Class group of the group ring of a cyclic group of order \(p^ 2\) over the ring of integers in a quadratic number field
type: Item
instance of: scholarly article
title: Class group of the group ring of a cyclic group of order \(p^ 2\) over the ring of integers in a quadratic number field
zbMATH Open document ID: 0591.12012
DOI: 10.1016/0021-8693(86)90104-3
author: Soroosh Homayouni
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathfrak O}\Gamma_ n\) be the group ring of a cyclic group \(\Gamma_ n\) of order \(p^{n+1}\) (p an odd prime) over the ring of integers \({\mathfrak O}\) in a quadratic number field k (unramified at p), cl(\({\mathfrak O}\Gamma_ n)\) the locally free class group of \(\Gamma_ n\), cl(m) the class group of the maximal order m. It is well known that there exists the surjective homomorphism cl(\({\mathfrak O}\Gamma_ n)\to cl(m)\). The author investigates the kernel D(\({\mathfrak O}\Gamma_ n)\) of this homomorphism for \(n=1\). He shows that \(| D({\mathfrak O}\Gamma_ 1)|\) has a large p part which is minimal when p does not divide the class number h of \(k(\xi_ 0)\) where \(\xi_ 0\) is a primitive p-th root of unity. In particular the author proves that if \((p,h)=1\) then \(| D({\mathfrak O}\Gamma_ 1)| =\Omega | D({\mathfrak O}\Gamma_ 0)|\) where for k real \(p^{p-2}\) divides \(\Omega\) and for k imaginary \(p^{p-3}\) divides \(\Omega\).
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3948418
zbMATH Keywords: group ring
zbMATH Keywords: cyclic group
zbMATH Keywords: quadratic number field
zbMATH Keywords: locally free class group
zbMATH Keywords: maximal order
zbMATH Keywords: class number
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90104-3
OpenAlex ID: W1989041020
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description: scientific article; zbMATH DE number 3948418