Computable Categoricity for Algebraic Fields with Splitting Algorithms

TL;DR

This paper characterizes when algebraic fields with splitting algorithms are computably categorical, linking this property to the decidability of element orbit relations and relative computable categoricity.

Contribution

It establishes a precise criterion for computable categoricity in algebraic fields with splitting algorithms, connecting it to orbit decidability and relative computable categoricity.

Findings

Computably categorical iff orbit relation is decidable

Orbit relation decidability is equivalent to relative computable categoricity

Provides a characterization for algebraic fields with splitting algorithms

Abstract

A computably presented algebraic field $F$ has a \emph{splitting algorithm} if it is decidable which polynomials in $F[X]$ are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of elements of $F$ belong to the same orbit under automorphisms. We also show that this criterion is equivalent to the relative computable categoricity of $F$.