Model Completeness for Henselian Fields with finite ramification valued in a $Z$-Group

TL;DR

This paper establishes model-completeness results for Henselian valued fields with finite ramification and value group as a Z-group, extending to certain algebraic extensions of p-adic fields, under conditions on residue fields.

Contribution

It proves model-completeness of Henselian valued fields with specific value group and residue field conditions, including finite ramification cases and algebraic extensions of p-adic fields.

Findings

Model-completeness of Henselian valued fields with Z-group value groups.

Characterization of model-completeness for algebraic extensions of Q_p.

Necessary and sufficient conditions for model-completeness of certain pseudo-algebraically closed fields.

Abstract

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. We apply this to prove that every infinite algebraic extension of the field of $p$-adic numbers $\Bbb Q_p$ with finite ramification is model-complete in the language of rings. For this, we give a necessary and sufficient condition for model-completeness of the theory of a perfect pseudo-algebraically closed field with pro-cyclic absolute Galois group.