  Let $K/\Q$ be a cyclic extension of number fields with Galois group $G$. We
study the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree
bigger than one in the class group of $K$. In particular, we explore such
extensions $K/\Q$ for which there exist an integer $f>1$ such that the ideal
classes of primes $\mathfrak{p}$ of $K$ of residue degree $f$ generate the full
class group of $K$. It is shown that there are many such fields. These results
are used to obtain information on class group of $K$; like rank of
$\ell-$torsion of the class group, factors of class number, fields with class
group of certain exponents, and even structure of class group in some cases.
Moreover, such $f$ can be used to construct annihilators of the class groups.
