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On dp-finite fields

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On dp-finite fields

Shelah’s conjecture predicts that any infinite NIP field iseither separably closed, real closed or admits a non-trivial henselianvaluation. Recently, Johnson proved that Shelah’s conjecture holds forfields of finite dp-rank, also known as dp-finite fields. The aim of these two talks is to give an introduction to dp-rank in some algebraic structures and an overview of Johnson’s work.
In the first talk, we define dp-rank (which is a notion of rank in NIP theories) and give examples of dp-finite structures. In particular, we discuss the dp-rank of ordered abelian groups and use them to construct multitude of examples of dp-finite fields. We also prove that every dp-finite field is perfect and sketch a proof that any valued field of dp-rank 1 is henselian.
In the second talk, we give an overview of Johnson’s proof that everyinfinite dp-finite field is either algebraically closed, real closed oradmits a non-trivial henselian valuation. Crucially, this relies on the notion of a W-topology, a natural generalization of topologies arising from valuations, and the construction of a definable W-topology on asufficiently saturated unstable dp-finite field.

- Séminaire Géométrie et théorie des modèles

Détails :

Orateur / Oratrice : Yatir Halevi et Franziska Jahnke
Date : 2 décembre 2021
Horaire : 9h00 - 12h00
Lieu : Zoom