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

TL;DR

This paper introduces a new algorithm to determine the p-capitulation type and related invariants of number fields with p-class rank two, supported by extensive computational results for real quadratic fields.

Contribution

It develops a comprehensive algorithm for p-capitulation and second p-class group determination, including implementation and extensive computational data for real quadratic fields.

Findings

Computed Artin patterns for 34631 real quadratic fields.

Provided statistical insights into second 3-class groups.

Analyzed the structure of 3-class field towers.

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).