Nakano, Shin On ideal class groups of algebraic number fields. (English) Zbl 0559.12004 J. Reine Angew. Math. 358, 61-75 (1985). The purpose of this paper is to prove the following theorem (and some related results): given 3 natural numbers \(r_ 1\), \(r_ 2\), and \(n\), consider number fields of degree \(r_ 1+2 r_ 2\) and with \(r_ 1\) real embeddings. In this family there exists an infinite subfamily of fields whose ideal class group contains a subgroup isomorphic to the product of \(r_ 2+1\) copies of \(\mathbb Z/n\mathbb Z\). For example: there exist infinitely many totally real number fields of given degree with their class numbers divisible by \(n\). The proof consists in considering fields defined by a root of an equation of type \(\prod^{r_ 1+2r_ 2}_{0}(X-A_ i)+C^ n=0\): the condition on the class number and the number of real embeddings is translated in nine different conditions. A step by step method proves that one can verify the conditions for infinitely many fields; one of the main tool is the density theorem about prime numbers. Reviewer: R.Gillard Cited in 4 ReviewsCited in 15 Documents MSC: 11R29 Class numbers, class groups, discriminants Keywords:prescribed subgroup of ideal class group; infinite subfamily of fields; class numbers PDFBibTeX XMLCite \textit{S. Nakano}, J. Reine Angew. Math. 358, 61--75 (1985; Zbl 0559.12004) Full Text: DOI Crelle EuDML