= Frobenius permutation and roots modulo a prime
{c}
{title2=$f\bmod p\text{ has a root}\iff\operatorname{Frob}_p\text{ fixes a root}$}
Let a monic integer <polynomial> have distinct roots in its <splitting field>. Outside the finitely many primes dividing its <polynomial discriminant> or ramifying in that field, its roots reduce to distinct residue roots. The <Frobenius automorphism> permutes them by $a\mapsto a^p$; a reduced root belongs to $\mathbb F_p$ exactly when it is fixed. Distinctness of reduction identifies this with a fixed root of the Galois permutation itself. Combining this observation with a <derangement in a transitive group action> and the <Chebotarev density theorem> proves that an irreducible polynomial of degree at least two has rootless reduction at a positive <Dirichlet density> of primes. The degree bound is essential: every monic linear integer polynomial has a root modulo every prime.
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