Construction and classification of p-ring class fields modulo p-admissible conductors

TL;DR

This paper classifies p-ring class fields over quadratic fields using Galois cohomology and differential principal factorization, providing a comprehensive view of their structure and construction via Magma computations.

Contribution

It introduces a novel classification method for p-ring class fields by analyzing entire multiplets of associated dihedral fields, enhancing understanding of their Galois structure.

Findings

Deeper insight into class field structures through multiplet classification

Explicit construction of dihedral field multiplets using Magma routines

New connections between Galois cohomology and differential principal factorizations

Abstract

Each p-ring class field K(f) modulo a p-admissible conductor f over a quadratic base field K with p-ring class rank r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet M(K(f))=[(N(c,i))_{1<=i<=m(c)}]_{c|f} of dihedral fields N(c,i) with various conductors c|f having p-multiplicities m(c) over K such that sum_{c|f} m(c)=(p^r(f)-1)/(p-1). The advanced viewpoint of classifying the entire collection M(K(f)), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields, and the actual construction of the multiplet M(K(f)) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.