Let $p$ be a fixed prime number. Roughly speaking, I am interested in the following ratio


$$
\frac{ |\{\text{ all CM number fields of degree }2g}|}
{ |\{\text{CM fields of degree 2g, such that p splits completely in K}\}|}
$$


A possible definition could be the following:
let $d\_{K}$ be the discriminant of K,
then we can define this ratio as 


$$ \lim\_{d \to \infty} \frac
{|\{ \text{all CM fields of degree 2g and}\ d\_{K} \le d\}|}
{|\{ \text{CM fields of degree 2g such that p splits completely and}\
 d\_{K} \le d \}|}.$$


Was it studied by anyone? 
I would appreciate any reference.



Density of fields with a decomposition condition
Let $K$ be a CM field with maximal real subfield $k$. Most of the time the normal closure
of $k$ will have the symmetric group $S\_g$ as Galois group; thus the number of fields
$k$ in which $p$ splits completely has density $1/g!$ among all of them by density theorems due to Kronecker, Frobenius and/or Chebotarev. In about half of the cases, $p$ will also split in $K$, giving $1/(2 \cdot g!)$ as a rough estimate. 


It is not necessarily true, however, that the density of fields with the symmetric Galois group is $1$ when the fields are ordered by discriminant. It is my impression that the 
corresponding problems are still open for $g > 5$, although there might be conjectural densities in the articles by Malle etc.


Thus you should be able to get a definitve answer for $g \le 4$ using known results
(check the articles by Cohen et al. on the distribution of Galois groups), and
perhaps $g = 5$ using recent advances e.g. by Bhargava. 


