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arXiv · 2109.01217

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

Lian Duan·Kelly Emmrich·Ning Ma·Xiyuan Wang
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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 $$1\to Cl_K\to \text{Gal}(H_K/k)\to\text{Gal}(K/k)\to 1$$ 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.

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Lian Duan, Kelly Emmrich, Ning Ma, Xiyuan Wang. 2021-09-02. Nonsplitting of the Hilbert exact sequence and the principal Chebotarev density theorem. https://arxiv.org/abs/2109.01217

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