Nonsplitting of the Hilbert exact sequence and the principal Chebotarev density theorem

TL;DR

This paper introduces a finite calculation method to verify the nonsplitting of a key exact sequence in algebraic number theory using the principal Chebotarev density theorem, linking ideal class groups and prime ideal densities.

Contribution

It provides explicit equations and a novel approach to determine the structure of ideal class groups via principal densities, advancing understanding of Galois extensions.

Findings

Verification of nonsplitting via finite calculations

Explicit formulas for principal densities in terms of invariants

Determination of ideal class group structure from densities

Abstract

Let $K/k$ be a finite Galois extension of number fields, and let $H_K$ be the Hilbert class field of $K$. We find a way to verify the nonsplitting of the short exact sequence $$1\to Cl_K\to \text{Gal}(H_K/k)\to\text{Gal}(K/k)\to 1$$ by finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of $k$ that factor into the product of principal prime ideals in $K$. We also find explicit equations to express the principal density in terms of the invariants of $K/k$. In particular, we prove that the group structure of the ideal class group of $K$ can be determined by reading the principal densities.