Eisenstein-prime Obstruction Sieve for Monogenicity

TL;DR

This paper demonstrates that within certain pure number field families, the likelihood of non-monogenic fields satisfying local obstructions is negligible, using a novel Eisenstein-prime obstruction sieve method.

Contribution

It introduces the Eisenstein-prime obstruction sieve, a new technique to analyze monogenicity in parametric families of number fields, showing that local obstructions do not imply global non-monogenicity.

Findings

In pure fields $K_m$, the set of $m$ with $g(m)>1$ but no local obstruction has density zero.

Monogenicity and $ extit{α}$-monogenicity have the same density in the studied family.

The method applies to other Eisenstein parameter families, broadening its impact.

Abstract

Alp\"oge--Bhargava--Shnidman showed that even a strengthened \emph{no local obstruction} condition for monogenicity does not force a global power integral basis: in the full spaces of cubic and quartic fields, a positive proportion are non-monogenic yet satisfy this ABS fixed-sign condition. This raises a natural family-level question: does the same phenomenon persist inside one-parameter families, where the local structure varies in a highly constrained way? In this paper we answer this in the negative for the pure fields $K_m=\mathbb Q(\alpha)$ with $\alpha^n=m$ ($n\ge 4$) and $m$ square-free. Writing $g(m)=[\mathcal O_{K_m}:\mathbb Z[\alpha]]$, we prove that the set of square-free $m$ for which $g(m)>1$ but $K_m$ has no ABS local obstruction has natural density $0$. Consequently, in the pure family monogenicity and $\alpha$--monogenicity have the same natural density. The proof isolates a reusable mechanism, which we call the Eisenstein-prime obstruction sieve. The argument is packaged in an abstract template and transfers to other Eisenstein parameter families.