On indecomposable algebras of exponent \(2\).
type: Item
instance of: scholarly article
title: On indecomposable algebras of exponent \(2\).
zbMATH Open document ID: 1170.16014
published in: Israel Journal of Mathematics
publication date: 2008-09-03T00:00:00Z
review text: For every integer \(n\geq 3\), the author constructs a central division algebra of degree \(2^n\) (i.e., dimension~\(4^n\)) and exponent~\(2\) (and characteristic different from~\(2\)) that does not decompose into a tensor product of proper subalgebras, even after an odd-degree extension of the base field. The algebra is Brauer-equivalent to a tensor product of \(n+1\) quaternion algebras \(D\otimes(a_1,t_1)\otimes\cdots\otimes(a_n,t_n)\) over a field of iterated Laurent series \(F(\!(t_1)\!)\cdots(\!(t_n)\!)\), where \(D\) is a suitably constructed quaternion \(F\)-algebra split by the multiquadratic extension \(L=F(\sqrt{a_1},\dots,\sqrt{a_n})\) of degree~\(2^n\). This quaternion algebra \(D\) was first considered by the author [in J. Algebra 275, No. 2, 859-866 (2004; Zbl 1133.11027)]. Its Brauer class is not the sum of \(2\)-torsion elements split by proper subextensions, so \(_2\text{Br}(L/F)\neq{_2\text{Br}}(L_1/F)+{_2\text{Br}}(L_2/F)\) for any \(L_1,L_2\subsetneq L\). A similar property holds for relative Witt groups, using the norm form of \(D\). The first examples of indecomposable division algebras of degree~\(8\) and exponent~\(2\) were constructed by \textit{S. A. Amitsur, L. H. Rowen} and \textit{J.-P. Tignol}, [Isr. J. Math. 33, 133-148 (1979; Zbl 0422.16010)]. They are Brauer-equivalent to tensor products of four quaternion algebras by \textit{J.-P. Tignol}, [C. R. Acad. Sci., Paris, Sér. A 286, 875-876 (1978; Zbl 0374.16006)].
reviewed by: Jean-Pierre Tignol
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 5320447
zbMATH Keywords: division algebras
zbMATH Keywords: multiquadratic extensions
zbMATH Keywords: quaternion algebras
zbMATH Keywords: tensor products
zbMATH Keywords: Brauer equivalence
zbMATH Keywords: relative Witt groups
author: A. S. Sivatski
MaRDI profile type: Publication
full work available at URL: s11856-008-0035-0
OpenAlex ID: W2001978568
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DOI: 10.1007/S11856-008-0035-0
description: scientific article; zbMATH DE number 5320447