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On the second 2-class group \(\mathrm{Gal}(K_2^{(2)} / K)\) of some imaginary quartic cyclic number field \(K\). (English) Zbl 1428.11190

Summary: Let \(H_8\) denote the quaternion group of order 8 and put \(G = \mathbb{Z} / 2 \mathbb{Z} \times H_8\). Let \(K\) be some imaginary quartic cyclic number field whose 2-class group is of type \((2, 2, 2)\). In this paper, we prove in particular that \(G\) is realizable over \(K\), i.e., \(G \simeq \mathrm{Gal}(K_2^{(2)} / K)\) where \(K_2^{(2)}\) is the second Hilbert 2-class field of \(K\). Then we study the capitulation problem of the 2-ideal classes of \(K\) in the fourteen intermediate unramified extensions between \(K\) and its first Hilbert 2-class field. Additionally, these fourteen unramified extensions are constructed, and the abelian type invariants of their 2-class groups and the length of the 2-class field tower of \(K\) are given.

MSC:

11R16 Cubic and quartic extensions
11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
11R32 Galois theory
11R37 Class field theory
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