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Weak Presentations of Computable Fields

Carl G. Jockusch, Jr and Alexandra Shlapentokh
The Journal of Symbolic Logic
Vol. 60, No. 1 (Mar., 1995), pp. 199-208
DOI: 10.2307/2275517
Stable URL: http://www.jstor.org/stable/2275517
Page Count: 10
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Weak Presentations of Computable Fields
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Abstract

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 $f: \mathbb{Q} \rightarrow \omega$ 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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