摘要
Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).
Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).
