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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 theoverring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and apositive integer $n$, we also obtain a complete description of the ideal ofpolynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ interms of its primary components.

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