α-monogeneity of pure number fields: Criterion and density

Let [Formula: see text], let [Formula: see text], and let [Formula: see text], where [Formula: see text] and [Formula: see text] is irreducible over [Formula: see text]. We study when the natural order [Formula: see text] is the full ring of integers [Formula: see text]. For the pure family [Formula: see text], we give a short proof, using only Dedekind’s index criterion, of the equivalence [Formula: see text] Equivalently, the prime support of [Formula: see text] is [Formula: see text] We then compute the natural density of the corresponding parameters in the one-parameter family [Formula: see text]: [Formula: see text] We also give an arithmetic-progression refinement, a density-theoretic independence statement for the local obstruction sets at primes dividing [Formula: see text], and discriminant-ordered counts of the corresponding fields.