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