Nontriviality of rings of integral‐valued polynomials

Abstract Let be a subset of , the ring of all algebraic integers. A polynomial is said to be integral‐valued on if for all . The set of all integral‐valued polynomials on forms a subring of containing . We say that is trivial if , and nontrivial otherwise. We give a collection of necessary and sufficient conditions on in order for to be nontrivial. Our characterizations involve, variously, topological conditions on with respect to fixed extensions of the ‐adic valuations to ; pseudo‐monotone sequences contained in ; ramification indices and residue field degrees; and the polynomial closure of in .