Rational separability over a global field Original Research Article

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(R1) is Turing reducible to j(R2) if and only if there exists a finite collection of non-constant rational functions {Gi} over F such that for every xϵR1 for some i, Gi(x) ϵR2. 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.