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Computation of Galois groups associated to the 2-class towers of some imaginary quadratic fields with 2-class group \(C_2 \times C_2\times C_2\). (English) Zbl 1177.11096

Let \(k\) be a complex quadratic number field. It is well known that \(k\) has an infinite \(2\)-class field tower if the \(2\)-rank of its large enough, and that it is finite if the \(2\)-class group has type \((2,2)\). For \(2\)-class groups of medium size, studying the finiteness of the \(2\)-class field tower is a challenging problem. E. Benjamin, C. Snyder and the reviewer [J. Number Theory 103, No. 1, 38–70 (2003; Zbl 1045.11077)] have studied families of complex quadratic number fields with small \(2\)-class field towers, and exhibited a family of fields with class group of type \((2,2,2)\) whose \(2\)-class field towers have length at least \(3\).
In [ANTS-VII, Lect. Notes Comput. Sci. 4076, 1–10 (2006; Zbl 1143.11369)], N. Boston and the author explained how to show that the smallest field in this family, \(\mathbb Q(\sqrt{-3135}\,)\), has \(2\)-class field tower of length exactly \(3\); in this article, the same conclusion is proved (modulo GRH) for 19 out of the 30 smallest fields in this family.
As a final result, the author observes that imaginary quadratic number fields with \(2\)-class groups of type \((2,2,2)\) and infinite \(2\)-class field towers have root discriminants \(\geq 78.3\) (even \(\geq 90.2\) under GRH).
In the current paper, GRH is used for computing class numbers of fields of degree \(8\) and \(16\). Since a lot of these fields are biquadratic extensions of other fields, the dependence on GRH could be removed by computing these class numbers (and, using more information such as the capitulation of ideal classes, even the class group) using Kuroda’s class number formula; see e.g the reviewer’s [Acta Arith. 66, No. 3, 245–260 (1994; Zbl 0807.11052)].

MSC:

11Y40 Algebraic number theory computations
11R11 Quadratic extensions
11R32 Galois theory
11R37 Class field theory
12F10 Separable extensions, Galois theory
20D15 Finite nilpotent groups, \(p\)-groups

Software:

Magma
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References:

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