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 Int(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 Z[X] whose fixed divisor is divisible by pn in terms of its primary components.