On the tensor product of division algebras
type: Item
instance of: scholarly article
title: On the tensor product of division algebras
zbMATH Open document ID: 0712.16014
DOI: 10.1007/BF01198639
author: Pasquale Mammone
published in: Archiv der Mathematik
publication date: 1992-01-01T00:00:00Z
review text: Let \(F_ 0\) be a field containing \(\xi\) a primitive n-th root of unity and \(F_ 1:=F_ 0(X,Y,Z)\) for X,Y,Z variables over \(F_ 0\). Let \(F:=F_ 1((t_ 1))((t_ r))\) be the Laurent series field in two variables. Consider the following two division algebras \(D_ 1\) and \(D_ 2:\) \(D_ 1:=(X(1-Y),t_ 1)_{n,F}\otimes (X(1-Z),t_ 1)_{n,F}\text{ and } D_ 2:=(X,Y,Z)_{n,F}\)where, as usual, for \(a,b\in F^*\) the symbol \((a,b)_{n,F}\) denotes the cyclic algebra generated by two elements i,j such that \(i^ n=a\); \(j^ n=b\) and \(ij=\xi ji.\) In this note it is proved that \(index(D_ 1\otimes D_ 2^{\otimes n})<index(D_ 1)\cdot index(D_ 2)\) for all \(n<\exp onent\)(D\({}_ 2)\) but nevertheless \(D_ 1\) and \(D_ 2\) have no common subfield properly containing F. This result gives a negative answer to a question of Rowen, see \textit{L. H. Rowen}'s book: Ring theory, vol. II (Academic Press, 1988; Zbl 0651.16002), p. 217. The above construction also gives a negative answer to the following question of Max Knus: Let \(A_ 1,A_ 2,A_ 3\) be three quaternion division algebras over a field f. If \(A_ 1\otimes_ FA_ 2\otimes_ FL\) is a division algebra for any quadratic extension L of F contained in \(A_ 3\), is \(A_ 1\otimes A_ 2\otimes A_ 3\) a division algebra?
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 4171152
zbMATH Keywords: tensor product
zbMATH Keywords: Schur index
zbMATH Keywords: Laurent series field
zbMATH Keywords: division algebras
zbMATH Keywords: cyclic algebra
zbMATH Keywords: exponent
MaRDI profile type: Publication
cites work: On the Wedderburn norm condition for cyclic algebras
cites work: The Sequential Design of Experiments for Infinitely Many States of Nature
cites work: A New Construction of Noncrossed Product Algebras
cites work: The Henselization of a valued division algebra
cites work: Symmetric bilinear forms and quadratic forms
cites work: Q5796078
cites work: The Index of a Brauer Class on a Brauer-Severi Variety
cites work: Q4040049
cites work: Totally Ramified Valuations on Finite-Dimensional Division Algebras
full work available at URL: https://doi.org/10.1007/bf01198639
OpenAlex ID: W2006823771
description: scientific article; zbMATH DE number 4171152