The Artin Symbol as a Canonical Capitulation Map

TL;DR

The paper introduces a canonical, order-preserving map linking class group subgroups to subfields of the Hilbert class field, providing a new perspective on capitulation and a strengthened version of the Hilbert 94 Theorem.

Contribution

It presents a novel, canonical capitulation map that enhances understanding of subgroup structures and capitulation phenomena in Galois number fields.

Findings

Established a canonical map between subgroup lattices

Proved the map is a capitulation map for primes in certain classes

Provided a stronger version of the generalized Hilbert 94 Theorem

Abstract

We show that there is a canonical, order preserving map $\psi$ of lattices of subgroups, which maps the lattice $\Sub(A)$ of subgroups of the ideal class group of a galois number field $\K$ into the lattice $\Sub(\KH/\K)$ of subfields of the Hilbert class field. Furthermore, this map is a capitulation map in the sense that all the primes in the classes of $A' \subset A$ capitulate in $\psi(A')$. In particular we have a new, strong version of the generalized Hilbert 94 Theorem, which confirms the result of Myiake and adds more structure to (part) of the capitulation kernel of subfields of $\KH$.