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Principalization algorithm via class group structure
For an algebraic number field K with 3-class group of type (3,3),
the structure of the 3-class groups of the four unramified cyclic
cubic extension fields , , of K is calculated with the
aid of presentations for the metabelian Galois group of the second Hilbert 3-class field of K. In the case of a
quadratic base field it is shown that the structure
of the 3-class groups of the four -fields frequently
determines the type of principalization of the 3-class group of K in
. This provides an alternative to the classical
principalization algorithm by Scholz and Taussky. The new algorithm, which is
easily automatizable and executes very quickly, is implemented in PARI/GP and
is applied to all 4596 quadratic fields K with 3-class group of type (3,3) and
discriminant to obtain extensive statistics of their
principalization types and the distribution of their second 3-class groups
on various coclass trees of the coclass graphs G(3,r), , in the sense of Eick, Leedham-Green, and Newman.Comment: 33 pages, 2 figures, presented at the Joint CSASC Conference, Danube
University, Krems, Austria, September 201
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