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.Comment: 33 pages, 2 figures, presented at the Joint CSASC Conference, Danube University, Krems, Austria, September 201

p-Capitulation over number fields with p-class rank two

Theoretical foundations of a new algorithm for determining the p-capitulation type kappa(K) of a number field K with p-class rank rho=2 are presented. Since kappa(K) alone is insufficient for identifying the second p-class group G=Gal(F(p,2,K) | K) of K, complementary techniques are developed for finding the nilpotency class and coclass of G. An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(tau(K),kappa(K)) of all 34631 real quadratic fields K=Q(squareroot(d)) with discriminants 0<d<100000000 and 3-class group of type (3,3). The results admit extensive statistics of the second 3-class groups G=Gal(F(3,2,K) | K) and the 3-class field tower groups H=Gal(F(3,K) | K).Comment: 13 pages, 4 tables, contributed presentation at the 2nd International Conference on Groups and Algebras (ICGA) in Suzhou, China, July 25-27, 201