Chebotarev density theorem

Statement

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$ , and let $C\subseteq G$ be a conjugacy class. The Chebotarev density theorem says that the unramified finite places $v$ of $K$ whose Frobenius conjugacy class $\operatorname{Frob}_v$ equals $C$ have natural density

\frac{|C|}{|G|}.

In particular, every conjugacy class occurs at infinitely many unramified places.

Consequence for Galois representations

Frobenius elements at unramified places are dense, up to conjugacy, in the finite quotients of the absolute Galois group. Consequently two continuous semisimple $\ell$ -adic representations with equal characteristic polynomials of Frobenius at all but finitely many places are isomorphic. This is the uniqueness mechanism behind compatible systems and many formulations of local–global compatibility.

Function fields

There is a corresponding theorem for global function fields. When the constant field grows inside $L$ , Frobenius classes and degrees satisfy a compatibility condition; equidistribution is stated degree by degree in the permitted congruence classes. Omitting this constant-field qualification can make the naive number-field wording false.

References

Nikolai G. Chebotarev, “Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören,” Mathematische Annalen 95 (1926), 191–228. EuDML.
Jean-Pierre Serre, Lectures on $N_X(p)$ , Research Notes in Mathematics 11, CRC Press, 2012, §3.