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1.1. Background and motivation. Let K be a number field and denote by H its group of ideal classes. Since H is finite an interesting question one may ask is how its size, the class number of K, denoted by h, varies as K varies in some natural family of number fields. This question is in general very difficult to answer. As an example, an unproven conjecture of Gauss is that there are infinitely many real quadratic number fields with h = 1. Here we will be interested in the opposite extreme. That is, we want to study how large h can possibly be as K varies. Littlewood [13] addresses this question in one of the simplest cases, for K an imaginary quadratic field. His work makes use of the class number formula for imaginary quadratic fields: h = |d|1/2π−1L(1, χ),

Non-abelian number fields with very large class numbers