Computation of \(K_ 2\mathbb{Z}[\sqrt {-6}]\)
type: Item
instance of: scholarly article
title: Computation of \(K_ 2\mathbb{Z}[\sqrt {-6}]\)
zbMATH Open document ID: 0808.11066
DOI: 10.1016/0022-4049(94)90124-4
author: Hourong Qin
published in: Journal of Pure and Applied Algebra
publication date: 1995-03-09T00:00:00Z
review text: Bass and Tate's method of computing \(K_ 2 O_ F\) consisting essentially in finding a finite set of primes \(S\) containing the set \(S_ \infty\) of archimedean primes, and such that: \[ K_ 2 O_ F= \text{Ker} (K_ 2^ S F\to \prod_{v\in S- S_ \infty} k(v)^*), \] where \(k(v)\) is the residue field and \(K_ 2^ S F\) is the subgroup of \(K_ 2 F\) generated by symbols of \(S\)-units. This method was carried out by Tate for all the Euclidean imaginary quadratic fields and for \(\mathbb{Q} (\sqrt{-15})\), using a series of technical lemmas. In the present paper, these lemmas are improved (also in a very technical way) and yield the triviality of \(K_ 2 O_ F\) for \(F= \mathbb{Q}( \sqrt{-6})\). The author also states that the same method allows him to prove that \(K_ 2 O_ F\) is trivial for \(F= \mathbb{Q}( \sqrt{-5})\) and \(\mathbb{Q}( \sqrt{-19})\) (Skalba's results) and \(K_ 2 O_ F\simeq \mathbb{Z}/ 2\mathbb{Z}\) for \(F= \mathbb{Q} (\sqrt{- 35})\) (to appear).
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 699138
zbMATH Keywords: \(K_ 2\)-groups
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90124-4
OpenAlex ID: W133557241
cites work: Q4053733
cites work: Elements of small order in K2F
cites work: Conjectures on the dilogarithm
cites work: On Sylow 2-subgroups of K2OF for quadratic number fields F.
cites work: Generalization of Thue's theorem and computation of the group \(K_ 2 O_ F\)
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description: scientific article; zbMATH DE number 699138