Franziska Jahnke

DP-MINIMAL VALUED FIELDS

We show that dp-minimal valued fields are henselian and that a dp-minimal field admitting a definable type V topology is either real closed, algebraically closed or admits a non-trivial definable henselian valuation. We give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure.

Definable Henselian valuations

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.

Uniformly defining p-henselian valuations☆

Admitting a non-trivial

On the quantifier complexity of definable canonical Henselian valuations

p

Defining Coarsenings of Valuations

-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity,

Henselianity in the language of rings

p

A definable Henselian valuation with high quantifier complexity

-henselianity is an elementary property in the language of rings. We are interested in the question when a field admits a non-trivial 0-definable

Fields with almost small absolute Galois group

p

NIP henselian valued fields

-henselian valuation (in the language of rings). We give a classification of elementary classes of fields in which the canonical

A Conjectural Classification of Strongly Dependent Fields

p