Finding non-trivial elements in \(K_ 2({\mathcal O}_ F)\) for imaginary quadratic number fields \(F\)
type: Item
instance of: scholarly article
title: Finding non-trivial elements in \(K_ 2({\mathcal O}_ F)\) for imaginary quadratic number fields \(F\)
zbMATH Open document ID: 0776.19006
DOI: 10.1016/0022-4049(92)90119-Z
author: Thom Mulders
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: Let \(F\) be a number field, \(S\) a finite set of primes containing the Archimedean ones, \({\mathcal O}_ F\) (resp. \({\mathcal O}_ S\)) the ring of integers (resp. \(S\)-integers) of \(F\). When \(F\) contains a primitive \(n\)th root of unity, there is a canonical map \(I: \mu_ n\otimes\text{Cl}({\mathcal O}_ S)\to K_ 2({\mathcal O}_ S)/n\) [\textit{F. Keune}, \(K\)-theory 2, 625-645 (1989; Zbl 0705.19007)]. If \(S\) contains the divisors of \(n\), \(I\) is injective and its image, roughly speaking, comes from the wild kernel (this could also be proved by means of Poitou- Tate's duality). If \(F\) does not contain enough roots of unity, then of course one can go up some suitable cyclotomic extension, and then go down by transfer. By using the map \(I\), the author proved in his thesis (Nijmegen, 1991) that if \(F\) contains ``many'' units, then \(K_ 2({\mathcal O}_ F)\) can be generated by Dennis-Stein symbols. In the opposite situation (e.g., when \({\mathcal O}^ x_ F\) is finite), it is even hard to point out non-trivial elements in \(K_ 2({\mathcal O}_ F)\). In this paper, the author gives a new construction of \(I\) which appeals to matrices and lends itself to calculation. He is then able to compute non-trivial elements in \(K_ 2({\mathcal O}_ F)\), \(F\) being an imaginary quadratic field.
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 19F15
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 166142
zbMATH Keywords: symbols
zbMATH Keywords: class-group
zbMATH Keywords: matrices
zbMATH Keywords: non-trivial elements
zbMATH Keywords: imaginary quadratic field
MaRDI profile type: Publication
cites work: Solution of the congruence subgroup problem for \(\text{SL}_ n\) \((n\geq 3)\) and \(\text{Sp}_{2n}\) \((n\geq 2)\)
cites work: Bounded Elementary Generation of SL n (O)
cites work: On the structure of the \(K_ 2\) of the ring of integers in a number field
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Generating the tame and wild kernels by Dennis-Stein symbols
cites work: Higher algebraic K-theory: I
cites work: Q3921379
cites work: K-theory of finite groups and orders. Notes by E. Graham Evans
cites work: Relations between \(K_2\) and Galois cohomology
cites work: Q5690489
Recommended article: Conditions under which $K₂(
Recommended article: Subrings in imaginary quadratic fields which are not universal for GE2
Recommended article: On the generation of the tame kernel by Dennis-Stein symbols
Recommended article: Generating the tame and wild kernels by Dennis-Stein symbols
Recommended article: The 2-Sylow subgroup of the wild kernel of exceptional number fields
Recommended article: On tame and wild kernels of special number fields
Recommended article: On elements of given order in \(K_ 2F\)
Recommended article: Generalization of Thue's theorem and computation of the group \(K_ 2 O_ F\)
Recommended article: Computation of \(K_ 2\mathbb{Z}[\frac{1+\sqrt{-35}}{2}]\)
Recommended article: La théorie de Kummer et le $K_2$ des corps de nombres
description: scientific article; zbMATH DE number 166142