Primes of higher degree

Let $K/\mathbb {Q}$ be a Galois extension of number fields. We study the ideal classes of primes $\mathfrak {p}$ of $K$ of residue degree bigger than 1 in the class group of $K$. In particular, we explore those extensions $K/\mathbb {Q}$ for which there exists an integer $f \gt 1$ such that the ideal classes of primes $\mathfrak {p}$ of $K$ of residue degree $f$ generate the full class group of $K$. We show that there are many such fields. Then we use this approach to obtain information on the class group of $K$, like the rank of the $\ell $-torsion subgroup of the class group, factors of the class number, fields with class group of certain exponents, and even structure of the class group in some cases. Moreover, such $f$ can be used to construct annihilators of the class groups. In fact, for any extension $K/F$ (even non-abelian), if the class group of $K$ is generated by primes of relative degree $f$ for the extension $K/F$ and the class group of $F$ is trivial, this method can be used to construct ‘relative’ annihilators.