Densities of primes and realization of local extensions

Abstract:

In this paper we introduce new densities on the set of primes of a number field. If $K/K_0$ is a Galois extension of number fields, we associate to any element $x \in \mathrm {G}_{K/K_0}$ a density $\delta _{K/K_0,x}$ on the primes of $K$. In particular, the density associated to $x = 1$ is the usual Dirichlet density on $K$. We also give two applications of these densities (for $x \neq 1$): the first is a realization result à la the Grunwald-Wang theorem such that essentially, ramification is only allowed in a set of arbitrarily small (positive) Dirichlet density. The second concerns the so-called saturated sets of primes, introduced by Wingberg.