The following ideas are based on this article (PDF). It turns out that every integer-valued polynomial p(x) has the form

(*)

Indeed, the coefficients (omitting the constant term, which is zero) of polynomials , , ..., form a triangular matrix, so these polynomials form a basis, i.e., every polynomial of degree ≤ n with no constant term can be expressed as a linear combination of these polynomials, perhaps with complex coefficients. So, every polynomials from ℂ[x] of degree ≤ n can be written in the form (*) with complex . Next prove by induction on k that .