The level of division algebras over local and global fields
type: Item
instance of: scholarly article
title: The level of division algebras over local and global fields
zbMATH Open document ID: 0698.12013
DOI: 10.1016/0022-314X(89)90059-0
author: David B. Leep
author: Jean-Pierre Tignol
author: Nicole Vast
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11S45
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11E99
zbMATH DE Number: 4144090
zbMATH Keywords: local fields
zbMATH Keywords: global fields
zbMATH Keywords: quadratic forms
zbMATH Keywords: level
zbMATH Keywords: finite-dimensional central division algebra
MaRDI profile type: Publication
cites work: On ordered algebras
cites work: Q5532660
cites work: The class number of hereditary orders in non-Eichler algebras over global function fields
cites work: Q3963168
cites work: Q4339908
cites work: Q4063245
cites work: Q5675788
cites work: Sums of squares in division algebras
cites work: Q3808258
cites work: Q3832673
cites work: Q3961610
cites work: Q4061092
cites work: On the level of skew-fields
cites work: Q5796078
cites work: On Ordered Skew Fields
cites work: Q3029076
description: scientific article; zbMATH DE number 4144090