Computation of \(K_ 2\) for the ring of integers of quadratic imaginary fields.
type: Item
instance of: scholarly article
title: Computation of \(K_ 2\) for the ring of integers of quadratic imaginary fields.
zbMATH Open document ID: 1037.11077
DOI: 10.1007/BF02880134
author: Hong You
author: Chen Sheng
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: For a set of finite places, \(S\), of a number field \(F\) let \(K_2^S(F)\) be the subgroup of \(K_2(F)\) generated by symbols of the form \(\{a,b\}\) with \(a,b\in U_S= \{x\in F^*\mid v(x)\geq 0\) \(\forall v\not\in S\}\). For imaginary quadratic fields \textit{J. Tate}, in the Appendix to his paper with \textit{H. Bass} on `The Milnor ring of a global field' [Lect. Notes Math. 342, 429--446 (1973; Zbl 0299.12013)] showed how to determine a bound \(M= M(F)\in \mathbb{N}\), such that \(K_2(O_F) \subset K_2^S(F)\) whenever \(S\) contains all finite places of norm at most \(M\). The authors improve Tate's bound (by refining Tate's methods) for certain quadratic imaginary fields and use the improved bounds to show that the group \(K_2(O_F)\) is trivial when \(F= \mathbb{Q} (\sqrt {-43})\).
reviewed by: Kevin Hutchinson
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 19C99
Mathematics Subject Classification ID: 19F15
zbMATH DE Number: 1782130
zbMATH Keywords: tame kernel
MaRDI profile type: Publication
cites work: Q4053733
cites work: Generalization of Thue's theorem and computation of the group \(K_ 2 O_ F\)
cites work: Computation of \(K_ 2\mathbb{Z}[\sqrt {-6}]\)
cites work: Computation of \(K_ 2\mathbb{Z}[\frac{1+\sqrt{-35}}{2}]\)
cites work: Tame and wild kernels of quadratic imaginary number fields
full work available at URL: https://doi.org/10.1007/bf02880134
OpenAlex ID: W1539591967
description: scientific article; zbMATH DE number 1782130