Principalization algorithm via class group structure

Abstract: For an algebraic number field K with 3-class group (Cl_3(K)) of type (3,3), the structure of the 3-class groups (Cl_3(N_i)) of the four unramified cyclic cubic extension fields (N_i), (1\le i\le 4), of K is calculated with the aid of presentations for the metabelian Galois group (G_3^2(K)=Gal(F_3^2(K) | K)) of the second Hilbert 3-class field (F_3^2(K)) of K. In the case of a quadratic base field (K=\mathbb{Q}(\sqrt{D})) it is shown that the structure of the 3-class groups of the four (S_3)-fields (N_1,\ldots,N_4) frequently determines the type of principalization of the 3-class group of K in (N_1,\ldots,N_4). 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 (-10^6<D<10^7) to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups (G_3^2(K)) on various coclass trees of the coclass graphs G(3,r), (1\le r\le 6), in the sense of Eick, Leedham-Green, and Newman.