Categoricity Properties for Computable Algebraic Fields

by Denis R. Hirschfeldt, Ken Kramer, Russell Miller, and Alexandra Shlapentokh

Status: published in the Transactions of the American Mathematical Society 367 (2015) 3981 - 4017.

Availability: journal version and preprint

Abstract. We examine categoricity issues for computable algebraic fields. Such fields behave nicely for computable dimension: we show that they cannot have finite computable dimension greater than 1. However, they behave less nicely with regard to relative computable categoricity: 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 Π04-complete.