Let \(f\in\ZZ[x]\) be monic of degree \(n\) with nonzero discriminant \(\Delta\), and let \(G\) be the Galois group of \(f\) over \(\QQ\), regarded as a subgroup of \(S_n\) through its action on the roots of \(f\). Let \(p\) be a prime with \(p\notdivides\Delta\), and let \(f\equiv f_1f_2\cdots f_m\pmod p\) be the factorization into monic irreducibles in \(\FF_p[x]\). Then \(G\) contains a permutation of cycle type \((\deg f_1, \deg f_2, \ldots, \deg f_m)\). Equivalently, the Galois group of \(f\bmod p\) over \(\FF_p\), which is cyclic and generated by the Frobenius automorphism, is isomorphic to a subgroup of \(G\) through a bijection between the roots of \(f\bmod p\) and the roots of \(f\).
The Chebotarev density theorem strengthens this: for each cycle type occurring in \(G\), the set of primes \(p\notdivides\Delta\) for which \(f\bmod p\) factors with that cycle type has natural density equal to the proportion of elements of \(G\) of that cycle type.