Nontriviality of rings of integral-valued polynomials

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