Class numbers of large degree nonabelian number fields

TL;DR

This paper proves unconditionally that a specific large degree nonabelian number field of degree 120 has class number one, using computational and algebraic methods without relying on GRH.

Contribution

It demonstrates the first unconditional proof of class number one for a degree 120 nonabelian number field, employing novel computational techniques.

Findings

The degree 120 nonabelian field has class number one.

The proof combines computational algorithms with algebraic arguments.

Techniques can be applied to other nonabelian number fields.

Abstract

If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number field of degree 120 has class number one. This field is the unique $A_5 \times C_2$ extension of the rationals that is ramified only at 653 with ramification index 2. It is the largest degree number field unconditionally proven to have class number 1. The proof uses the algorithm of Gu\`ardia, Montes and Nart to calculate an integral basis and then finds integral elements of small prime power norm to establish an upper bound for the class number; further algebraic arguments prove the class number is 1. It is possible to apply these techniques to other nonabelian number fields as well.