Definable henselian valuations and absolute Galois groups

Thesis

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. Since the existence of a parameter-free definable henselian valuation on a field ensures that henselianity is elementary in ${cal L}_{ring}$, we also study henselianity as an ${cal L}_{ring}$-property.

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.

Authors: Jahnke, F

• Publication date: 2014
• DOI: 10.5287/ora-yrkjg2zmp
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: absolute Galois theory; Henselian; model theory of fields; definability; valuations
• Subjects: Mathematical logic and foundations