A new proof of McKenna's theorem

Author:
Guang Xin Zeng

Journal:
Proc. Amer. Math. Soc. **102** (1988), 827-830

MSC:
Primary 12D15

DOI:
https://doi.org/10.1090/S0002-9939-1988-0934851-6

MathSciNet review:
934851

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Abstract: Using a new and simpler method, the following result is shown: Let be a formally real field with core which has only a finite number of orderings. Then has the Weak Hilbert Property if and only if is dense in every real closure of . This result contains the main theorem of McKenna in [**1**]

**[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**, https://doi.org/10.1007/BF01343896**[6]**Nathan Jacobson,*Basic algebra. I*, W. H. Freeman and Co., San Francisco, Calif., 1974. MR**0356989**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0934851-6

Keywords:
Ordered fields,
formally real fields with a core,
the Weak Hilbert Property,
definite rational functions

Article copyright:
© Copyright 1988
American Mathematical Society