Finite Undecidability in Fields I: NIP Fields

TL;DR

This paper proves that all NIP henselian nontrivially valued fields are finitely undecidable, and under the NIP Fields Conjecture, all NIP fields share this property, advancing the understanding of decidability in field theories.

Contribution

It extends existing constructions to classify NIP henselian fields as finitely undecidable and links this to the broader NIP fields conjecture.

Findings

All NIP henselian nontrivially valued fields are finitely undecidable.

Assuming the NIP Fields Conjecture, all NIP fields are finitely undecidable.

The work connects model-theoretic properties with decidability in field theories.

Abstract

A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if $\mbox{Cons}(\Sigma)$ is undecidable for every nonempty finite $\Sigma \subseteq \mbox{Th}(K; \mathcal{L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author's PhD thesis.