The level of division algebras over local and global fields
Let \(D\) be a division ring. The authors use the following definitions: 1) The level \(s(D)\) of \(D\) is the least integer \(s\) such that \(-1\) is a su m of s squares in \(D\), \(s(D)=\infty\) if \(-1\) is not a sum of squares; 2) The sublevel \(\underline{s}(D)\) of \(D\) is the least integer \(\sigma\) such that 0 is a sum of \(\sigma+1\) nonzero squares in \(D\), \(\underline{s}(D)=\infty\) otherwise; 3) The product-level \(s_{\pi}(D)\) is the least integer \(s\) such that \(- 1 \) is a sum of \(s\) products of squares in \(D\), \(s_{\pi}(D)=\infty\) otherwise. If \(D\) is commutative, trivially \(s(D)=\underline{s}(D)=s_{\pi}(D)\). The authors investigate the relations between these different notions of ``level'' in the case that \(D\) is a finite-dimensional central division algebra over a local or a global field. Their main results are the following theorems: A. Let \(D\) be a central division algebra over a local field \(F\). (i) \(\deg D\) even \(\Rightarrow s(D) = \underline{s}(D) = s_{\pi}(D)=1\), (ii) \(\deg D\) odd, \(D\neq F \Rightarrow s(D)=\min(3,s(F))\), \(\underline{s}(D)=s_{\pi}(D) = 2\). B. Let \(D\) be a central division algebra over a global field \(F\). (i) \(\deg D\) even \(\Rightarrow s_{\pi}(D)=1\), \(\b s(D)=1\) or \(2\), \(s(D)=1\) or \(2\), (ii) \(\deg D\) odd, \(D\neq F \Rightarrow s(D) = \min(3,s(F))\), \(\underline{s}(D) = s_{\pi}(D) = 2\). Moreover, the authors describe the conditions under which \(\underline{s}(D)\) resp. \(s(D)=1\) in the even degree case.
finite-dimensional central division algebra
scientific article; zbMATH DE number 4144090
scientific article