Statistics of action of Galois group of number field on primes over unramified rational primes

Question

Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}_K$ over $p$. Do we know any statistical information about the distribution of isomorphism classes of these actions as $p$ ranges over all unramified primes? By this I mean that we define two actions of $G$ on sets $X$, $Y$ to be isomorphic if there exists a $G$-equivariant bijection from $X$ to $Y$, and ask how often a given isomorphism class occurs (for your favorite definition of “often”). Have questions in this vein been studied before? If we can allow the base field to be a number field different from $\mathbb{Q}$ that’s even better.

Answer 1

This is an elaboration of Chris Wuthrich's comment. Let $p$ be unramified (i.e. $p$ does not divide the discriminant of the Galois extension $K / \mathbb{Q}$), and let $\mathfrak{P}$ be a prime in $\mathcal{O}_K$ over $p$ with decomposition group $D_\mathfrak{P}\leq G$. The primes in $\mathcal{O}_K$ over $p$ correspond bijectively to the left cosets in $G/D_\mathfrak{P}$, and the action of $G$ on them corresponds to the left action of $G$ on $G/D_\mathfrak{P}$. Now $D_\mathfrak{P}$ is the cyclic group generated by $\mathrm{Frob}_\mathfrak{P}$, and changing $\mathfrak{P}$ results in conjugating $\mathrm{Frob}_\mathfrak{P}$. Hence the question boils down to how often the conjugacy class of $\mathrm{Frob}_\mathfrak{P}$ equals a given conjugacy class $C\subset G$. By the Chebotarev density theorem, the density of such primes $p$ equals $|C|/|G|$.