Units in \({\mathbb{Z}}[Q_ p]\)
type: Item
instance of: scholarly article
title: Units in \({\mathbb{Z}}[Q_ p]\)
zbMATH Open document ID: 0621.16011
DOI: 10.1016/0022-314X(87)90080-1
author: E. Kleinert
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Let p be an odd prime and \(Q_ p=<x,y|\) \(x^ p=y^ 2\), \(yxy^{- 1}=x^{-1}>\) the quaternion group with 4p elements. It is known that the trivial units \(\pm Q_ p\) have a torsion-free normal complement H in \(U(Z[Q_ p])\) [see: \textit{G. H. Cliff}, \textit{S. K. Sehgal}, \textit{A. R. Weiss}, J. Algebra 73, 167-185 (1981; Zbl 0484.16004)]. In this paper H is described explicitly in terms of matrices over algebraic number fields. A related paper by the same author is J. Number Theory 13, 541-561 (1981; Zbl 0473.16006), and a lecture of it will facilitate the study of the present article. Here, unit groups of group rings are analyzed using facts from the theory of orders and from integral representation theory. The paper is well organized and well written.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C10
zbMATH DE Number: 4006465
zbMATH Keywords: quaternion group
zbMATH Keywords: trivial units
zbMATH Keywords: torsion-free normal complement
zbMATH Keywords: unit groups
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
cites work: Units of integral group rings of metabelian groups
cites work: Zur Zahlentheorie der Quaternionen-Algebren.
cites work: Einheiten in Z(D//2m)
cites work: Q4061092
cites work: Q5562643
description: scientific article; zbMATH DE number 4006465