On a ratio between relative class numbers
Let \(K/k\) be any extension of CM-fields of finite degrees. Let \(h_ K^ -\) and \(h_ k^ -\) denote the relative class numbers of \(K\) and \(k\), respectively. It is known that the denominator of \(h_ K^ -/h_ k^ -\) is a 2-power. The present paper first proves that the denominator of \(h_ K^ -/h_ k^ -\) divides 4 if \(K\) is abelian over the rational field, discussing next this result more precisely. Some sufficient conditions for \(h_ K^ -/h_ k^ -\) to be an integer and a (sufficient) condition for \(h_ K^ -/h_ k^ -\) to have denominator 4 are also given, together with simple examples of \(K/k\) such that the denominator of \(h_ K^ -/h_ k^ -\) equals 2. The last part of the paper generalizes the above-mentioned fact that the denominator of \(h_ K^ -/h_ k^ -\) is a 2-power, as follows: For any extension \(F/E\) of finite algebraic number fields, let \(h_{F/E}\) denote the order of the kernel of the norm map from the ideal class group of \(F\) into that of \(E\). Let \(K_ 2/K_ 1\) and \(k_ 2/k_ 1\) be Galois extensions of finite algebraic number fields such that \(K_ 1\supseteq k_ 1\), \(K_ 2=K_ 1k_ 2\), and the Galois closure of \(K_ 1/k_ 1\) is disjoint from \(k_ 2\) over \(k_ 1\). Then every prime divisor of the denominator of \(h_{K_ 2/K_ 1}/h_{k_ 2/k_ 1}\) is a divisor of \([K_ 2:K_ 1]\).
scientific article; zbMATH DE number 126410