Infinite Moments of Class Groups for Solvable Fields with a Normal Abelian Subgroup

We apply the class field theory and Minkowski bound to obtain an upper bound estimate for the number of solutions to the restricted ramifications when the Galois group is solvable. Together with suitable conditions on the solvable group and the ordering of number fields, we could prove an upper bound on specific field-counting problems, hence the infinite moment of the class groups. In particular, for non-Galois cubic fields ordered by the product of ramified primes, we could show that the $\mathbb{Z}\mathrm{/}3\mathbb{Z}$-moment is infinite with the results on the $\mathbb{Z}\mathrm{/}3\mathbb{Z}$-moment of quadratic number fields and the field-counting on cubic fields ordered by the generalized discriminant.