Let K/k be a finite Galois extension of number fields, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> be the Hilbert class field of K. We find a way to verify the nonsplitting of the short exact sequence <Equation ID="Equ18"> <EquationSource Format="TEX">$\begin{aligned} 1\rightarrow Cl_K\rightarrow \textrm{Gal}(H_K/k){\rightarrow }\textrm{Gal}(K/k)\rightarrow 1 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mn>1</mn> <mo stretchy="false">→</mo> <mi>C</mi> <msub> <mi>l</mi> <mi>K</mi> </msub> <mo stretchy="false">→</mo> <mtext>Gal</mtext> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mi>K</mi> </msub> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mtext>Gal</mtext> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>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.

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

期刊论文
出版模式：

开放获取
发表日期：

2025年12月12日

查看全文

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

摘要

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 $\begin{aligned} 1\rightarrow Cl_K\rightarrow \textrm{Gal}(H_K/k){\rightarrow }\textrm{Gal}(K/k)\rightarrow 1 \end{aligned}$ $\begin{matrix} 1 \to C l_{K} \to Gal (H_{K} / k) \to Gal (K / k) \to 1 \end{matrix}$ 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.