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Transfer principles for pseudo real closed e-fold ordered fields

Published online by Cambridge University Press:  12 March 2014

Şerban A. Basarab
Şerban A. Basarab*
Affiliation:
Institute of Mathematics, Bucharest University, Bucharest, Romania Increst, Department of Mathematics, Bucharest, Romania
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Extract

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].

The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.

Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 4 , December 1986 , pp. 981 - 991
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

[1]Ax, J., The elementary theory of finite fields, Annals of Mathematics, ser. 2, vol. 88 (1968), pp. 239–271.CrossRefGoogle Scholar
[2]Basarab, Ş., Definite functions on algebraic varieties over ordered fields, Revue Roumaine de Mathematiques Pures et Appliquées, vol. 29 (1984), pp. 527–535.Google Scholar
[3]Basarab, Ş., On some classes of Hilbertian fields, Resultate der Mathematik, vol. 7 (1984), pp. 1–34.CrossRefGoogle Scholar
[4]Basarab, Ş., The elementary theory of pseudo real closed e-fold ordered fields, Research Report No. AIFD-32, Institute of Informatics, Bucharest, 1983.Google Scholar
[5]Basarab, Ş., Profinite groups with involutions and pseudo real closed fields, Research Report No. AIFD-38, Institute of Informatics, Bucharest, 1983.Google Scholar
[6]Basarab, Ş., The absolute Galois group of a pseudo real closed field with finitely many orders, Journal of Pure and Applied Algebra, vol. 38 (1985), pp. 1–18.CrossRefGoogle Scholar
[7]Chatzidakis, Z., Model theory of profinite groups, Ph.D. Thesis, Yale University, New Haven, Connecticut, 1984.Google Scholar
[8]Cherlin, G., van Den Dries, L. and Acintyre, A. M, The elementary theory of regularly closed fields, Journal für die Reine und Angewandte Mathematik (to appear).Google Scholar
[9]van Den Dries, L., Model theory of fields, Thesis, Utrecht, 1978.Google Scholar
[10]Éršov, Ú. L., Regularly r-closed fields, Doklady Akadémii Nauk SSSR, vol. 266 (1982), pp. 538–540; English translation, Soviet Mathematics Doklady, vol. 26 (1982), pp. 363–366.Google Scholar
[11]Fried, M., Haran, D. and Jarden, M., Galois stratification over Frobenius fields, Advances in Mathematics, vol. 51 (1984), pp. 1–35.CrossRefGoogle Scholar
[12]Jarden, M., The elementary theory of ω-free Ax fields, Inventiones Mathematicae, vol. 38 (1976), pp. 181–206.CrossRefGoogle Scholar
[13]Jarden, M., The elementary theory of large e-fold ordered fields, Acta Mathematica, vol. 149 (1982), pp. 239–260.CrossRefGoogle Scholar
[14]Jarden, M. and Kiehne, U., The elementary theory of algebraic fields of finite corank, Inventiones Mathematicae, vol. 30 (1975), pp. 275–294.CrossRefGoogle Scholar
[15]McKenna, K., Pseudo Henselian and pseudo real closed fields, Ph. D. Thesis, Yale University, New Haven, Connecticut.Google Scholar
[16]Lubotzky, A. and van den Dries, L., Subgroups of free profinite groups and large subfields of , Israel Journal of Mathematics, vol. 39 (1981), pp. 25–45.CrossRefGoogle Scholar
[17]Prestel, A., Pseudo real closed fields, Set theory and model theory (Bonn, 1979), Lecture Notes in Mathematics, vol. 782, Springer-Verlag, Berlin, 1981, pp. 127–156.CrossRefGoogle Scholar
[18]Sacks, G., Saturated model theory, Addison-Wesley, Reading, Massachusetts, 1972.Google Scholar
[19]van der Waerden, B. L., Moderne Algebra, Vol. I, Springer-Verlag, Berlin, 1930.CrossRefGoogle Scholar