Dp-finite fields VI: the dp-finite Shelah conjecture

TL;DR

This paper proves the dp-finite case of the Shelah conjecture, showing that dp-finite fields are either finite, real closed, algebraically closed, or admit a definable henselian valuation, advancing the classification of such fields.

Contribution

It establishes the dp-finite Shelah conjecture, demonstrating that dp-finite fields have a definable henselian valuation unless they are finite, real closed, or algebraically closed.

Findings

Dp-finite fields admit a non-trivial definable henselian valuation unless finite, real closed, or algebraically closed.

Dp-finite valued fields are henselian.

Unstable dp-finite expansions of fields have a unique definable V-topology.

Abstract

We prove the dp-finite case of the Shelah conjecture on NIP fields. If K is a dp-finite field, then K admits a non-trivial definable henselian valuation ring, unless K is finite, real closed, or algebraically closed. As a consequence, the conjectural classification of dp-finite fields holds. Additionally, dp-finite valued fields are henselian. Lastly, if K is an unstable dp-finite expansion of a field, then K admits a unique definable V-topology.