Extension fields and integer-valued polynomials
Let \(A\) be a Dedekind domain with finite residue fields and quotient field \(K\), and let \(\text{Int}(A)=\{f\in K[X]\mid f(A)\subset A\}\) be the ring of integer valued polynomials for \(A\). If \(L\) is a finite separable extension of \(K\) and \(B\) is the integral closure of \(A\) in \(L\), one may form \(\text{Int}(B)\) and ask how it is related to \(\text{Int}(A)\). The paper under review shows that \(\text{Int}(A)\subseteq \text{Int}(B)\) if and only if all maximal ideals of \(A\) split completely in \(L\) and that in this case \(\text{Int}(B)\) is the integral closure of \(\text{Int}(A)\) in \(L(X)\). The proof proceeds by studying essential valuations on \(K(X)\) (i.e. valuations whose valuation ring is the localization of \(\text{Int}(A)\) at a prime ideal) and analyzing the extensions of these to \(L(X)\).
scientific article; zbMATH DE number 1268042