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

Let K / k be a finite Galois extension of number fields, and let HK H K be the Hilbert class field of K . We find a way to verify the nonsplitting of the short exact sequence aligned 1→ ClK→ Gal(HK/k)→ Gal(K/k)→ 1 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.