Computable valued fields

TL;DR

This paper explores the computability properties of valued fields, establishing effectiveness conditions for extending valuations, and analyzing the computable dimension of algebraically closed and p-adically closed valued fields.

Contribution

It introduces new effectiveness conditions related to Hensel's lemma and divisibility in value groups, and demonstrates the computable dimension of certain valued fields.

Findings

Existence of a computable formally p-adic field not embeddable into any computable p-adic closure.

Effectiveness condition on divisibility relation ensures embedding into computable p-adic closures.

Algebraically closed and p-adically closed valued fields of infinite transcendence degree have computable dimension ω.

Abstract

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and $p$-adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally $p$-adic field which does not embed into any computable $p$-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and $p$-adically closed valued fields of infinite transcendence degree have the Mal'cev property, we show that they have computable dimension $\omega$.