---
title: "A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so"
authors:
  - "Levent Alpoge"
  - "Manjul Bhargava"
  - "Ari Shnidman"
year: 2024
venue: "Mathematische Annalen"
arxiv: "2107.05514"
doi: null
source: "https://arxiv.org/abs/2107.05514"
topics:
  - "arithmetic-geometry"
  - "number-theory"
seed_rank: 1207
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 9
lineage: arithmetic-geometry
cites:
  - title: "A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so"
    url: "https://arxiv.org/abs/2011.01186"
    year: 2020
    arxiv: "2011.01186"
    doi: "10.1007/s00208-024-03054-w"
see:
  []
---

# A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

## One-sentence takeaway

Positive proportion of quartic fields (and of quartic rings of integers) fail to be monogenic / fail to be invariant orders of binary quartics, with no local obstruction — the quartic upgrade of the cubic paper.

## Why it's lovely

Monogenic means O_K = Z[alpha]. Local-global failure for a classical invariant.

## Problem

After the cubic case, do quartics also fail monogeneity globally while locally fine?

## Design

Build on the cubic comparison of descents; Bhargava parametrizations of quartic rings / binary quartics.

## Evidence

arXiv:2107.05514, 12 Jul 2021. Math. Ann. 2024 (MaRDI / homepage journal). DOI not confirmed at ingest — do not invent; cite arXiv plus journal name.

## Limitations

Proportions, not a classification. See 1208 for cubics.

## Implications for Broadside

Journal version wins; preprint cited.

## Bottom line

Mint one card, journal year 2024.

## Links

- arXiv: [2107.05514](https://arxiv.org/abs/2107.05514)