Polynomial in $\mathbb{Z}[x]$ which is irreducible but not prime.

Question

I know that in an integral domain, prime implies irreducible. Moreover, in a principal integral domain, these notions are equivalent. The ring of polynomials $\mathbb{Z}[x]$ is not a principal integral domain (if it were, an irreducible ideal $I$ would be maximal, and the quotient $\mathbb{Z}[x]/I$ would be a field, but the quotient by the irreducible ideal $(x)$ is $\mathbb{Z}$ which is not a field). Is there a polynomial in $\mathbb{Z}[x]$ which is irreducible but not prime?

Answer 1

As said in comment, you cannot find such an example in $\mathbb{Z}[X]$. In other rings, this could be possible : for example, $2 \in \mathbb{Z}[2i]$ is irreducible but not prime.