Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields
type: Item
instance of: scholarly article
title: Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields
zbMATH Open document ID: 1100.11040
author: Cristian D. Popescu
published in: Journal of Number Theory
publication date: 2006-03-08T00:00:00Z
review text: For a field \(k\) and a natural number \(m\) denote by \(\text{Br}_m(k)\) the \(m\)-torsion subgroup of the Brauer group Br\((k)\) of \(k\). In their paper ``Relative Brauer groups and \(m\)-torsion'', Proc. Am. Math. Soc. 130, No. 5, 1333--1337 (2002; Zbl 1099.11066) \textit{E. Aljadeff} and \textit{J. Sonn} raised the question whether Br\(_m(k)\) is algebraic, i.e., equal to the kernel \(\text{Br}(L/k)\) of the restriction homomorphism \(\text{Br}(k)\to \text{Br}(L)\) for some algebraic separable extension \(L/k\), and examples were given there such that \(\text{Br}_m(k)\) is not algebraic. In the paper under review the author considers this question in the case where \(k\) is a global field of characteristic \(p\), i.e., \(k\) is a finite extension of the rational function field in one variable over the field with \(p\) elements, and obtains the following result in this case: If \(L/k\) is a Galois extension of local degree \(m\) at every place of \(k\) then \(\text{Br}_m(k)=\text{Br}(L/k)\). Motivated by this result the author investigates the existence of Galois extensions \(L/k\) such that their local degrees at every place of \(k\) are equal to a given natural number \(m\). One of his results in this respect is as follows: There is an abelian Galois extension \(L/k\) with this property provided \(m\) is coprime to the order of the non-\(p\)-part of the Picard group of \(k\).
reviewed by: Hans Opolka
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5009834
zbMATH Keywords: class field theory
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.07.002
OpenAlex ID: W1966036485
cites work: Relative Brauer groups and \(m\)-torsion
cites work: Q4692374
cites work: Q3741682
cites work: On the \(n\)-torsion subgroup of the Brauer group of a number field.
cites work: Multiplicative independence in function fields
cites work: Q4298518
DOI: 10.1016/J.JNT.2004.07.002
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description: scientific article; zbMATH DE number 5009834