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A transfer theorem for Henselian valued and ordered fields

Published online by Cambridge University Press:  12 March 2014

Rafel Farré
Rafel Farré*
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo, 5, 08028 Barcelona, Spain, E-mail: farre@ma2.upc.es
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Abstract

In well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [Bl] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 3 , September 1993 , pp. 915 - 930
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[A-S]Artin, E. and Screier, O., Algebraische Konstruktion reeller Körpern, Abhandlungen am dem Mathematischen Seminar der Unhersität Hamburg, vol. 5 (1972), pp. 85–99.CrossRefGoogle Scholar
[A-Kl]Ax, J. and Kochen, S., Diophantine problems over local fields. I, American Journal of Mathematics, vol. 87 (1965), pp. 605–630.CrossRefGoogle Scholar
[A-K2]Ax, J. and Kochen, S., Diophantine problems over local fields. II, American Journal of Mathematics, vol. 87 (1965), pp. 631–648.CrossRefGoogle Scholar
[Bl]Becker, E., Hereditarily pythagorean fields and orderings of higher level, IMPA Lecture Notes, vol. 29, Rio de Janeiro, 1978.Google Scholar
[B2]Becker, E., Summen n-ter Potenzen in Körpern, Journal für die Reine und Angewandte Mathematik, vol. 307–308 (1979), pp. 8–30.Google Scholar
[B-H-R]Becker, E., Harman, J. and Rosenberg, H., Signatures of fields and extension theory, Journal für die reine and angewandte Mathematik, vol. 330 (1982), pp. 53–75.Google Scholar
[D-F]Delon, F. and Farré, R.. Some model theory for almost real closed fields, preprint, 1992.Google Scholar
[En]Endler, O., Valuation theory, Universitext 8, Springer-Verlag, Berlin, Heidelberg, and New York, 1972.CrossRefGoogle Scholar
[Ch-K]Chang, C. C. and Keisler, H. J., Model theory, Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, London, 1973.Google Scholar
[E]Ershov, J. L., On the elementary theory of maximal normed fields, Soviet Mathematics Doklady, vol. 6 (1965), pp. 1390–1393.Google Scholar
[J]Jacob, B., The model theory of Pythagorean fields, Ph.D. Thesis, Princeton University, Princeton, 1979.Google Scholar
[K-P]Kuhlmann, F. V. and Prestel, A., On places of algebraic function fields, Journal für die Reine und angewandte Mathematik, vol. 353 (1984), pp. 181–195.Google Scholar
[Po]Powers, V., Characterizing reduced Witt rings of higher level, Pacific Journal of Mathematics, vol. 128 (1987), pp. 333–347.CrossRefGoogle Scholar
[Ri]Ribenboim, P., Théorie des Valuations, Les presses de l’Université de Montréal, vol. 9, Montréal, 1965.Google Scholar