Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

Annals of Pure and Applied Logic 164 (12):1236-1246 (2013)
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Abstract

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring

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Citations of this work

Model theory of adeles I.Jamshid Derakhshan & Angus Macintyre - 2022 - Annals of Pure and Applied Logic 173 (3):103074.
Definable Henselian valuations.Franziska Jahnke & Jochen Koenigsmann - 2015 - Journal of Symbolic Logic 80 (1):85-99.
An undecidability result for the asymptotic theory of p-adic fields.Konstantinos Kartas - 2023 - Annals of Pure and Applied Logic 174 (2):103203.
Definable Henselian valuation rings.Alexander Prestel - 2015 - Journal of Symbolic Logic 80 (4):1260-1267.

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References found in this work

On definable subsets of p-adic fields.Angus MacIntyre - 1976 - Journal of Symbolic Logic 41 (3):605-610.
Model Theory: An Introduction.David Marker - 2003 - Bulletin of Symbolic Logic 9 (3):408-409.
The Elementary Theory of Finite Fields.James Ax - 1973 - Journal of Symbolic Logic 38 (1):162-163.

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