Categoricity Properties for Computable Algebraic Fields
  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. 
Denis Hirschfeldt, Ken Kramer, Russell Miller and Alexandra   Shlapentokh
math.LO math.AC math.NT