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Categoricity Properties for Computable Algebraic Fields

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

Availability: PDF

**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
Π^{0}_{4}-complete.

drh@math.uchicago.edu