On the class numbers of certain Hilbert class fields (Q1313517)

The main theorem, which is a refinement of the author's earlier work [Osaka J. Math. 28, 55-62 (1991; Zbl 0722.11055)] states: Let \(p\) be an odd prime and assume the Galois extensions of number fields \(L/K/k\) satisfy: (1) the degree \([K:k]\) is prime to \(p\); (2) \(L/K\) is an unramified \(p\)-extension. Let \(\varepsilon: 1\to \mathbb{Z}/ p\mathbb{Z}\to E\to \text{Gal} (L/k)\to 1\) be a non-split central embedding extension. Then there exists a Galois extension \(M/k\) such that (i) \(M/k\) gives a proper solution to the central embedding problem \((L/k, \varepsilon)\) and (ii) \(M/k\) is unramified. As an application of the main theorem, the author obtains the following result: Let \(l\) and \(p\) be distinct odd primes and assume that \(p\) has even order modulo \(l\). Let \(K/k\) be an abelian \(l\)-extension. If the class number of \(K\) is divisible by \(p\), then the class number of the maximal unramified abelian \(p\)-extension over \(K\) is also divisible by \(p\).