Model completeness for finitely ramified Henselian fields valued in a $\mathbb{Z}$-group and for pseudo-algebraically closed fields

Derakhshan, Jamshid¹ ; Macintyre, Angus²

¹ St. Hilda’s College, University of Oxford, Cowley Place, Oxford OX4 1DY, UK
² School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road Edinburgh EH9 3FD, UK

Confluentes Mathematici, Tome 15 (2023), pp. 137-147

Résumé

We prove that the theory of a finitely ramified Henselian valued field of characteristic zero with perfect residue field of positive characteristic whose value group is a $\mathbb{Z}$-group is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. This extends results of Ax–Kochen [4], Macintyre [15], Ziegler [22], Basarab [5], and Prestel–Roquette [17].

We also prove that the theory of a perfect pseudo-algebraically closed (PAC) field $K$ such that the absolute Galois group $\mathrm{Gal}\left(K\right)$ is pro-cyclic is model-complete in the language of rings if and only if every finite algebraic extension of $K$ is generated by elements that are algebraic over the prime subfield of $K$.

From these we deduce that every infinite algebraic extension of the field of $p$-adic numbers ${\mathbb{Q}}_p$ with finite ramification is model-complete in the language of rings.

Our proofs of model completeness for Henselian fields use only basic model-theoretic and algebraic tools including Cohen’s structure theorems for complete local rings and basic results on coarsenings of valuations. These enable us to obtain short proofs of model completeness in the language of rings without any need to extend the ring language.

Our proofs on PAC fields use the model theory of the absoute Galois group dual to the field and elementary invariants given by Cherlin–van den Dries–Macintyre [8] for the theory of PAC fields generalizing Ax’s results for the pseudo-finite case [3].

Détail

Reçu le : 2022-05-05
Révisé le : 2023-06-21
Accepté le : 2023-11-09
Publié le : 2024-11-13

DOI : 10.5802/cml.96

Classification : 03C10, 12E30, 12J20, 12L12, 13J15
Keywords: model theory, model completeness, Henselian valued fields, pseudo-algebraically closed fields

Licence : CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits