Two classes of ideals determined by integer-valued polynomials

Abstract

If D is a domain with quotient field K, let Int(D) = {f(X) ∈ K[X] | f(d) ∈ D for every d ∈ D} be the ring of integer-valued polynomials over D. It is well known that the binomial polynomials GX n H = X(X−1)...(X−n+1) n! form a basis of Int(ZZ) as a free ZZ-module and that for every prime integer p, the Fermat polynomials fp(X) = 1 p (Xp − X) are in Int(ZZ). If the domain D contains ZZ, for each nonnegative integer n, set C(n) = {α ∈ K | α GX n H ∈ Int(D)}, and for every prime integer p, set E(p) = {α ∈ K | α · fp(X) ∈ Int(D)} . Each C(n) and E(p) is an ideal of D which we explicitly determine when D is a Dedekind domain.

Keywords

• integer-valued polynomial
• dedekind domain