  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 $x \in {\rm Gal}_{K/K_0}$ a density $\delta_{K/K_0,x}$ on primes of
$K$. In particular, the density associated to $x = 1$ is the usual Dirichlet
density on $K$. After establishing some properties of these densities, we use
them to show that the maximal solvable extension of a number field unramified
outside an almost Chebotarev set realize the maximal local extension at each
prime lying outside this set.
