p-Capitulation over Number Fields with p-Class Rank Two

Theoretical foundations of a new algorithm for determining the p-capitulation type K ( )  of a number field K with p-class rank = 2  are presented. Since K ( )  alone is insufficient for identifying the second p-class group p K K = 2 Gal(F | ) G 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 , K K K = τ AP( ) ( ( ) ( ))  of all 34631 real quadratic fields K d  = ( ) with discriminants d < < 8 0 10 and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups K K = 2 3 Gal(F | ) G and the 3-class field tower groups G K K ∞ = 3 Gal(F | ) .