Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture

TL;DR

This paper generalizes the Artin transfer pattern to heterogeneous multiplets of ramified extensions and uses this to computationally verify Scholz's conjecture on subfield units in dihedral fields.

Contribution

It introduces a generalized Artin pattern for ramified multiplets and applies it to dihedral fields to verify Scholz's conjecture computationally.

Findings

Generalized Artin transfer pattern for ramified extensions

Verification of Scholz's conjecture for specific dihedral fields

Computational evidence supporting the conjecture

Abstract

The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms$T_{K,N_i}:\,\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(N_i)$ is generalized for heterogeneous multiplets of ramified extensions. By application to quadratic subfields K of dihedral fields N of degree 2p with an odd prime p, a conjecture of Scholz concerning the index of subfield units, $(U_N:U_0)$, for ramified extensions N/K with conductor f>1 is verified computationally.