Sums of squares in division algebras
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.
scientific article; zbMATH DE number 3873538
scientific article