Generators of central simple algebras
type: Item
instance of: scholarly article
title: Generators of central simple algebras
zbMATH Open document ID: 0908.16018
author: Zinovy Reichstein
published in: Journal of Algebra
publication date: 1999-03-17T00:00:00Z
review text: Let \(A\) be a central simple algebra with centre \(K\). Given a subfield \(k\) of \(K\) and \(z_1,\dots,z_m\in A\), the subalgebra \(k(z_1,\dots,z_m)\) of \(A\) is defined by taking the \(k\)-algebra \(B\) generated by \(z_1,\dots,z_m\) and localizing at all elements of \(B\cap K^\times\). The author asks which algebras can be generated in this way, for a given \(m\), and in answer proves the following theorem. Let \(A\) be a central simple algebra of degree \(n\) over its centre \(K\). Given an integer \(m\geq 2\), suppose that \(K\) is a finitely generated extension field of \(k\) such that \(\text{tr.deg}_k(K)\leq(m-1)n^2\). Then \(A=k(z_1,\dots,z_m)\) for some \(z_1,\dots,z_m\in A\). He notes the special cases \(n=1\), giving a form of the primitive element theorem and the case \(\text{tr.deg}_k(K)\leq 4\), showing that a noncommutative algebra can be generated by two elements over \(k\). He gives two proofs, a fairly direct one for characteristic zero and one that is valid generally. He also notes that the result is a converse of a theorem of \textit{C. Procesi} [Rings with polynomial identities, Dekker, New York (1973; Zbl 0262.16018), VIII. 3].
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16S15
zbMATH DE Number: 1213804
zbMATH Keywords: central simple algebras
zbMATH Keywords: finitely generated field extensions
zbMATH Keywords: primitive element theorem
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: jabr.1998.7492
OpenAlex ID: W2035349570
Wikidata QID: Q117380884
cites work: Q3937313
cites work: Q5588717
cites work: Q5602178
cites work: Q5677619
cites work: On Automorphisms of Matrix Invariants
cites work: An embedding property of universal division algebras
cites work: Q4040049
DOI: 10.1006/JABR.1998.7492
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description: scientific article; zbMATH DE number 1213804