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Eisenstein-prime Obstruction Sieve for Monogenicity

This paper proves that for families of pure fields $\mathbb{Q}(\sqrt[n]{m})$ , the phenomenon where fields satisfy all local monogenicity conditions but lack a global power integral basis is rare, occurring with natural density zero.

Original authors: Khai-Hoan Nguyen-Dang

Published 2026-02-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Khai-Hoan Nguyen-Dang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master locksmith trying to figure out if a specific, high-tech digital lock can be opened with a single, perfect "master key."

In the world of mathematics, this is the problem of Monogenicity. A "lock" is a number field (a complex mathematical structure), and a "master key" is a single number that can generate the entire structure through simple arithmetic.

Here is the breakdown of what this paper is doing, using the analogy of The Master Key and the Local Flaws.

1. The Mystery: Local vs. Global

Imagine you are inspecting a series of high-tech locks. You look at each lock under a microscope. You check the tiny gears, the electronic sensors, and the tumblers. You find that at every single tiny level—the "local" level—the lock looks like it should be able to accept a master key. There are no tiny scratches or broken pins that would prevent it.

However, when you actually try to use the master key, it fails! The lock won't turn.

This is a famous problem in math (called the ABS phenomenon). Even if a lock has no "local" flaws, it might still have a "global" flaw—a structural issue that only appears when you try to use the whole key at once.

2. The Question: Does this happen in "Families"?

The researcher, Khai-Hoan Nguyen-Dang, wanted to know if this "hidden flaw" happens in specific, predictable families of locks.

He focused on a specific type of lock called "Pure Fields" (the Eisenstein families). These are locks that are built using a very specific, repetitive blueprint. Because they follow a strict pattern, you would expect them to behave predictably. You might think, "If these locks are built so consistently, then if they look good locally, they must work globally!"

3. The Discovery: The "Eisenstein-Prime" Sieve

The paper proves that for these specific families, the "local-to-global" rule actually works.

He discovered that if a lock in this family looks perfect under the microscope (no local obstructions), it is almost certainly going to work with a master key (it is monogenic). He proved that the "hidden flaws" that plague other types of locks are incredibly rare in this specific family—so rare that they effectively don't exist in the grand scheme of things (they have "density zero").

To prove this, he invented a new tool called the Eisenstein-Prime Obstruction Sieve.

The Analogy of the Sieve:
Imagine you have a giant bucket of sand, and you want to find all the grains that are "perfect." Instead of checking every grain, you use a special sieve.

He identified certain "special primes" (the Eisenstein primes) that act like tiny, invisible sensors.
If a lock is built using one of these special primes, that prime acts as a "witness."
If the lock has a flaw, the prime will "scream" (mathematically, it creates a local obstruction).
He showed that because these "witness" primes are so common and well-distributed, almost every lock that could have a hidden flaw will eventually be caught by one of these screaming primes.

4. Why does this matter?

In mathematics, we love to find patterns. This paper tells us that geometry matters.

If you look at a chaotic, messy crowd of different locks, you can't predict much. But if you look at a disciplined, marching army of identical locks (the Eisenstein family), the rules become much stricter and more reliable.

The "TL;DR" Summary:
In most mathematical structures, "looking good locally" doesn't guarantee "working well globally." But in this specific, highly-structured family of numbers, the author proved that if it looks good locally, it almost certainly works globally. He did this by creating a mathematical "sieve" that catches the rare exceptions.

Technical Summary: Eisenstein-Prime Obstruction Sieve for Monogenicity

Author: Khai-Hoan Nguyen-Dang
Date: February 12, 2026
Subject Area: Algebraic Number Theory (Monogenicity, Sieve Methods, Kummer Theory)

1. The Problem: Local-to-Global Principles in Monogenicity

The central problem addressed is the relationship between monogenicity (the existence of a power integral basis $\mathcal{O}_K = \mathbb{Z}[\theta]$ ) and local solvability (the existence of local power bases at every prime $\ell$ ).

