Rational separability over a global field
type: Item
instance of: scholarly article
title: Rational separability over a global field
zbMATH Open document ID: 0857.12004
DOI: 10.1016/0168-0072(95)00023-2
author: Alexandra Shlapentokh
published in: Annals of Pure and Applied Logic
publication date: 1997-03-11T00:00:00Z
review text: 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 \(K\).
Mathematics Subject Classification ID: 12L05
Mathematics Subject Classification ID: 03D45
Mathematics Subject Classification ID: 03D20
zbMATH DE Number: 908469
zbMATH Keywords: rational separability
zbMATH Keywords: finitely generated field
zbMATH Keywords: total recursive functions
zbMATH Keywords: weak presentations
zbMATH Keywords: Turing reducible
zbMATH Keywords: rational functions
zbMATH Keywords: algebraic function field
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
full work available at URL: 0168-0072(95)00023-2
OpenAlex ID: W2080103284
cites work: Q5810622
cites work: Q3027108
cites work: Q4063245
cites work: Weak presentations of computable fields
cites work: Q5675787
cites work: Computable Algebra, General Theory and Theory of Computable Fields
cites work: Diophantine equivalence and countable rings
cites work: Non-standard extensions of weak presentations
cites work: Algebraic and Turing separability of rings
description: scientific article; zbMATH DE number 908469