1. Classical scenario:The Galois group of the 2nd Hilbert 3class fieldof a quadratic fieldThe original purpose of Brigitte Nebelung's thesis [4] , written in 1989 under the supervision of Wolfram Jehne at Cologne, was to gain a thorough overview of all 2stage metabelian 3groups G with commutator factor group G/G' of type (3,3) and to apply the results to the following problems which were first posed by Arnold Scholz and Olga TausskyTodd in 1933 (inspired by the Furtwängler / Artin proof of the principal ideal theorem) for imaginary quadratic fields K [1] but arise for any algebraic number field K with 3class group Syl_{3}C(K) of type (3,3):1. to determine the Galois group G(K_{2}K) of the 2^{nd} Hilbert 3class field K_{2} of K over K, 2. to find the structure of the 3class group Syl_{3}C(K_{1}) of the 1^{st} Hilbert 3class field K_{1} of K.
This application is due to class field theory, since K_{1} is the maximal abelian unramified 3extension of K and thus the subgroup U = G(K_{2}K_{1}) of G = G(K_{2}K) with factor group G/U = G(K_{1}K) = Syl_{3}C(K) = (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_{2}K_{1}) = Syl_{3}C(K_{1}) is abelian, i.e., G'' = 1. Warning: Although taking G = G(K_{n}K), for some integer n >= 3, similarly yields G' = G(K_{n}K_{1}) and G/G' = (3,3), the Galois group G = G(K_{n}K) of the n^{th} Hilbert 3class field K_{n} of K over K is not a 2stage metabelian 3group any longer, in general, since G'' = G(K_{n}K_{2}) != 1, if 3 divides the class number of K_{2}. Top recent applications of the present theory have been developed in the following two articles: 

A. The Galois group of the 2nd Hilbert 3class field over a real quadratic field  
B. The Galois group of the 2nd Hilbert 3class field over a complex quadratic field  

< Navigation Center < 
< Back to Algebra < 