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Let $K/k$ be a finite Galois extension of number fields, and let $H_K$ be theHilbert class field of $K$. We find a way to verify the nonsplitting of theshort 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 principalversion of the Chebotarev density theorem, which represents the density of theprime 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 termsof the invariants of $K/k$. In particular, we prove that the group structure ofthe ideal class group of $K$ can be determined by reading the principaldensities.

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