---
tag: 7X92G8CQ33
kind: concept
name: Decomposition group
msc: [11R32, 11S20]
sources:
  - url: https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions
---

## 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|0XTY86BZKY]] 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|H5145REFMW]].
It is the Galois group of the local extension obtained by completing at $\mathfrak{P}$.

## Explanations

### L1 human

The Galois group of $L/K$ moves the prime ideals of $L$ that lie over a given prime of $K$ around among themselves.
The decomposition group of one of them is the set of symmetries that leave it in place.
If the prime splits into many pieces, few symmetries fix a given piece, and the decomposition group is small.
In $\Q(i)/\Q$, complex conjugation swaps the two primes $(2 + i)$ and $(2 - i)$ over $5$, so their decomposition groups are trivial.
It fixes the prime $(3)$, so the decomposition group of $(3)$ is the whole Galois group.