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Proceedings of the American Mathematical Society

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A new proof of McKenna's theorem

Author: Guang Xin Zeng
Journal: Proc. Amer. Math. Soc. 102 (1988), 827-830
MSC: Primary 12D15
MathSciNet review: 934851
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Abstract: Using a new and simpler method, the following result is shown: Let $ (K,C)$ be a formally real field with core $ C$ which has only a finite number of orderings. Then $ (K,C)$ has the Weak Hilbert Property if and only if $ K$ is dense in every real closure of $ (K,C)$. This result contains the main theorem of McKenna in [1]

References [Enhancements On Off] (What's this?)

  • [1] Kenneth McKenna, New facts about Hilbert’s seventeenth problem, Model theory and algebra (A memorial tribute to Abraham Robinson), Springer, Berlin, 1975, pp. 220–230. Lecture Notes in Math., Vol. 498. MR 0401720
  • [2] 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
  • [3] A. Prestel, Lectures on formally real fields, IMPA Lecture Notes, No. 22, Rio de Janeiro.
  • [4] 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
  • [5] Abraham Robinson, On ordered fields and definite functions, Math. Ann. 130 (1955), 275–271. MR 0075932,
  • [6] Nathan Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974. MR 0356989

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Keywords: Ordered fields, formally real fields with a core, the Weak Hilbert Property, definite rational functions
Article copyright: © Copyright 1988 American Mathematical Society