Frobenius cycle types (reviewed)

If $K$ is a degree $n$ extension of $\mathbb{Q}$, $\hat K$ its normal closure and $G=\text{Gal}(\hat K/\mathbb{Q})$, then $G$ acts on the set of $n$ embeddings of $K\to \hat K$ giving an embedding $G\to S_n$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $p$ a prime number. Then \[ p\mathcal{O}_K = P_1^{e_1}\cdots P_g^{e_g}\] where the $P_i$ are distinct prime ideals of $\mathcal{O}_K$. The prime $p$ is unramified if $e_i=1$ for all $i$.

Suppose hereafter that $p$ is unramified. For each $P_i$, there is a unique element of $G$ that fixes $P_i$ and acts on the quotient $\mathcal{O}_K/P_i$ via the Frobenius automorphism $x \mapsto x^p$; this element is the Frobenius element associated to $P_i$. The Frobenius elements associated to the different $P_i$ are conjugate to each other, so their images in $S_n$ all have the same lengths of cycles in their disjoint cycle decompositions. This is the Frobenius cycle type of $p$.

Alternatively, for each prime $P_i$, its residue degree $f_i$ is defined by $|\mathcal{O}_K/P_i| = p^{f_i}$. The list of $f_i$ is the same partition of $n$ as the cycle decomposition described above.