Statistics of genus numbers of cubic fields

TL;DR

This paper analyzes the distribution of genus numbers in cubic fields, showing that most have genus number one, and explores properties like norm-Euclideanity among these fields.

Contribution

It provides the first precise proportions of cubic fields with specific genus numbers and computes the average genus number, advancing understanding of their distribution.

Findings

Approximately 96.23% of cubic fields have genus number one.

The exact proportion of cubic fields with each genus number is determined.

A positive proportion of totally real cubic fields with genus number one are not norm-Euclidean.

Abstract

We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean.