A new proof of McKenna’s theorem

Zeng, Guang Xin

doi:10.1090/S0002-9939-1988-0934851-6

A new proof of McKenna’s theorem
HTML articles powered by AMS MathViewer

by Guang Xin Zeng PDF

Proc. Amer. Math. Soc. 102 (1988), 827-830 Request permission

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

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
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

Lectures on formally real fields

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
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