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

Abstract

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).