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

Daniel C Mayer

doi:10.4236/jamp.2016.47135

Abstract

Theoretical foundations of a new algorithm for determining the p-capitulation type u(K) of a number field K with p-class rank ?=2 are presented. Since u(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),u(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0<d<108 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).

Highlights

For each 1 ≤ i ≤ n, let jLi|K :Cl p K → Cl p Li denote the class extension homomorphism induced by the ideal extension monomorphism ([2] 1, p. 74)
In the case u ≥ v > 1, the p + 1 subgroups of index p of Clp K and the p + 1 subgroups of order p of Clp K can be renumerated completely independently of each other, which can be expressed by two independent permutations σ,τ ∈ S p+1
This is the implementation of Theorem 2.1 in MAGMA [8]

Summary

Introduction

If p ≥ 3 is an odd prime, and K = ( d ) is a quadratic field with fundamental discriminant d := dK and p-class rank ≥ 1, there arise the following possibilities for the p-capitulation kernel in any of the unramified cyclic relative extensions Li | K of degree p, which are absolutely dihedral extensions Li | of degree 2p, according to

Theoretical Prerequisites

Computational Techniques

Findings

Interpretation of Numerical Results