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Suppose $K$ a number field. Does every element of the class group $C_K$ have a prime ideal in its class? More generally, is this true for ray class groups?

CCC
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  • In brief, "yes": this is the immediate generalization of Dirichlet's theorem to (Hecke...) characters on finite generalized class groups and their $L$-functions. For this case of finite families, simply the non-vanishing of all the relevant $L$-functions at $1$ is sufficient, and more serious arguments (Tauberian theorems) are not needed. So I should ask in what context you need clarification. – paul garrett Aug 05 '16 at 21:31
  • The reason I care is: in class field theory, for an unramified extension L/K the Artin map C_K->Gal(L/K) has the property that the ramification index of a prime p is the order of its image in Gal(L/K), so split primes get killed. But I want to argue further that the kernel is generated by the split primes, which would follow from the fact I asked about. Maybe there is a cheaper way? – CCC Aug 05 '16 at 21:52
  • Related: https://math.stackexchange.com/questions/27308. In fact, the set of prime ideals of $O_K$ in a given ideal class of the class group $C_K$ is a set of positive natural density $1/|C_K|$. (See the comments there and Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, corollary 6 from §7.2, page 346 (notations on page 93).). – Watson Dec 02 '18 at 10:22
  • Typically, a prime $p$ of $O_K$ is principal iff it splits in the Hilbert class field $H_K$, and the density of such primes is $1/[H_K : K] = 1/|C_K|$ ; thus the class of $O_K$ contains infinitely many primes (those primes are the principal primes). – Watson Dec 02 '18 at 10:22

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