 # 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  in Casablanca, Maroc, was the first to observe that Nebelung's results  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:  Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989  Aïssa Derhem, Sur les corps cubiques cycliques de conducteur divisible par deux premiers, Casablanca, 2002  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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