Ax-Kochen-Ershov principles for finitely ramified henselian fields

TL;DR

This paper extends the Ax-Kochen-Ershov principles to finitely ramified henselian valued fields, showing how their theories are determined by the residue field and value group, with implications for embeddings and existential theories.

Contribution

It establishes new model-theoretic principles for finitely ramified henselian fields, linking their structure to that of residue fields and value groups, including embedding and existential properties.

Findings

The theory of the valued field is determined by the enriched residue field and value group.

The existential theory of the valued field is determined by the positive existential theory of the residue field.

Embeddings are existentially closed if induced embeddings of residue field and value group are existentially closed.

Abstract

We study the model theory of finitely ramified henselian valued fields of fixed initial ramification, obtaining versions of the Ax-Kochen-Ershov principle as follows. We identify the induced structure on the residue field and show that once the residue field is endowed with this structure, the theory of the valued field is determined by the theories of the enriched residue field and the value group. Similarly, we show that the existential theory of the valued field is determined by the positive existential theory of the enriched residue field. We also prove that an embedding of finitely ramified henselian valued fields is existentially closed as soon as the induced embeddings of value group and residue field are existentially closed. This last result requires no enrichment of the residue field, in analogy to the corresponding result for model completeness, which holds by results of Ershov and Ziegler.