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.