Multiplicative groups in some infinite algebraic number fields
type: Item
instance of: scholarly article
title: Multiplicative groups in some infinite algebraic number fields
zbMATH Open document ID: 0659.12001
DOI: 10.1007/BF01190665
author: Kuniaki Horie
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: An abelian group A is called regular when the factor group of A modulo the torsion subgroup of A is a free abelian group. This paper proves that if F is an algebraic number field whose Galois closure contains only a finite number of roots of unity, then the multiplicative group of F is regular. In particular, the multiplicative group of every totally real algebraic number field becomes regular. It also follows that the unit group of every CM-field is regular. On the other hand, the multiplicative group of the maximal CM-field turns out to be isomorphic to a direct sum of a regular group and countably many infinite copies of the additive group of the rational numbers. Incidentally, it is proved that for any prime number p, if K is a CM- field with all roots of unity of p-power orders, then the Galois group of the maximal p-extension of K over K is isomorphic to a free pro-p-group with countably many free generators.
Mathematics Subject Classification ID: 11R99
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 4077374
zbMATH Keywords: multiplicative group of algebraic number field
zbMATH Keywords: totally real algebraic number field
zbMATH Keywords: unit group
zbMATH Keywords: CM-field
MaRDI profile type: Publication
cites work: A note on Kummer extensions
cites work: Q5775616
cites work: Théoremes de finitude en cohomologie galoisienne
full work available at URL: https://doi.org/10.1007/bf01190665
OpenAlex ID: W1997679266
description: scientific article; zbMATH DE number 4077374