Computable valued fields

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 valuatio...

Full description

Saved in:
Bibliographic Details
Published in:Archive for mathematical logic 2018-08, Vol.57 (5-6), p.473-495
Main Author: Harrison-Trainor, Matthew
Format: Article
Language:English
Subjects:
Algebra
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 ω .
ISSN:0933-5846
1432-0665
DOI:10.1007/s00153-017-0589-9