On the Generalization of Hilbert’s 17th Problem and Pythagorean Fields

• 作者: Yuji Shimizuike
• 来源期刊: 理论数学进展(英文)
• 学科: 数学
• 关键词: Hilbert’s 17th PROBLEM Preorderings nth RADICAL PYTHAGOREAN FIELDS Round QUADRATIC Forms
• 年，卷（期）: 2013,（7）
• 页码范围: 1-4
• 页数: 4页
• 分类号: O1

摘要

The notion of preordering, which is a generalization of the notion of ordering, has been introduced by Serre. On the other hand, the notion of round quadratic forms has been introduced by Witt. Based on these ideas, it is here shown that 1) a field F is formally real n-pythagorean iff the nth radical, RnF is a preordering (Theorem 2), and 2) a field F is n-pythagorean iff for any n-fold Pfister form ρ. There exists an odd integer l(>1) such that l×ρ is a round quadratic form (Theorem 8). By considering upper bounds for the number of squares on Pfister’s interpretation, these results finally lead to the main result (Theorem 10) such that the generalization of pythagorean fields coincides with the generalization of Hilbert’s 17th Problem.