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

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.

26 pages

References in corpus (3)

• Strong ordered Abelian groups and dp-rank
• Dp-finite fields II: the canonical topology and its relation to henselianity
• Dp-finite fields V: topological fields of finite weight