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

Computable Categoricity for Algebraic Fields with Splitting Algorithms