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Does irreducibility of a polynomial over the integers imply that there is a positive integer $n$ such that the polynomial is irreducible mod $n$?

I know that $n$ cannot be prime as there polynomials irreducible over the integers which are reducible mod $p$ for every prime $p$. (This is related to my previous question.)

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In this American Mathematical Monthly article Driver, Leonard, and Williams give examples of irreducible quartics that are reducible modulo every $n$. One such example is $x^4-72x^2+4$. What is new in the paper is that relatively elementary tools are used. The fact that there are such polynomials was known earlier, I don't know how much earlier.

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