1111.1211Nov 4, 2011In 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 This claim was based on a theorem from another paper, and that theorem has now been retracted by its authors.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
Categoricity Properties for Computable Algebraic Fields
Denis Hirschfeldt, Ken Kramer, Russell Miller, Alexandra Shlapentokh
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 Π40-complete.