##
**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)].

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)].

Reviewer: Franz Lemmermeyer (Jagstzell)

### 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 |

### Keywords:

class group; Hilbert class field; class field tower; complex quadratic field; Galois groups### Software:

Magma
Full Text:
DOI

### References:

[1] | Benjamin, E.; Lemmermeyer, F.; Snyder, C., Imaginary quadratic fields with \(\operatorname{Cl}_2(k) = (2, 2, 2)\), J. number theory, 103, 38-70, (2003) · Zbl 1045.11077 |

[2] | Bosma, W.; Cannon, J.J., Handbook of magma functions, (1996), School of Mathematics and Statistics, University of Sydney |

[3] | Boston, N.; Leedham-Green, C.R., Explicit computation of Galois p-groups unramified at p, J. algebra, 256, 402-413, (2002) · Zbl 1016.11051 |

[4] | Boston, N.; Nover, H., Computing pro-p Galois groups, (), 1-10 · Zbl 1143.11369 |

[5] | Bush, M.R., Computation of Galois groups associated to the 2-class towers of some quadratic fields, J. number theory, 100, 313-325, (2003) · Zbl 1039.11091 |

[6] | Eick, B.; Koch, H., On maximal 2-extensions of \(\mathbb{Q}\) with given ramification, (), (English version) · Zbl 1188.11059 |

[7] | Hajir, F., On a theorem of Koch, Pacific J. math., 176, 15-18, (1996) · Zbl 0879.11066 |

[8] | Hajir, F.; Maire, C., Tamely ramified towers and discriminant bounds for number fields. II, J. symbolic comput., 33, 415-423, (2002) · Zbl 1086.11051 |

[9] | O’Brien, E.A., The p-group generation algorithm, J. symbolic comput., 9, 677-698, (1990) · Zbl 0736.20001 |

[10] | Shafarevich, I.R., Extensions with prescribed ramification points, IHES publ. math., 18, 71-95, (1964), (in Russian) · Zbl 0199.09707 |

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.