Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	马宁，State University of New York at Buffalo
Inviter:	王标
Title:	Nonsplitting of the Hilbert Exact Sequence and the Principal Chebotarev Density Theorem
Language:	Chinese
Time & Venue:	2023.03.03 09:00-10:00 Zoom ID: 839 2782 7695 Password: 1728
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 $$ 1\to Cl_K\to \Gal(H_K/k){\to}\Gal(K/k)\to 1 $$ 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.