Work

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

Year: 2025

Type: article

Abstract: Let K / k be a finite Galois extension of number fields, and let $$H_K$$ H K be the Hilbert class field of K . We find a way to verify the nonsplitting of the short exact sequence $$\begin{aligned} 1\rightarrow Cl_K\rightarrow \textrm{Gal}(H_K/k){\rightarrow }\textrm{Gal}(K/k)\rightarrow 1 \end{aligned}$$ 1 → C l K → Gal ( H K / k ) → Gal ( K / k ) → 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.

Source: The Ramanujan Journal

DOI: 10.1007/s11139-025-01251-y

Authors: Lian Duan, Ning Ma, Kelly O’Connor, Xiyuan Wang

Institutions: ShanghaiTech University, University at Buffalo, State University of New York, University of Wisconsin–La Crosse, The Ohio State University

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990

OpenAlex ID: W4417283965