Centennial 2004

Descending Central Series of

2-Stage Metabelian 3-Groups

2. A modern point of view:

The Galois group of the 1st Hilbert 3-class field

of the 3-genus field of a cyclic cubic field

In 2002, Aïssa Derhem [2] in Casablanca, Maroc, was the first to observe that Nebelung's results [1] concerning 2-stage metabelian 3-groups can be applied equally well to the following questions that arise for the absolute 3-genus field K* = (K|Q)* of a cyclic cubic number field K, whose conductor f has exactly 2 prime divisors, whence K* is a bicyclic bicubic field:

1. to determine the absolute Galois group G(K*1|Q) of the 1st Hilbert 3-class field K*1 of K* over Q,
2. to find the structure of the 3-class group Syl3C(K*) of K*.

G(K*1|K*1) = 1
G' = G(K*1|K*) = Syl3C(K*)
G/G' = G(K*|Q) = (3,3)
G = G(K*1|Q)

This application is due to genus field theory, since K* is the maximal abelian 3-extension of Q that is unramified over K and thus the subgroup U = G(K*1|K*) of G = G(K*1|Q) with factor group G/U = G(K*|Q) = (3,3) must be the minimal subgroup of G with abelian factor group, i. e., must coincide with the commutator subgroup G' of G. Further, G is a 2-stage metabelian 3-group, since G' = G(K*1|K*) = Syl3C(K*) is abelian, i.e., G'' = 1.

Top recent applications of the present theory have been developed in the following article:
The Galois group of the 1st Hilbert 3-class field of the 3-genus field of a cyclic cubic field

[1] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem
Inauguraldissertation, Köln, 1989

[2] Aïssa Derhem,
Sur les corps cubiques cycliques
de conducteur divisible par deux premiers
Casablanca, 2002

[3] Daniel C. Mayer,
Class Numbers and Principal Factorizations of Families
of Cyclic Cubic Fields with Discriminant d < 1010
Univ. Graz, Computer Centre, 2002

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