  We study the distributions of the splitting primes in certain families of
number fields. The first and main example is the family Pn,N of integer
polynomials monic of degree n with height less or equal then N, and then let N
go to infinity. We prove an average version of the Chebotarev Density Theorem
for this family. In particular, this gives Central Limit Theorem for the number
of primes with given splitting type in some ranges. As an application, we
deduce some estimates for the torsion in the class groups and for the average
of ramified primes.
