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]**K. McKenna,*New facts about Hilbert's seventeenth problem*, Lecture Notes in Math., vol. 498, Springer, 1975 pp. 220-230. MR**0401720 (53:5547)****[2]**T. Y. Lam,*The theory of ordered fields*, Lecture Notes in Pure and Appl. Math., Vol. 55, Dekker, New York, 1980, pp. 1-152. MR**584611 (82e:12033)****[3]**A. Prestel,*Lectures on formally real fields*, IMPA Lecture Notes, No. 22, Rio de Janeiro.**[4]**-,*Sums of squares over fields*, Soc. Brasil Mat., Rio de Janeiro, 1979, pp. 33-44 MR**82a**: 12014. MR**572053 (82a:12014)****[5]**A. Robinson,*On ordered fields and definite functions*, Math. Ann.**130**(1955), 257-271. MR**0075932 (17:822a)****[6]**N. Jacobson,*Basic algebra*. I, Freeman, San Francisco, Calif., 1974, pp. 291-299. MR**0356989 (50:9457)**

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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