Categoricity Properties for Computable Algebraic Fields

TL;DR

This paper investigates the conditions under which computable algebraic fields are uniquely characterized up to isomorphism, providing a criterion for relative computable categoricity, constructing a specific example, and analyzing the complexity of categoricity.

Contribution

It introduces a structural criterion for relative computable categoricity, constructs a computably categorical but not relatively computably categorical field, and establishes the complexity class of computable categoricity.

Findings

A structural criterion for relative computable categoricity.

Existence of a computably categorical field not relatively computably categorical.

Computable categoricity is $ ext{Pi}^0_4$-complete.

Abstract

We examine categoricity issues for computable algebraic fields. We give a structural criterion for relative computable categoricity of these fields, and use it to construct a field that is computably categorical, but not relatively computably categorical. Finally, we show that computable categoricity for this class of fields is $\Pi^0_4$-complete.