Adapting Rabin's Theorem for Differential Fields

Harrington extended the first half of Rabin’s Theorem to differential fields, proving that every computable differential field can be viewed as a computably enumerable subfield of a computable presentation of its differential closure. For fields F , the second half of Rabin’s Theorem says that this subfield is Turing-equivalent to the set of irreducible polynomials in F [X]. We investigate possible extensions of this second half, asking both about the degree of the differential field K within its differential closure and about the degree of the set of constraints for K, which forms the closest analogue to the set of irreducible polynomials.