Computable Categoricity for Algebraic Fields with Splitting Algorithms

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 $F$.