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Title: Definable henselian valuations and absolute Galois groups
Author: Jahnke, Franziska Maxie
ISNI:       0000 0004 5349 8885
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2014
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This thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.
Supervisor: Koenigsmann, Jochen Sponsor: European Union (Marie Curie Grant, Network Maloa)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematical logic and foundations ; Valuations ; Henselian ; Absolute Galois Theory ; Model Theory of Fields ; Definability