Sums of squares in division algebras
type: Item
instance of: scholarly article
title: Sums of squares in division algebras
zbMATH Open document ID: 0548.16019
DOI: 10.1007/BF01160457
author: David B. Leep
author: Daniel B. Shapiro
author: A. R. Wadsworth
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173617
review text: Suppose F is a field of characteristic not 2, and A is a central simple (finite dimensional) F-algebra. Let \(\Sigma A^ 2\) denote the set of sums of squares of elements in A. The question is whether \(\Sigma A^ 2=A\). The authors find examples of fields F and F-algebras A where \(\Sigma A^ 2\neq A\). In fact, whenever F is not a SAP field (e.g. \(F={\mathbb{Q}}(x))\) then there are quaternion algebra examples over F. This answers a question of Jacobson. Such examples seem to be quite rare. If \(A\neq\Sigma A^ 2\) then F must be formally real, A must be a division algebra and no subfield \(L\supseteq F\) in A can have an F-automorphism of order \(>2\). Consequently, if such an algebra A is a crossed product, the Galois group of a Galois maximal subfield must be an elementary abelian 2-group. A major step in the proofs of these results is the following Theorem: Let A be a central simple F-algebra. Then 0 is a sum of nonzero squares in A if and only if \(\Sigma A^ 2=A\). This theorem is proved using a characterization (due to Kasch, Asano and others) of the linear subspaces of A which are invariant under all inner automorphisms. This result on invariant subspaces also provides the tool needed to determine \(\Sigma A^ 2\) when F has characteristic 2. In that case, \(\Sigma A^ 2=\{a\in A: tr(a)\in F^ 2\}.\) Here tr: \(A\to F\) is the reduced trace.
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 3873538
zbMATH Keywords: sums of squares
zbMATH Keywords: quaternion algebra
zbMATH Keywords: division algebra
zbMATH Keywords: F-automorphism
zbMATH Keywords: crossed product
zbMATH Keywords: Galois group
zbMATH Keywords: Galois maximal subfield
zbMATH Keywords: central simple F- algebra
zbMATH Keywords: inner automorphisms
MaRDI profile type: Publication
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OpenAlex ID: W1984075028
description: scientific article; zbMATH DE number 3873538