On divisibility by 2 of the relative class numbers of imaginary number fields (Q2548625)

\(K\) and \(F\) are algebraic number fields of finite degree. It was proved by H. Yokoi that if \(K/F\) is a cyclic extension such that \(K\) and the absolute class field \(\tilde F\) of \(F\) are disjoint over \(F\) and \(K\) has only one ramified prime divisor over \(F\), then the class number \(h_F\) of \(F\) is equal to the ambiguous class number \(a_{K/F}\) of \(K/F\). First, we prove an analogous result in relation to his result (\S 2 Theorem 1). Next, suppose \(K\) is imaginary and \(K = K^J\), where \(J\) is a substitution from a complex number \(\alpha\) to the complex conjugate number \(\overline\alpha\) and let \(K_0\) be the maximal real subfield of \(K\). Then we give necessary conditions to make the relative class number of \(K/K_0\) odd (\S 2 Lemma 2). From this Lemma 2, the well known property of cyclotomic field K=P_{p^n}\( that the relative class number of \)K/K_0\( is odd if and only if the class number of \)K\( is odd, follows easily, where \)P_{p^n}\( is the cyclotomic field generated by a primitive \)p^n\(-th root of unity over the rational number field \)P\( for a prime \)p\( and a natural number \)n\(. \par Finally, suppose \)K\( is totally imaginary, \)K = K^J\(, and the maximal real subfield \)K_0\( of \)K\( is totally real. Then we give necessary conditions to make the relative class number of \)K/K_0\( odd (\S 2 Theorem 2). This Theorem 2 is a generalization of {\mathit H. Hasse}'s Satz 42 in [\"Uber die Klassenzahl abelscher Zahlk\"orper Berlin: Akademie-Verlag (1952; Zbl 0046.26003)]. In \S 3 applying Theorem 2 to an absolutely cyclic imaginary number field, we give necessary and sufficient conditions to make the relative class number odd.\)