J.-P. Serre, “Extensions of Ordered Fields,” Comptes Rendus de l’Académie des Sciences, Paris, Vol. 229, 1949, pp. 576-577.

has been cited by the following article:

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

  AUTHORS: Yuji Shimizuike

  KEYWORDS: Hilbert’s 17th Problem; Preorderings; nth Radical; Pythagorean Fields; Round Quadratic Forms

  JOURNAL NAME: Advances in Pure Mathematics, Vol.3 No.7A, October 24, 2013

  ABSTRACT: 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.