Polynomial with integer values

Question

I'm looking for a polynomial $P \in \mathbb Q[x]$ with $P(\mathbb Z) \subset \mathbb Z$, with $P \notin \mathbb Z[x]$

I found that $f_n := \frac{1}{n}x(x-1)(x-2)\dots(x-(n-1))$ is such an element.

My question is : are these the "only" solutions ?

More precisely, if $I$ denote the ideal generated by the $f_n$, my question is :

Is there a such polynomial $P$ with $P \notin I$ ??

edit : I forgot to precise the condition $P \notin \mathbb Z[x]$

Answer 1

Polynomials with integer values are a subring of $\mathbf Q[x]$, and the polynomials $f_n$ are a basis for this ring as a $\mathbf Z$-module. The ideal $I$ is the ring itself, since $f_0=1$.

As a ring, though contained in the PID $\mathbf Q[x]$, it is non-noetherian. Actually, it is a Prüfer domain (non-noetherian generalization of a Dedekind domain) of Krull dimension $2$. Its finitely generated ideals are invertible and generated by at most $2$ elements.