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Defining Coarsenings of Valuations

Part of: Arithmetic rings and other special rings Model theory Connections with logic General field theory Topological fields

Published online by Cambridge University Press:  10 January 2017

Franziska Jahnke  and
Jochen Koenigsmann
Franziska Jahnke
Affiliation:
Institut für Mathematische Logik, Einsteinstrasse 62, 48149 Münster, Germany (franziska.jahnke@uni-muenster.de)
Jochen Koenigsmann
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK (koenigsmann@maths.ox.ac.uk)
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Abstract

We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.

Keywords

valuationsHenselian valued fieldsdefinable valuations

MSC classification

Secondary: 03C40: Interpolation, preservation, definability 12J10: Valued fields 03C60: Model-theoretic algebra 12L12: Model theory 12E30: Field arithmetic 13F30: Valuation rings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 665 - 687
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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