3. Most recent point of view:The Galois group of the 1st Hilbert 3class fieldof the relative 3genus field of a sextic S_{3} fieldIn 2003 again, Aïssa Derhem [2] in Casablanca, Maroc, was the first to observe that Nebelung's results [1] concerning 2stage metabelian 3groups can be applied equally well to the following questions that arise for the relative 3genus field K* = (Kk_{0})* of a sextic S_{3} field K, whose conductor f over its quadratic subfield k_{0} has exactly 2 prime divisors, whence K* is a bicyclic bicubic relative extension of k_{0}:1. to determine the relative Galois group G(K*_{1}k_{0}) of the 1^{st} Hilbert 3class field K*_{1} of K* over k_{0}, 2. to find the structure of the 3class group Syl_{3}C(K*) of K*.
This application is due to genus field theory, since K* is the maximal abelian 3extension of k_{0} that is unramified over K and thus the subgroup U = G(K*_{1}K*) of G = G(K*_{1}k_{0}) with factor group G/U = G(K*k_{0}) = (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 2stage metabelian 3group, since G' = G(K*_{1}K*) = Syl_{3}C(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 3class field of the relative 3genus field of a sextic S_{3} field  

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