# 2-Stage Metabelian 3-Groups

## 2. A modern point of view:

### 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*.

 K*1 | K* | Q
 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
 References: [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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