Rational separability over a global field
Let \(F\) be a finitely generated field and let \(j:F\to \mathbb{N}\) be a weak presentation of \(F\) (i.e. an isomorphism from \(F\) onto a field whose universe is a subset on \(\mathbb{N}\) and such that all the field operations are extendable to total recursive functions). Then if \(R_1\) and \(R_2\) are recursive subrings of \(F\), for all weak presentations \(j\) of \(F\), \(j(R_1)\) is Turing reducible to \(j(R_2)\) iff there exists a finite collection of nonconstant rational functions \(\{G_i\}\) over \(F\) such that for every \(x\in R_1\), for some \(i\), \(G_i(x) \in R_2\). The scope of this paper is to investigate under what circumstances such a collection of rational functions exists. The main result is that in the case when \(R_1 \nsubseteq R_2\) 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 \(R_2\) do not have factors of relative degree 1 in some simple extension of \(K\).
scientific article; zbMATH DE number 908469