Published online by Cambridge University Press: 12 March 2014
It is shown that for any computable field K and any r.e. degree a there is an r.e. set A of degree a and a field F ≃ K with underlying set A such that the field operations of F (including subtraction and division) are extendible to (total) recursive functions. Further, it is shown that if a and b are r.e. degrees with b ≤ a, there is a 1-1 recursive function f: ℚ → ω such that f(ℚ) ∈ a, f(ℤ) ∈ b, and the images of the field operations of ℚ under f can be extended to recursive functions.
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