Dec groups for arbitrarily high exponents
type: Item
instance of: scholarly article
title: Dec groups for arbitrarily high exponents
zbMATH Open document ID: 0801.16016
author: B. A. Sethuraman
published in: Pacific Journal of Mathematics
publication date: 1994-12-01T00:00:00Z
review text: Let \(K/F\) be an abelian Galois extension of fields. The group \(\text{Dec}(K/F)\) is the subgroup of the relative Brauer group \(\text{Br}(K/F)\) generated by the relative Brauer groups \(\text{Br}(L/F)\) of all the cyclic extensions \(L/F\) contained in \(K\). This group was introduced by the reviewer [J. Algebra 70, 420-436 (1980; Zbl 0473.16004)] in relation with the construction of indecomposable division algebras of prime exponent. If \(K/F\) is elementary abelian of degree 4, then it is known that \(\text{Dec}(K/F)\) is the 2-torsion subgroup in \(\text{Br}(K/F)\). In the present paper, the author explicitly constructs for each prime \(p\) and each integer \(n\geq 1\) (\(n \geq 2\) if \(p = 2\)) a field \(F\), an abelian extension \(K/F\) with Galois group \(({\mathbb{Z}}/p^ n \mathbb{Z})\times (\mathbb{Z}/p\mathbb{Z})\) and a central simple algebra \(A\) of exponent \(p\) split by \(K\) whose Brauer class is not in \(\text{Dec}(K/F)\). The base field \(F\) is a rational function field in three variables over a field of characteristic zero containing sufficiently many roots of unity. The methods of proof are essentially valuation-theoretic.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 12E12
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 537390
zbMATH Keywords: abelian Galois extensions
zbMATH Keywords: relative Brauer groups
zbMATH Keywords: cyclic extensions
zbMATH Keywords: indecomposable division algebras of prime exponent
zbMATH Keywords: central simple algebras
zbMATH Keywords: Brauer class
zbMATH Keywords: rational function fields
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
full work available at URL: pjm.1994.162.373
OpenAlex ID: W2063334420
DOI: 10.2140/PJM.1994.162.373
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description: scientific article; zbMATH DE number 537390