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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 all $e_i=1$. For each $P_i$, we have an element of $G$, its Frobenius element. These are conjugate, so when mapped to $S_n$, they must have the same cycle type from their disjoint cycle decompositions. This is the Frobenius cycle type.

Alternatively, we note that 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.

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  • Review status: beta
  • Last edited by John Jones on 2012-03-21 12:11:44.421000