Principalization algorithm via class group structure

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.

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