Linear theories of global fields with absolute values
Logic
2026-03-27 v1 Number Theory
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.
Keywords
Cite
@article{arxiv.2603.24720,
title = {Linear theories of global fields with absolute values},
author = {Arno Fehm and Pierre Touchard},
journal= {arXiv preprint arXiv:2603.24720},
year = {2026}
}
Comments
16 pages