Rational separability over a global field

Annals of Pure and Applied Logic 79 (1):93-108 (1996)
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Let F be a finitely generated field and let j : F → N be a weak presentation of F, i.e. an isomorphism from F onto a field whose universe is a subset of N and such that all the field operations are extendible to total recursive functions. Then if R1 and R2 are recursive subrings of F, for all weak presentations j of F, j is Turing reducible to j if and only if there exists a finite collection of non-constant rational functions {Gi} over F such that for every x ε R1 for some i, Gi ε R2. We investigate under what circumstances such a collection of rational functions exists and conclude that in the case when R1 R2 are both holomorphy rings and F is of characteristic 0 or is an algebraic function field over a perfect field of constants, the existence of the above-described collection of rational functions is equivalent to the requirement that the non-archimedean primes which do not appear as poles of elements of R2 do not have factors of relative degree 1 in some simple extension of K



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Generalized weak presentations.Alexandra Shlapentokh - 2002 - Journal of Symbolic Logic 67 (2):787-819.
Weak presentations of non-finitely generated fields.Alexandra Shlapentokh - 1998 - Annals of Pure and Applied Logic 94 (1-3):223-252.

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Weak Presentations of Computable Fields.Carl G. Jockusch & Alexandra Shlapentokh - 1995 - Journal of Symbolic Logic 60 (1):199 - 208.
Diophantine equivalence and countable rings.Alexandra Shlapentokh - 1994 - Journal of Symbolic Logic 59 (3):1068-1095.

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