Model theory of finite-by-Presburger Abelian groups and finite extensions of $p$-adic fields

TL;DR

This paper introduces finite-by-Presburger abelian groups, proves their theory is model-complete, and applies these results to local fields, providing a new proof of their model completeness in the ring language.

Contribution

It defines finite-by-Presburger groups, proves their theory is model-complete, and interprets higher residue rings of local fields, offering a new proof of local field model completeness.

Findings

Finite-by-Presburger groups have a model-complete theory.

Certain quotients of local field multiplicative groups are finite-by-Presburger.

The paper provides a new proof of local field model completeness.

Abstract

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are finite-by-Presburger and interpret the higher residue rings of the local field. We apply these results to give a new proof of the model completeness in the ring language of a local field of characteristic zero (a result that follows also from work of Prestel-Roquette).