On computable field embeddings and difference closed fields

TL;DR

This paper explores the conditions under which computable automorphisms of fields can be extended to their algebraic closures and applies these findings to effective embeddings of computable difference fields into difference closed fields.

Contribution

It introduces new methods for extending automorphisms in computable fields and advances the understanding of embeddings into difference closed fields.

Findings

Characterization of extendability of automorphisms

Effective methods for embedding difference fields

Insights into computability in algebraic closures

Abstract

We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of computable difference fields into computable difference closed fields.