On the structure of the \(K_ 2\) of the ring of integers in a number field
type: Item
instance of: scholarly article
title: On the structure of the \(K_ 2\) of the ring of integers in a number field
zbMATH Open document ID: 0705.19007
DOI: 10.1007/BF00535049
author: Frans Keune
published in: \(K\)-Theory
publication date: 1989-01-01T00:00:00Z
review text: Let \(K_ 2({\mathcal O}_ F)\) (resp. \(W_ F)\) be the tame (resp. wild) kernel in \(K_ 2(F)\) for a number field F, and let \(C\ell (F)\) be the S- class group of F (S containing infinite and p-adic places of F). When F contains the group \(\mu_ r\) of \(p^ r\)-th roots of unity, \textit{J. Tate}'s results [Invent. Math. 36, 257-274 (1976; Zbl 0359.12011)] give an explicit description of \(K_ 2({\mathcal O}_ F)/p^ r\) by means of \(\mu_ r\otimes C\ell (F)\). In the general case, the author uses Galois descent (in \(F_ r=F(\mu_ r)\) over F) to interprete \(K_ 2({\mathcal O}_ F)/p^ r\) in terms of \(C\ell (F_ r)\); in fact he uses Galois- coinvariants via the transfer and ``Hilbert 90 theorem'' of \textit{A. S. Merkur'ev} and \textit{A. A. Suslin} [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)]. The author obtains mainly such an interpretation for \(W_ F\). He gives also a second approach using more specific class field theory.
reviewed by: Georges Gras
Mathematics Subject Classification ID: 19F05
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 4156734
zbMATH Keywords: ideal class groups
zbMATH Keywords: p-power cyclotomic extensions
zbMATH Keywords: tame kernel
zbMATH Keywords: wild kernel
zbMATH Keywords: Hilbert 90 for \(K_ 2\)
zbMATH Keywords: Galois-coinvariants
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00535049
OpenAlex ID: W2072591267
description: scientific article; zbMATH DE number 4156734