# On maximal immediate extensions of valued fields

## Anna Blaszczok

### University of Katowice

A valued field extension is called immediate if the corresponding value group and residue field extensions are trivial. A better understanding of the structure of such extensions turned out to be important for questions in algebraic geometry, real algebra and the model theory of valued fields.

In this talk we focus mainly on the problem of the uniqueness of maximal immediate extensions.

Kaplansky proved that under a certain condition, which he called “hypothesis A”, all maximal immediate extensions of the valued field are isomorphic. We study a more general case, omitting one of the conditions of hypothesis~A. We describe the structure of maximal immediate extensions of valued fields under such weaker assumptions. This leads to another condition under which fields in this class admit unique maximal immediate extensions.

We further prove that there is a class of fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree. We introduce a classification of Artin-Schreier defect extensions and describe its importance for the construction of such maximal immediate extensions.

We present also the consequences of the above results and and of the model theory of tame fields for the problem of uniqueness of maximal immediate extensions up to elementary equivalence.