Categoricity Properties for Computable Algebraic Fields

Abstract

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 ${\Pi }_4^0$-complete.

In an earlier version of this article, the authors claimed to have shown that computable algebraic fields cannot have finite computable dimension greater than 1. This claim was based on a theorem from another paper, and that theorem has now been retracted by its authors. Therefore, we have removed that claim from the current version of this article