Allowable groups and cogalois extensions
type: Item
instance of: scholarly article
title: Allowable groups and cogalois extensions
zbMATH Open document ID: 0839.12003
DOI: 10.1016/0022-4049(94)00126-X
author: Fernando Barrera Mora
author: Martha Rzedowski-Calderón
author: Gabriel D. Villa Salvador
published in: Journal of Pure and Applied Algebra
publication date: 1996-06-19T00:00:00Z
review text: Let \(K/F\) be an extension of number fields. The torsion of \(K/F\) is defined by \({\mathcal T} (K/F) : = \{a \in K \mid a^m \in F^*\) for some \(m \in \mathbb{N}\}\). Then \(\text{cog} (K/F) : = {\mathcal T} (K/F)/F^* \subseteq K^*/F^*\) is called the `cogalois group of \(K/F\)'. \textit{C. Greither} and \textit{D. K. Harrison} [J. Pure Appl. Algebra 43, 257-270 (1986; Zbl 0607.12015)] introduced the concept of cogalois extensions. \(K/F\) is called `cogalois' if \(|\text{cog} (K/F) |\leq [K : F]\) and \(K = F({\mathcal T} (K/F))\). Let \(r\) be an integer and let \(m\) be a divisor of \(r\) such that each prime divisor of \(r\) divides \(m\) and \(4 |m\) if \(4 |r\). Set \(D_{r,m} : = \{x \bmod r \in (\mathbb{Z}/r \mathbb{Z})^* \mid x \equiv 1 \bmod m\}\). We call a group \(G\) `allowable' if there is an exact sequence \(1 \to H \to G \to D_{r,m} \to 1\), with \(H\) an abelian group, the exponent of \(G\) divides \(r\) and \(D_{r,m}\) acts on \(H\) by scalar multiplication. The paper under review gives a necessary and sufficient condition on the field \(F\) in order to have a Galois and cogalois extension \(K/F\) with Galois group an arbitrary allowable group \(G\). This condition depends on the structure of the abelian group \(G/G'\) \((G' =\) commutator subgroup of \(G)\) and certain roots of unity. An allowable group is nilpotent, hence a direct product of its Sylow groups, which are also allowable. From this we get a reduction to the case of allowable \(p\)-groups.
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 825330
zbMATH Keywords: cogalois group
zbMATH Keywords: Galois extension
zbMATH Keywords: allowable group
zbMATH Keywords: cogalois extensions
reviewed by: Martin Epkenhans
MaRDI profile type: Publication
cites work: On cogalois extensions
cites work: A Galois correspondence for radical extensions of fields
cites work: Q5840058
cites work: Q3232305
full work available at URL: 0022-4049(94)00126-x
OpenAlex ID: W2046402965
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description: scientific article; zbMATH DE number 825330