TITLE: Irreducible cubics modulo primes
QUESTION [7 upvotes]: 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?

REPLY [11 votes]: No. It follows from 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.