Construction of valuations from \(K\)-theory
type: Item
instance of: scholarly article
title: Construction of valuations from \(K\)-theory
zbMATH Open document ID: 0963.12003
author: Ido Efrat
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: This expository paper gives a unified presentation of some explicit constructions of certain valuations on a field. In particular, given a prime number \(p\), a field \(K\) of characteristic \(\neq p\), and a subgroup \(T\) of \(K^\times\) containing \((K^\times)^p\) and \(-1\), suppose \(T\) satisfies the two conditions (i) if \(x \in K^\times \backslash T\) and \(y \in T\backslash K^p\), then \(\{x,y\} \neq 0\) and (ii) if the cosets of \(x,y \in K^\times\) in \(K^\times /T\) are \( F_p\)-linearly independent then \(\{x,y\} \neq 0\), where \(\{x,y\}\) denotes an element in the second Milnor \(K\)-group \(K_2^M(K)\) of \(K\). Then a valuation ring \(O\) on \(K\) is constructed having non-\(p\)-divisible value group, residue characteristic \(\neq p\), and for which the principal units are \(p\)th powers. The approach is elementary in that it uses neither cohomology nor noncommutative division rings, nor is the existence of primitive \(p\)th roots of unity in the field \(K\) assumed. The cohomological structures used in many previous proofs of the existence of such a valuation are replaced by the second Milnor \(K\)-group.
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 19F99
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 1489500
zbMATH Keywords: valuation theory
zbMATH Keywords: K-theory
zbMATH Keywords: expository paper
MaRDI profile type: Publication
full work available at URL: mrl.1999.v6.n3.a7
OpenAlex ID: W2060587529
DOI: 10.4310/MRL.1999.V6.N3.A7
description: scientific article; zbMATH DE number 1489500