On the ideal class groups of ray class fields of algebraic number fields (Q1431077)

Let \(C(k)\) denote the ideal class group of an algebraic number field \(k\), and let \(k_m\) denote the ray class field of \(k\) modulo \(m\), where \(m\in\mathbb Z\), \(m>1\). The author first proves the following theorem: Let \(L/k\) be an abelian extension with a subextension \(K/k\) of degree \(n\). Then \(C(L)\) has a subgroup isomorphic to \(C(K)^{nh(k)}\). Next, he constructs an infinite family of quadratic fields \(\mathbb Q(\sqrt m)\) such that \(C(\mathbb Q(\sqrt m))\) has an arbitrarily large cyclic subgroup. As an application, he obtains an infinite number of positive integers \(m\) with special properties such that \(C(k_m)\) has an arbitrarily large cyclic subgroup. Taking \(k=\mathbb Q\), he gets the result for the maximal real subfields of a family of cyclotomic fields. This generalizes earlier results of the author [Nagoya Math. J. 113, 147--151 (1989; Zbl 0696.12007)]. There are further applications of the theorem along the same lines.