On the torsion in \(K_2\) of a field
type: Item
instance of: scholarly article
title: On the torsion in \(K_2\) of a field
zbMATH Open document ID: 1162.11054
author: Kejian Xu
author: Min Liu
published in: Science in China. Series A
publication date: 2008-09-23T00:00:00Z
review text: Let \(F\) be either a number field or a function field over an algebraically closed field \(k\), \(K_2(F)\) the Milnor \(K_2\)-group of \(F\), and \(G_n(F) = \{\{a, \Phi_n(a)\} \in K_2(F) \; | \; a, \Phi_n(a) \in F^*\}\) where \(\Phi_n(a)\) is \(n\)-th cyclotomic polynomial. Assume that \(n \neq 1,4,8,12\) is a positive integer having a square factor and \(\mathrm{char}\,k \nmid n\) in the function field case. The authors prove that \(G_n(F)\) is not a subgroup of \(K_2(F)\). The proofs use the Faltings theorem in the number field case and the results of Manin, Grauert, Samuel and Li on Mordell conjecture in the function field case. In contrast, it is known that if \(n = 1,2,3,4\) or 6, and \(F \neq \mathbb{F}_2\), then \(G_n(G)\) is a subgroup of \(K_2(F)\).
reviewed by: Vasyl I. Andriychuk
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 19F15
zbMATH DE Number: 5346103
zbMATH Keywords: number field
zbMATH Keywords: function field
zbMATH Keywords: Milnor \(K_2\)-group
zbMATH Keywords: Mordell conjecture
MaRDI profile type: Publication
full work available at URL: s11425-008-0071-6
OpenAlex ID: W1885239255
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Relations between \(K_2\) and Galois cohomology
cites work: Torsion in \(K_ 2\) of fields
cites work: Elements of small order in K2F
cites work: On elements of given order in \(K_ 2F\)
cites work: On the dynamical origin of asymptotic t2 dispersion of a nondiffusive tracer in incompressible laminar flows
cites work: A conjecture on a class of elements of finite order in \(K_2F_{\mathfrak p}\)
cites work: A class of torsion elements in \(K_{2}\) of a local field
cites work: On the torsion in \(K_ 2\) of local fields
cites work: Game Theory and Eigensystems
cites work: The torsion elements in K2of some local fields
cites work: The subgroups of finite orders in \(K_2\mathbb{Q}\)
cites work: Some elements of finite order in \(K_{2}\mathbb Q\)
cites work: Some elements of finite order in \(K_2\mathbb{Q}\)
cites work: SOME DIOPHANTINE EQUATIONS OVER AND WITH APPLICATIONS TO OF A FIELD
cites work: On the elements of prime power order in K2of a number field
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: On Browkin's conjecture about the elements of order five in \(K_2(\mathbb Q)\)
cites work: Elements of small orders in \(K_2F\). II
cites work: Rational points of algebraic curves over function fields
cites work: Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper
cites work: Complements à un article de Hans Grauert sur la conjecture de Mordell
cites work: Bounds on the number of non-rational subfields of a function field
cites work: On the group of automorphisms of a function field of genus at least two
cites work: Q5723265
cites work: Q3710637
cites work: Q2776773
cites work: Q3994813
DOI: 10.1007/S11425-008-0071-6
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description: scientific article; zbMATH DE number 5346103