A P-adic structure which does not interpret an infinite field but whose Shelah completion does

TL;DR

This paper presents a p-adic structure that, while its Shelah completion interprets the p-adic numbers, does not interpret an infinite field, challenging assumptions about interpretability in model theory.

Contribution

It provides a novel example of a p-adic structure with unique interpretability properties, influencing the understanding of NIP structures and their geometric theories.

Findings

Shelah completion interprets $\

$\

challenges existing beliefs about field interpretation in NIP structures.

Abstract

We give a $p$-adic example of a structure whose Shelah completion interprets $\mathbb{Q}_p$ but which does not (provided an extremely plausible conjecture holds) interpret an infinite field. In the final section we discuss the significance of such examples for a possible future geometric theory of $\mathrm{NIP}$ structures.