The number of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor

TL;DR

This paper counts maximal quartic $D_4$-fields with monogenic cubic resolvent, ordered by conductor and discriminant, providing asymptotic formulas and extending understanding of their distribution.

Contribution

It provides the first asymptotic counts of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor and discriminant.

Findings

Asymptotic number of such fields ordered by conductor

Asymptotic number of such fields ordered by discriminant

Extension of previous enumeration results

Abstract

In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic $D_4$-field. By a theorem of M. Wood, such quartic orders may be regarded as quartic $D_4$-fields whose ring of integers has a monogenic cubic resolvent. We shall give the asymptotic number of such objects when ordered by conductor, as well as estimate the asymptotic number of such objects when ordered by discriminant.