Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power

We characterize the fixed divisor of a polynomial $f(X)$ in $\Z[X]$ by looking at the contraction of the powers of the maximal ideals of the overring $\IZ$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\Z[X]$ whose fixed divisor is divisible by $p^n$ in terms of its primary components.