On the class numbers of certain Hilbert class fields. (English) Zbl 0806.11053

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 \(p\).


11R37 Class field theory
11R32 Galois theory
12F10 Separable extensions, Galois theory
11R20 Other abelian and metabelian extensions
11R29 Class numbers, class groups, discriminants


Zbl 0722.11055
Full Text: DOI EuDML


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