Nontriviality of rings of integral-valued polynomials

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