Class numbers of large degree 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 Guardia, 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.