Recent work by Alpöge–Bhargava–Shnidman (ABS) demonstrated that in the full spaces of cubic and quartic fields, a "strengthened" local condition—where the sign of the index form is fixed globally (ABS fixed-sign local solvability)—does not guarantee global monogenicity. This implies a failure of the local-to-global principle in high-dimensional parameter spaces.

The author investigates whether this phenomenon persists in one-parameter families, specifically the pure Eisenstein families $K_m = \mathbb{Q}(\sqrt[n]{m})$ for $n \geq 4$ . In these families, the local structure is highly constrained by a distinguished generator $\alpha = \sqrt[n]{m}$ . The core question is: Do monogenicity and $\alpha$ -monogenicity (where $\mathcal{O}_K = \mathbb{Z}[\alpha]$ ) differ in density within these families?

2. Methodology: The Eisenstein-Prime Obstruction Sieve

The paper introduces a novel, modular mechanism termed the Eisenstein-prime obstruction sieve. This is not a case-by-case computation but a general three-step pipeline:

Local Rigidity: The author proves that for a residue-degree 1 Discrete Valuation Ring (DVR) $A$ over $R$ , the unit values of the local index form $f_\omega$ on all $R$ -generators form exactly one coset in $R^\times / (R^\times)^N$ . This "rigidifies" the local obstruction into a single Kummer class $\kappa(A/R, \omega)$ .
One-Prime Obstruction Certificates: In Eisenstein families, any prime $q$ dividing the parameter $m$ is an "Eisenstein prime" for the polynomial $X^n - m$ . The author shows that a single such prime can certify a local obstruction. Specifically, if $q \equiv 1 \pmod{2N}$ , the local solvability at $q$ is equivalent to a residue-field condition: the index $g(m)$ must be an $N$ -th power modulo $q$ .
Sieve Closure via Chebotarev: Using Kummer theory, the condition $g \notin (\mathbb{F}_q^\times)^N$ is translated into a splitting condition in a Kummer extension $L/K$ . By the Chebotarev Density Theorem, there exists a positive-density set of primes $P_g$ that satisfy this "obstructing" condition. A classical sieve argument is then used to show that almost all parameters $m$ (in a density sense) are divisible by at least one prime in $P_g$ , thereby forcing a local obstruction.

3. Key Contributions and Results

The paper provides several landmark results regarding the density of monogenic fields:

Density Identification (Main Result): For all $n \geq 4$ , the author proves that in the pure family, monogenicity and $\alpha$ -monogenicity have the same natural density. Specifically, the set of fields that are monogenic but not $\alpha$ -monogenic has density zero.
Explicit Monogenic Density: By combining the sieve results with known criteria for $\alpha$ -monogenicity, the author provides an explicit formula for the density of monogenic pure fields:
$\delta(\{m : \mathcal{O}_{K_m} \text{ is monogenic}\}) = \frac{6}{\pi^2} \prod_{p|n} \frac{p}{p+1}$
The "Negative" Answer to the ABS Question: While generic families show a positive proportion of ABS-unobstructed non-monogenic fields, the author proves that Eisenstein families behave differently. In these families, the ABS local condition does generically force monogenicity (at the density level).
Generalization to Scaled Families: The author proves that this sieve applies to a broader class of "scaled Eisenstein families" $f_t(X) = X^n + t h(X)$ , provided the index values are bounded and the associated Kummer extensions are non-trivial.
Counter-examples and Necessity: The paper includes constructions showing that the "Kummer non-triviality" hypothesis is essential. It demonstrates that in "thin" families (like primes), the local-to-global principle can still fail.

4. Significance

This paper is significant for several reasons:

Theoretical Bridge: It bridges the gap between explicit integral basis computations (classical) and the arithmetic statistics of monogenicity (modern).
Geometric Sensitivity: It reveals that the validity of the local-to-global principle for monogenicity is highly sensitive to the geometry of the parameter space. High-dimensional spaces allow for "global failures," whereas constrained one-parameter Eisenstein families exhibit a density-level local-to-global principle.
Reusable Framework: The "Eisenstein-prime obstruction sieve" is presented as a reusable template that can be transferred to other families of number fields, offering a powerful tool for studying the distribution of arithmetic properties in parameterized families.