Unramified generator criterion with separable reduction
= Unramified generator criterion with separable reduction
A finite extension of complete discretely valued fields is <unramified> exactly when it has an integral generator whose reduced <minimal polynomial of an algebraic element> is separable. In one direction, lift a residue-field primitive element. In the other, the factorization form of <Hensel's lemma> makes the separable reduction irreducible, forcing residue degree to equal the entire extension degree.