Is there a small finite (perhaps of cardinality two or three) collection of cubic polynomials $p\_1, \dotsc, p\_k \in \mathbb{Z}[x]$ such that for every prime $p$ at least one of these is irreducible?



Irreducible cubics modulo primes
No. It follows from [Chebotarev's density theorem](https://en.wikipedia.org/wiki/Chebotarev's_density_theorem) that for any polynomials $p\_1,\dots,p\_k\in\mathbb{Z}[x]$ there are infinitely many primes that split all these polynomials. Simply, apply this theorem for the number field $K$ generated by the roots of $p\_1,\dots, p\_k$, and note that if an unramified prime completely splits in $K$ then each $p\_i$ splits into linear factors modulo that prime.


