Finding definable henselian valuations
(Joint work with Jochen Koenigsmann.) There has been a lot of recent progress in the area of definable henselian valuations. Here, a valuation is called definable if its valuation ring is a first-order definable subset of the field in the language of rings. Applications of results concerning definable henselian valuations typically include showing decidability of the theory of a field or facts about its absolute Galois group.
We study the question of which henselian fields admit definable henselian valuations with and without parameters. In equicharacteristic 0, we give a complete characterization of henselian fields admitting parameter-definable (non-trivial) henselian valuations. We also give a partial characterization result for the parameter-free case.
Franziska Jahnke received her DPhil from Oxford University in 2013, under the supervision of Jochen Koenigsmann. She is now a research assistant in Münster, Germany, studying the model theory of fields.