$\alpha$-monogeneity of pure number fields: criterion and density

For pure extensions $K=\mathbb{Q}(\alpha )$ with ${\alpha }^n=m$, we give a short proof, based only on Dedekind’s index theorem, of the $\alpha$-monogeneity criterion: $\mathbb{Z}[\alpha ]={\mathcal{O}}_K$ if and only if $m$ is square-free and ${\nu }_p(m^p-m)=1$ for every prime $p\mid n$. We then derive an explicit natural density ${\delta }_n=\frac{6}{{\pi }^2}{\prod }_{p\mid n}\frac{p}{p+1}$, independence across primes, refinements in arithmetic progressions, and discriminant-order asymptotics.