APM  Vol.3 No.7 A , October 2013
On the Generalization of Hilbert’s 17th Problem and Pythagorean Fields
Author(s) Yuji Shimizuike*
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.


Cite this paper
Y. Shimizuike, "On the Generalization of Hilbert’s 17th Problem and Pythagorean Fields," Advances in Pure Mathematics, Vol. 3 No. 7, 2013, pp. 1-4. doi: 10.4236/apm.2013.37A001.
References
[1]   R. Elman and T. Y. Lam, “Quadratic Forms under Algebraic Extensions,” Mathematische Annalen, Vol. 219, No. 1, 1976, pp. 21-42.
http://dx.doi.org/10.1007/BF01360856

[2]   E. Becker, “Hereditarily Pythagorean Field and Orderings of Higher Level,” Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 1978.

[3]   K. Koziol, “Quadratic Forms over Quadratic Extensions of Generalized Local Fields,” Journal of Algebra, Vol. 118, No. 1, 1988, pp. 1-13.
http://dx.doi.org/10.1016/0021-8693(88)90043-9

[4]   K. Szymiczek, “Generalized Hilbert Fields,” Journal für die Reine und Angewandte Mathematik, Vol. 329, 1981, pp. 58-65.

[5]   K. Szymiczek, “Generalized Rigid Elements in Fields,” Pacific Journal of Mathematics, Vol. 129, No. 1, 1987, pp. 171-186. http://dx.doi.org/10.2140/pjm.1987.129.171

[6]   J. L. Yucas, “Quadratic Forms and Radicals of Fields,” Acta Arithmetica, Vol. 39, No. 4, 1977, pp. 313-322.

[7]   D. Kijima and M. Nishi, “Kaplansky’s Radical and Hilbert Theorem 90 II,” Hiroshima Mathematical Journal, Vol. 13, No. 1, 1983, pp. 29-37.

[8]   E. Witt, “über Quadratische Formen in Korpern,” Collected Papers (1999), Springer, 1967, pp. 35-40.

[9]   I. Kaplansky, “Frolich’s Local Quadratic Forms,” Journal für die Reine und Angewandte Mathematik, Vol. 239-240, 1969, pp. 74-77.

[10]   E. Artin, “über die Zerlegung Definiter Funktionen in Quadrate,” Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, Vol. 5, No. 1, 1927, pp. 100-115. http://dx.doi.org/10.1007/BF02952513

[11]   A. Pfister, “Quadratic Forms with Applications to Algebraic Geometry and Topology,” Cambridge University Press, Cambridge, 1995.

[12]   T. Y. Lam, “Ordering, Valuations and Quadratic Forms,” Regional Conference Series in Mathematics 52, American Mathematical Society, Providence, 1983.

[13]   T. Y. Lam, “Introduction to Quadratic Forms over Fields,” American Mathematical Society, Providence, 2005.

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

[15]   D. Kijima and Y. Shimizuike, “Quadratic Extensions of n-Pythagorean Fields,” Communications in Algebra, Vol. 23, No. 13, 1995, pp. 4851-4860.
http://dx.doi.org/10.1080/00927879508825504

[16]   Y. Shimizuike, “Some Characterizations of n-Pythagorean Fields,” Annual Meeting of Chugoku and Shikoku Area of the Mathematical Society of Japan at Kouchi University, 19 January 1992, pp. 1-3.

[17]   Y. Shimizuike, “Some Characterizations of n-Pythagorean Fields II,” Unpublished Manuscript, 1995, pp.1-8.

[18]   D. Kijima, “On the Abstract Witt Ring of n-Pythagorean fields,” Annual Meeting of Chugoku and Shikoku Area of the Mathematical Society of Japan at Kouchi University, 19 January 1992.

[19]   K. Kula, “Fields with Prescribed Quadratic Form Schemes,” Mathematische Zeitschrift, Vol. 167, No. 3, 1979, pp. 201-212.
http://dx.doi.org/10.1007/BF01174801

[20]   A. Pfister, “Quadratische Formen in Beliebigen Korpern,” Inventiones Mathematicae, Vol. 1, No. 2, 1966, pp. 116-132. http://dx.doi.org/10.1007/BF01389724

[21]   G. Krawczyk, “Closed Subgroups of a Quadratic Form Scheme,” Annales Mathematicae Silesianae, Vol. 3, No. 5, 1990, pp. 18-23.

[22]   Y. Shimizuike, “A Galois Correspondence in the Quadratic Form Theory,” in Preparation.

 
 
Top