TITLE:
p-Capitulation over Number Fields with p-Class Rank Two
AUTHORS:
Daniel C. Mayer
KEYWORDS:
Hilbert p-Class Field Tower, Maximal Unramified Pro-p Extension, p-Capitulation of Class Groups, Real Quadratic Fields (3, 3)
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.4 No.7,
July
25,
2016
ABSTRACT: 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(Fp2K∣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 0d8 and 3-class group of
type (3, 3). The results admit extensive statistics of the second 3-class
groups G=Gal(F32K∣K) and the 3-class field
tower groups G=Gal(F3∞K∣K).