The Frobenius and Cebatarov density theorems -- the factorization of an integer polynomial modulo various primes is controlled by the Galois group of the corresponding extensions. Even if you are really hard core about excluding analytic results, to the point that you won't accept statements about Dirichlet density, surely it is worth understanding why $x^3+2 x^2-x-1$ always factors into (linear)(linear)(linear) or stays irreducible, but is never (linear)(quadratic).