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

Abstract

Let $n\ge 2$, let $m\in \mathbb{Z}\setminus \{0\}$, and let $K=\mathbb{Q}(\alpha )$, where ${\alpha }^n=m$ and $X^n-m$ is irreducible over $\mathbb{Q}$. We study when the natural order $\mathbb{Z}[\alpha ]$ is the full ring of integers ${\mathcal{O}}_K$. For the pure family $X^n-m$, we give a short proof, using only Dedekind's index criterion, of the equivalence ${\mathcal{O}}_K=\mathbb{Z}[\alpha ]$ iff $m$ is square-free and $\nu \_p(m^p-m)=1$ for every prime $p\mid n$. Equivalently, the prime support of $[{\mathcal{O}}_K:\mathbb{Z}[\alpha ]]$ is $$\{p:{\nu }_p(m)\ge 2\}\cup \{p\mid n:{\nu }_p(m^p-m)\ge 2\}.$$ We then compute the natural density of the corresponding parameters in the one-parameter family $X^n-m$: $${\delta }_n=\frac{6}{{\pi }^2}\prod _{p\mid n}\frac{p}{p+1}.$$ We also give an arithmetic-progression refinement, a density-theoretic independence statement for the local obstruction sets at primes dividing $n$, and discriminant-ordered counts of the corresponding fields.

13 pages, accepted version