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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of preordered fields with the weak Hilbert property
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by Guangxin Zeng PDF
Proc. Amer. Math. Soc. 104 (1988), 335-342 Request permission

Abstract:

By introducing a new notion "local density", we show the following result: Let $(K,S)$ be a preordered field, then $(K,S)$ has the weak Hilbert property if and only if $(K,S)$ is locally dense. From this, a theorem of McKenna in reference [1] and one of Prestel in reference [2] will be easily proved.
References
  • Kenneth McKenna, New facts about Hilbert’s seventeenth problem, Model theory and algebra (A memorial tribute to Abraham Robinson), Lecture Notes in Math., Vol. 498, Springer, Berlin, 1975, pp. 220–230. MR 0401720
  • Alexander Prestel, Sums of squares over fields, Proceedings of the 5th School of Algebra (Rio de Janeiro, 1978) Soc. Brasil. Mat., Rio de Janeiro, 1978, pp. 33–44. MR 572053
    • —, Lectures on formally real fields, IMPA Lecture Notes, no. 22, Soc. Brasil Rio de Janeiro, 1975.
  • Alexander Prestel, Pseudo real closed fields, Set theory and model theory (Bonn, 1979) Lecture Notes in Math., vol. 872, Springer, Berlin-New York, 1981, pp. 127–156. MR 645909
  • T. Y. Lam, The theory of ordered fields, Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979) Lecture Notes in Pure and Appl. Math., vol. 55, Dekker, New York, 1980, pp. 1–152. MR 584611
  • Abraham Robinson, On ordered fields and definite functions, Math. Ann. 130 (1955), 275–271. MR 75932, DOI 10.1007/BF01343896
  • Nathan Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974. MR 0356989
  • Guang Xing Zeng, Positive definite functions over formally real fields with core, Adv. in Math. (Beijing) 17 (1988), no. 3, 285–289 (Chinese, with English summary). MR 962689
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 104 (1988), 335-342
  • MSC: Primary 12D15; Secondary 11E25, 11E81, 12J15
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0962795-2
  • MathSciNet review: 962795