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DEFINABLE HENSELIAN VALUATIONS

Published online by Cambridge University Press:  13 March 2015

FRANZISKA JAHNKE  and
JOCHEN KOENIGSMANN
FRANZISKA JAHNKE
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK, EINSTEINSTR. 62, 48149 MÜNSTER, GERMANYE-mail: franziska.jahnke@wwu.de
JOCHEN KOENIGSMANN
Affiliation:
MATHEMATICAL INSTITUTE, WOODSTOCK ROAD, OXFORD OX2 6CG, UKE-mail: koenigsmann@maths.ox.ac.uk
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Abstract

In this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

Keywords

Primary: 03C4012E3012J10. Secondary: 03C6012L12Valuationshenselian valued fieldsdefinable valuationsabsolute Galois groups

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 1 , March 2015 , pp. 85 - 99
Copyright
Copyright © The Association for Symbolic Logic 2015 

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