois over \(\mathbb Q\). In the second section we study the connection between \(G = \Aut(K/k)\) and \(H= \mathrm{Cog}(K/k)\) in case \(K/k\) is both cogalois and galois. A somewhat surprising duality for a special class of nonabelian groups emerges, and we slightly enlarge the class of extensions where we can exhibit all intermediate fields. In an appendix we prove that \(\mathrm{Cog}(K/k)\) is always finite if \(K\) and \(k\) are algebraic number fields.''
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 06A15
zbMATH DE Number: 3981298
zbMATH Keywords: finite field extension
zbMATH Keywords: Cogalois theory
zbMATH Keywords: radical extensions
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90067-8
OpenAlex ID: W1970042668
cites work: Hopf algebras and Galois theory
cites work: Q5343374
cites work: Q3232305
cites work: The torsion group of a radical extension
cites work: The lattice of subfields of a radical extension
description: scientific article; zbMATH DE number 3981298