  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.
