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EXISTENTIAL ∅-DEFINABILITY OF HENSELIAN VALUATION RINGS

Published online by Cambridge University Press:  13 March 2015

ARNO FEHM
ARNO FEHM*
Affiliation:
FACHBEREICH MATHEMATIK UND STATISTIK, UNIVERSITY OF KONSTANZ, 78457 KONSTANZ, GERMANYE-mail:[email protected]
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Abstract

In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields.

Keywords

Henselian valuationexistential definabilityfinite fieldspseudo-algebraically closed fields
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 1 , March 2015 , pp. 301 - 307
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Anscombe, Will and Koenigsmann, Jochen, An existential ∅-definition of in , this Journal, vol. 79 (2014), no. 4, pp. 1336–1343.Google Scholar
Ax, James, On the undecidability of power series fields. Proceedings of the American Mathematical Society, vol. 16 (1965), p. 846.Google Scholar
Ax, James, The elementary theory of finite fields. Annals of Mathematics, vol. 88 (1968), no. 2, pp. 239–271.CrossRefGoogle Scholar
Ax, James and Kochen, Simon, Diophantine problems over local fields I. American Journal of Mathematics, vol. 87 (1965), no. 3, pp. 605–630.Google Scholar
Cluckers, Raf, Derakhshan, Jamshid, Leenknegt, Eva, and Macintyre, Angus, Uniformly defining valuation rings in henselian valued fields with finite or pseudo-finite residue fields. Annals of Pure and Applied Logic, vol. 164 (2013), pp. 1236–1246.Google Scholar
Efrat, Ido, Valuations, Orderings, and Milnor K-Theory, American Mathematical Society, Providence, 2006.CrossRefGoogle Scholar
Fehm, Arno, Subfields of ample fields. Rational maps and definability. Journal of Algebra, vol. 323 (2010), pp. 1738–1744.CrossRefGoogle Scholar
Fried, Michael D. and Jarden, Moshe, Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 11, Springer, Berlin, 2008.Google Scholar
Heinemann, B. and Prestel, A., Fields regularly closed with respect to finitely many valuations and orderings. Canadian Mathematical Society Conference Proceedings, vol. 4 (1984), pp. 297–336.Google Scholar
Helbig, Patrick, Existentielle Definierbarkeit von Bewertungsringen, Bachelor thesis, Konstanz, 2013.Google Scholar
Prestel, Alexander, Jensen, R. B. and Prestel, A., editors, Pseudo real closed fields, Set Theory and Model Theory, Proceedings, Bonn 1979, Springer, Berlin, 1981, pp. 127–156.Google Scholar