Decomposition group

Summary

Let $L/K$ be a Galois extension of number fields, and let $\mathfrak{P}$ be a prime of $L$. The decomposition group of $\mathfrak{P}$ is the subgroup of the Galois group that maps $\mathfrak{P}$ to itself. It surjects onto the Galois group of the residue field extension, and the kernel of this map is the inertia group. The decomposition group describes how $\mathfrak{P}$ splits, and it is the home of the Frobenius element. It is the Galois group of the local extension obtained by completing at $\mathfrak{P}$.