Linear theories of global fields with absolute values

TL;DR

This paper investigates the logical theories of global fields equipped with absolute value predicates, establishing decidability results for certain types and undecidability for others, and axiomatizing their existential theories.

Contribution

It provides a comprehensive analysis of the decidability of theories of global fields with various absolute value predicates, including simultaneous axiomatization.

Findings

Decidable theory for ultrametric and real archimedean absolute values

Undecidable theory for complex absolute value

Axiomatization of global fields with all non-complex absolute values

Abstract

We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a decidable theory, while with a complex absolute value the theory is always undecidable. We also study the existential theories and axiomatize k together with predicates for all non-complex absolute values on k simultaneously.