The 2-Sylow subgroup of \(K_2O_F\) for certain quadratic number fields
type: Item
instance of: scholarly article
title: The 2-Sylow subgroup of \(K_2O_F\) for certain quadratic number fields
zbMATH Open document ID: 1486.11142
author: Xia Wu
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: Let \(F\) be the quadratic number field \(\mathbb{Q}(\sqrt{d})\) where \(d\) is squarefree. Let \(E=F(\sqrt{d/2})\) when \(d\) is even. Suppose that all odd prime divisors of \(d\) are congruent to \(1\) modulo \(8\). The author uses results of H. Qin on the \(4\)-rank of \(K_2(\mathcal{O}_F)\) to determine the relationship between the \(4\)-rank of \(K_2(\mathcal{O}_F)\) and the \(4\)-rank of the ideal class group of \(F\) or of \(E\) under certain additional conditions on the odd prime divisors of \(d\) and the norm of the fundamental unit. The author then gives a simple unified proof of two conjectures of \textit{P. E. Conner} and \textit{J. Hurrelbrink} [Acta Arith. 73, No. 1, 59--65 (1995; Zbl 0844.11072)] which give necessary and sufficient conditions under which \(K_2(\mathcal{O}_F)\) is elementary abelian. These conjectures have already been proved by \textit{A. Vazzana} [Acta Arith. 81, No. 3, 253--264 (1997; Zbl 0905.11051)] in a less concise manner using different methods.
reviewed by: Kevin Hutchinson
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 19F15
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 7344966
zbMATH Keywords: tame kernel
zbMATH Keywords: ideal class group
zbMATH Keywords: quadratic number field
MaRDI profile type: Publication
full work available at URL: s11139-020-00251-4
OpenAlex ID: W3042735534
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DOI: 10.1007/S11139-020-00251-4
description: scientific article; zbMATH DE number 7344966