Multiplicative groups of fields modulo products of subfields
type: Item
instance of: scholarly article
title: Multiplicative groups of fields modulo products of subfields
zbMATH Open document ID: 0846.12003
DOI: 10.1016/0022-4049(94)00128-6
author: Roger Wiegand
author: Jean-Louis Colliot-Thélène
author: Robert Guralnick
published in: Journal of Pure and Applied Algebra
publication date: 1996-03-20T00:00:00Z
review text: Let \(K/k\) be a finite separable field extension. Let \(E_1, \dots, E_r\) be intermediate fields of \(K/k\). In this paper some properties of the quotient \(K^*/ E^*_1 \dots E^*_r\) are investigated. If \(r\leq 2\), then the \(n\)-torsion subgroup of \(K^*/ E^*_1 \dots E^*_r\) is finite for any \(n>0\). If \(E_1\) is finitely generated over its prime field, then \(K^*/ E^*_1\) is finite. Let \(\mu (K)\) denote the group of roots of unity contained in \(K\). Then \(K^*/ E^*_1 E^*_2\) is finite if and only if \(\mu (K)/ \mu(E_1) \mu(E_2)\) is finite. These results are no longer true if \(r>2\). Examples are given that \(K^*/ E^*_1 \dots E^*_r\) is trivial, finite, infinite torsion or has infinite torsion free rank. Some of these statements or examples are already contained in previous papers of the authors. The aim of the paper under review is a general approach to these finiteness results. The authors start with the general concept of \(G\)-modules and then they apply Galois cohomology and things like Hilbert 90.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12E30
zbMATH DE Number: 858645
zbMATH Keywords: fields modulo products of subfields
zbMATH Keywords: \(n\)-torsion subgroup
zbMATH Keywords: finiteness results
reviewed by: Martin Epkenhans
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00128-6
OpenAlex ID: W2028245855
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description: scientific article; zbMATH DE number 858645