# Monic polynomial irreducible modulo finitely many given primes

There are irreducible monic polynomials over $$\mathbb{Z}$$ that are reducible modulo every prime number $$p$$ (e.g. $$x^4+1$$). Given a finite non-empty set $$S$$ of primes is there a monic polynomial over $$\mathbb{Z}$$ that is irreducible modulo the primes in $$S$$ and is reducible modulo all other primes?

Perhaps surprisingly, the answer is no for every nonempty $$S$$.
If $$f(x)$$ is a degree-$$d$$ polynomial with integer coefficients, then its factorization modulo a prime $$p$$ is related to a conjugacy class of the Galois group of $$f$$ over $$\Bbb Q$$ (the Frobenius class). In particular, $$f$$ being irreducible when reduced modulo $$p$$ is equivalent to the Frobenius class corresponding to a cycle that permutes the $$d$$ roots of $$f$$.