---
tag: 72W4HXM98H
kind: definition
name: Splitting of prime ideals
msc: [11R04, 11R32]
defines: B7KNZCBFYT
uses: [MWBDFS2KJ5, 7NPY2CN5KB, 00MHHZ9SNE, H5RC67SD24, 0EAJ2XXADC, 0QMZAQ6QQ4]
sources:
  - url: https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions
---

## Definition

Let $L/K$ be a finite extension of [[algebraic number fields|MWBDFS2KJ5]] of [[degree|0EAJ2XXADC]] $n$, with rings of integers $\mathcal{O}_K \subseteq \mathcal{O}_L$.
Let $\mathfrak{p}$ be a nonzero [[prime ideal|00MHHZ9SNE]] of $\mathcal{O}_K$.
Since $\mathcal{O}_L$ is a [[Dedekind domain|7NPY2CN5KB]], the ideal $\mathfrak{p}\mathcal{O}_L$ factors uniquely as
$$\mathfrak{p}\mathcal{O}_L = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_g^{e_g}$$
with distinct nonzero prime ideals $\mathfrak{P}_i$ of $\mathcal{O}_L$ and integers $e_i \ge 1$.
The $\mathfrak{P}_i$ are the primes of $L$ *lying over* $\mathfrak{p}$.
The number $e_i$ is the [[ramification index|H5RC67SD24]] of $\mathfrak{P}_i$ over $\mathfrak{p}$, and the *residue degree* (or inertia degree) is $f_i = [\mathcal{O}_L/\mathfrak{P}_i : \mathcal{O}_K/\mathfrak{p}]$.
The prime $\mathfrak{p}$ is said to:

1. *split completely* in $L$ if $g = n$, that is, $e_i = f_i = 1$ for all $i$;
2. be *inert* in $L$ if $\mathfrak{p}\mathcal{O}_L$ is a prime ideal, that is, $g = 1$ and $e_1 = 1$;
3. be *ramified* in $L$ if $e_i > 1$ for some $i$.

## Remarks

The integers satisfy $\sum_{i=1}^g e_i f_i = n$, the [[fundamental identity|REJDQ6BTQV]].
If $L/K$ is a [[Galois extension|0QMZAQ6QQ4]], the Galois group permutes the $\mathfrak{P}_i$ transitively, so all $e_i$ are equal to some $e$, all $f_i$ to some $f$, and $efg = n$.

Example: for $K = \Q$, $L = \Q(i)$ and $\mathfrak{p} = 5\Z$, one has $5\Z[i] = (2+i)(2-i)$ with two distinct primes, each with residue field $\F_5$.
So $g = 2 = n$ and $5$ splits completely.

Non-example: $3$ does not split completely in $\Q(i)$.
The ideal $3\Z[i]$ is prime, because $x^2 + 1$ is irreducible modulo $3$; so $g = 1$, $f = 2$, and $3$ is inert.
The prime $2$ is ramified: $2\Z[i] = (1+i)^2$, with $e = 2$.