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

Abstract Let K / k be a finite Galois extension of number fields, and let $$H_K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> be the Hilbert class field of K . We find a way to verify the nonsplitting of the short exact sequence $$\begin{aligned} 1\rightarrow Cl_K\rightarrow \textrm{Gal}(H_K/k){\rightarrow }\textrm{Gal}(K/k)\rightarrow 1 \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mtext>Gal</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mtext>Gal</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>/</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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.