On the \(n\)-torsion subgroup of the Brauer group of a number field.
type: Item
instance of: scholarly article
title: On the \(n\)-torsion subgroup of the Brauer group of a number field.
zbMATH Open document ID: 1048.11089
author: Hershy Kisilevsky
author: Jack Sonn
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93880
review text: Let \(K\) be an algebraic number field Galois over \(\mathbb Q\), let \(\text{Br}(K)\) be the Brauer group, and let \(\text{Br}_n(K)\) be its \(n\)-torsion subgroup. The authors show that, providing \(n\) is prime to the class number of \(K\), then \(\text{Br}_n(K)\cong \text{Br}(L/K)\) is an algebraic relative Brauer group for an abelian extension \(L/K\). The proof depends on the following construction: given a prime number \(\ell\) not dividing the class number of \(K\) and a positive integer \(r\), there is an abelian extension \(L/K\) of exponent \(\ell^r\) with local degree equal to \(\ell^r\) at every finite prime. If \(\ell=2\), \(L\) can be taken to be totally complex.
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 2058864
MaRDI profile type: Publication
OpenAlex ID: W2320129096
cites work: Relative Brauer groups and \(m\)-torsion
cites work: Q3876028
cites work: Q3905382
cites work: Relative Brauer groups. III.
DOI: 10.5802/JTNB.397
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description: scientific article; zbMATH DE number 2058864