Three theorems on common splitting fields of central simple algebras
type: Item
instance of: scholarly article
title: Three theorems on common splitting fields of central simple algebras
zbMATH Open document ID: 0935.16007
DOI: 10.1007/BF02810681
author: Nikita Karpenko
published in: Israel Journal of Mathematics
publication date: 2000-05-07T00:00:00Z
review text: The following theorem generalizes results of \textit{A. A. Albert} [Bull. Am. Math. Soc. 37, 301-302 (1931; Zbl 0001.26701)], \textit{L. J. Risman} [Proc. Am. Math. Soc. 51, 35-36 (1975; Zbl 0309.16014)]: Theorem 1. Let \(A\) be a central simple algebra of prime degree \(p\) over a field \(F\). Given \(p-1\) central simple \(F\)-algebras \(\{B_i\}\) of degrees a power of \(p\), if each tensor product \(A\otimes B_i\) has zero-divisors, then there is a field extension \(E/F\) of degree \(\leq k\) splitting \(A\) and the \(B_i\), where \(k\) is the product of the degrees of the \(B_i\). -- For a second result define two central simple algebras \(A\), \(B\) of prime degree \(p\) over a field to be linked if the tensor product of any tensor powers of \(A\) and \(B\) has index \(\leq p\). If they have a common splitting field of finite degree not divisible by \(p^2\), they are called strongly linked. Now the author proves that for algebras \(A\), \(B\) of odd prime degree that are not strongly linked, the base field can be extended so that they become linked but not strongly linked. -- Given a prime \(p\) and \(x_1,\dots,x_n\in\text{Br}(F)\), the Brauer group of \(F\), let \(\alpha\colon\mathbb{Z}^n\to\mathbb{Z}\) be such that for any \(i_1,\dots,i_n\in\mathbb{Z}\) the Schur index \(\text{ind}(i_1x_1+\cdots+i_nx_n)\) is \(p^a\), where \(a=\alpha(i_1,\dots,i_n)\); then the elements \(x_1,\dots,x_n\) are said to have behaviour \(\alpha\) and \(\alpha\) is called a behaviour. For each behaviour \(\alpha\) an integer \(\beta\) is defined in terms of the Brauer-Severi varieties of the corresponding \(x_i\) and the author proves, for given \(p\), \(\alpha\), \(\beta\), (i) for any field \(F\), any \(x_1,\dots,x_n\in\text{Br}(F)\) with behaviour \(\alpha\) have a common splitting field \(E/F\) such that \(v_p([E:F])\leq\beta\) and (ii) there is a field \(F\) and \(x_1,\dots,x_n\in\text{Br}(F)\) with behaviour \(\alpha\) such that all their common splitting fields \(E/F\) satisf…
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16K50
zbMATH DE Number: 1356473
zbMATH Keywords: central simple algebras of prime degree
zbMATH Keywords: tensor products
zbMATH Keywords: field extensions
zbMATH Keywords: splitting fields
zbMATH Keywords: Brauer groups
zbMATH Keywords: Schur index
zbMATH Keywords: Brauer-Severi varieties
zbMATH Keywords: generic algebras
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02810681
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description: scientific article; zbMATH DE number 1356473