3. Most recent point of view:The Galois group of the 1st Hilbert 3-class fieldof the relative 3-genus field of a sextic S3 fieldIn 2003 again, 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 relative 3-genus field K* = (K|k0)* of a sextic S3 field K, whose conductor f over its quadratic subfield k0 has exactly 2 prime divisors, whence K* is a bicyclic bicubic relative extension of k0:1. to determine the relative Galois group G(K*1|k0) of the 1st Hilbert 3-class field K*1 of K* over k0, 2. to find the structure of the 3-class group Syl3C(K*) of K*.
This application is due to genus field theory, since K* is the maximal abelian 3-extension of k0 that is unramified over K and thus the subgroup U = G(K*1|K*) of G = G(K*1|k0) with factor group G/U = G(K*|k0) = (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: |
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| The Galois group of the 1st Hilbert 3-class field of the relative 3-genus field of a sextic S3 field | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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