---
tag: X8VKW5Y1CZ
kind: statement
role: theorem
name: Chebotarev's density theorem
msc: [11R45, 11R44]
uses: [MWBDFS2KJ5, 0QMZAQ6QQ4, 0XTY86BZKY, 0DEBXAJ5SA, H5145REFMW, H5RC67SD24, M54ADM0CVF, 0GNVN3CA2P, KXS5DQDFYD, B7KNZCBFYT]
about: [H5145REFMW]
sources:
  - url: https://en.wikipedia.org/wiki/Chebotarev_density_theorem
same_as:
  wikidata: Q1425529
  wikipedia: Chebotarev density theorem
---

## Statement

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[Galois group|0XTY86BZKY]] $G$.
Let $C \subseteq G$ be a [[conjugacy class|0DEBXAJ5SA]].
For a nonzero prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ [[unramified|H5RC67SD24]] in $L$, let $\mathrm{Frob}_{\mathfrak{p}}$ be its [[Frobenius|H5145REFMW]] conjugacy class, and let $N(\mathfrak{p})$ be its [[norm|M54ADM0CVF]].
Then
$$\lim_{x \to \infty} \frac{\#\set{\mathfrak{p} : N(\mathfrak{p}) \le x, \ \mathfrak{p} \text{ unramified in } L, \ \mathrm{Frob}_{\mathfrak{p}} = C}}{\#\set{\mathfrak{p} : N(\mathfrak{p}) \le x}} = \frac{\abs{C}}{\abs{G}}.$$

## Remarks

The proof first treats abelian extensions, where the statement reduces, through class field theory, to the non-vanishing at $s = 1$ of Hecke $L$-functions, as in [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].
The general case is reduced to the cyclic case by passing to the fixed field of the cyclic subgroup generated by an element of $C$.
Chebotarev's original proof also adjoins roots of unity to reduce to cyclotomic extensions.

Example: for $K = \Q$ and $L = \Q(\zeta_n)$, the group $G$ is $(\Z/n\Z)^\times$ and $\mathrm{Frob}_p$ is the class of $p$ modulo $n$.
The theorem gives that the primes are equidistributed among the $\varphi(n)$ classes modulo $n$ coprime to $n$, which is Dirichlet's theorem in its density form.

Example: taking $C = \set{1}$, the primes of $K$ that split completely in $L$ have density $1/[L : K]$.
For $L = \Q(i)$, half the primes split, namely those $\equiv 1 \pmod 4$.

## Explanations

### L1 human

In a Galois extension, each unramified prime has a Frobenius symmetry that records how it splits.
The theorem says that these symmetries are spread out evenly over the Galois group, in proportion to the size of each conjugacy class.
So in the long run, every possible splitting pattern happens with a predictable frequency.

### L2 human

The theorem generalises [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]] from cyclotomic extensions of $\Q$ to all Galois extensions.
It is used to show that a Galois extension is determined by the set of primes that [[split completely|B7KNZCBFYT]] in it.
It also gives the density of primes for which a polynomial has a given factorisation pattern modulo $p$, by looking at the cycle types in its Galois group.
Effective versions with explicit error terms depend on zero-free regions of [[Dedekind zeta functions|KXS5DQDFYD]].