Eisenstein-prime Obstruction Sieve for Monogenicity

TL;DR

This work proves a density-zero obstruction phenomenon for monogenicity within Eisenstein-parameter families of pure fields: for $K_m=\mathbb{Q}(\alpha)$ with $\alpha^n=m$ and square-free $m$, the set of indices with $g(m)>1$ that still satisfy the ABS fixed-sign local solvability condition is vanishingly small. Central to the argument is the Eisenstein-prime obstruction sieve, a three-step process combining local index-form coset rigidity, a one-prime $N$th-power obstruction at Eisenstein primes, and a Chebotarev-based sieve closure that yields a positive-density obstructing set of primes. The authors obtain density-equality results between monogenic and $\alpha$-monogenic fields in the pure family, and provide an explicit monogenic density formula $\delta=\frac{6}{\pi^2}\prod_{p\mid n}\frac{p}{p+1}$, highlighting a density-level local-to-global principle in these Eisenstein families. They extend the framework to scaled Eisenstein families, demonstrate necessary hypotheses via thin-parameter counterexamples, and exhibit positive-density ABS-unobstructed subfamilies with fixed nontrivial index, illustrating the nuanced interaction between local conditions and global monogenicity across parameter spaces.

Abstract

Alpöge--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 $K_m=\mathbb Q(α)$ with $α^n=m$ ($n\ge 4$) and $m$ square-free. Writing $g(m)=[\mathcal O_{K_m}:\mathbb Z[α]]$, 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 $α$--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.

Theorems & Definitions (92)

• Theorem 1.1: Theorem \ref{['thm:density-zero-ABS']}
• Theorem 1.2: Theorem \ref{['thm:density-equality']}
• Theorem 1.3: Theorem \ref{['thm:scaled-density-zero-merged']}
• Definition 2.1
• Theorem 2.2: ABHS representability
• Theorem 2.3: ABHS: the index form cuts out monogenerators
• Lemma 2.4: Translation invariance
• proof
• Lemma 2.5: Unit wedge implies generation
• proof