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Quantifier elimination in tame infinite p-adic fields

Published online by Cambridge University Press:  12 March 2014

Ingo Brigandt
Ingo Brigandt*
Affiliation:
Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
*
Department of History and Philosophy of Science, University of Pittsburgh, 1017 Cathedral of Learning, Pittsburgh, PA 15260, USA, E-mail: inb1+@pitt.edu
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Abstract

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of ℚp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky's condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued fields is extended by the power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1493 - 1503
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Ax, J. and Kochen, S., Diophantine problems over localfields II. A complete set of axioms for p-adic number theory, American Journal of Mathematics, vol. 87 (1965), pp. 631–648.CrossRefGoogle Scholar
[2]Ax, J. and Kochen, S., Diophantine problems over localfields III. Decidable fields, Annals ofMathematics, vol. 83 (1966), pp. 437–456.Google Scholar
[3]Ershov, Yu. L., On elementary theories of local fields, Algebra i Logika, vol. 4 (1965), no. 2, pp. 5–30.Google Scholar
[4]Fried, M. D. and Jarden, M., Field arithmetic, Springer-Verlag, 1986.CrossRefGoogle Scholar
[5]Kiefe, C., Sets definable over finite fields: Their zeta-functions, Transactions of the American Mathematical Society, vol. 223 (1976), pp. 45–59.CrossRefGoogle Scholar
[6]Koenigsmann, J., Half-ordered fields, Ph.D. thesis, Universität Konstanz, Germany, 1993.Google Scholar
[7]Kuhlmann, F.-V., Henselian function fields and tame fields, unpublished manuscript, 1990.Google Scholar
[8]Kuhlmann, F.-V., Valuation theory of fields, abelian groups and modules, Habilitationsschrift, Universität Heidelberg, Germany, 1994.Google Scholar
[9]Kuhlmann, F.-V., Pank, M., and Roquette, P., Immediate and purely wild extensions of valued fields, Manuscripta Mathematica, vol. 55 (1986), pp. 39–67.CrossRefGoogle Scholar
[10]Macintyre, A., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605–610.Google Scholar
[11]Macintyre, A., Twenty years of p-adic model theory, Logic Colloquium, vol. 84 (1986), pp. 121–153.Google Scholar
[12]Prestel, A. and Roquette, P., Formally p-adic fields, Springer-Verlag, 1984.CrossRefGoogle Scholar
[13]Robinson, A. and Zakon, E., Elementary properties of ordered abelian groups, Transactions of the American Mathematical Society, vol. 96 (1969), pp. 222–236.CrossRefGoogle Scholar
[14]Weispfenning, V., Elimination of quantifiers for certain ordered and lattice-ordered abelian groups, Bulletin de la Société Mathématique de Belgique, vol. 33, series B (1981), pp. 131–155.Google Scholar