An explicit family of cubic number fields with large $2$-rank of the class group

TL;DR

This paper constructs infinite families of cubic number fields with large 2-rank class groups using algebraic geometry, specifically degree 1 del Pezzo surfaces, providing explicit examples and methods.

Contribution

It introduces a novel geometric approach to explicitly construct cubic number fields with large 2-rank class groups, expanding the understanding of class group structures.

Findings

Infinite families of cubic fields with class group 2-rank at least 8

Explicit construction method using degree 1 del Pezzo surfaces

Example illustrating the construction process

Abstract

We show how to construct infinite families of explicitly determined cubic number fields whose class group has a subgroup isomorphic to $(\mathbb{Z}/2)^8$ using degree $1$ del Pezzo surfaces. We illustrate the method and provide an example of such a family.