Units in \({\mathbb{Z}}[Q_ p]\) (Q1090392)

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.