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

TL;DR

This paper demonstrates that a positive proportion of quartic fields are not monogenic even without local obstructions, extending previous results from cubic fields and analyzing quartic rings and forms.

Contribution

It establishes the existence of non-monogenic quartic fields without local obstructions, building on prior work for cubic fields and examining quartic rings and forms.

Findings

A positive proportion of quartic fields are not monogenic without local obstructions.

Many quartic rings do not come from integral binary quartic forms despite no local obstructions.

Extension of previous cubic field results to quartic fields.

Abstract

We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.