Nomura, Akito On the class numbers of certain Hilbert class fields. (English) Zbl 0806.11053 Manuscr. Math. 79, No. 3-4, 379-390 (1993). 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 \(p\). Reviewer: C.Parry (Blacksburg) Cited in 3 ReviewsCited in 4 Documents MSC: 11R37 Class field theory 11R32 Galois theory 12F10 Separable extensions, Galois theory 11R20 Other abelian and metabelian extensions 11R29 Class numbers, class groups, discriminants Keywords:Hilbert class fields; Galois extension; central embedding problem; class number; abelian \(p\)-extension Citations:Zbl 0722.11055 PDF BibTeX XML Cite \textit{A. Nomura}, Manuscr. Math. 79, No. 3--4, 379--390 (1993; Zbl 0806.11053) Full Text: DOI EuDML OpenURL References: [1] Akagawa, Y., A tripling on the algebraic number field, Osaka. J. Math., 23, 151-179, (1986) · Zbl 0591.12013 [2] E. Artin and J. Tate, “Class Field Theory”, Math. Lecture Note Ser. W.A. Benjamin, 1967 · Zbl 0681.12003 [3] A. Babakhanian, “Cohomological Methods in Group Theory”, Dekker, 1972 · Zbl 0256.20068 [4] Crespo, T., Embedding problem with ramification conditions, Arch. Math., 53, 270-276, (1989) · Zbl 0648.12007 [5] K. Hoechsmann,Zum Einbettungsproblem, J. rine angew. Math.229 (1968), 81-106 · Zbl 0185.11202 [6] M. Ikeda,Zur Existenz eigentlicher galoisscher Körper beim Einbettungsproblem, Hamb. Abh.24 (1960), 126-131 · Zbl 0095.02901 [7] J. Neukirch,Über das Einbettungsproblem der algebraischen Zahlentheorie, Invent. Math.21 (1973), 59-116 · Zbl 0267.12005 [8] Neumann, O., Proper solutions of the embedding problem with restricted ramification, Acta. Arith., 33, 49-52, (1977) · Zbl 0364.12009 [9] Nomura, A., On the existence of unramified p-extensions, Osaka J. Math., 28, 55-62, (1991) · Zbl 0722.11055 [10] A. Nomura,On embedding problems with ramification conditions and their applications, Dissertation, Kanazawa University, 1991 [11] Shafarevich, I. R., Extensions with given points of ramification, AMS Translation, Ser.2, 59, 128-149, (1966) · Zbl 0199.09707 [12] M. Suzuki, “Group Theory I”, Grundlehren der math. vol.247, Springer-Verlag, 1980 [13] Uchida, K., Class numbers of cubic cyclic fields, J. Math. Soc. Japan, 26, 447-453, (1974) · Zbl 0281.12007 [14] Washington, L. C., Class Numbers of the Simplest Cubic Fields, Math. Comp., 48, 371-384, (1987) · Zbl 0613.12002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.