Chebotarev's density theorem

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. 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 $\equiv 1 \pmod 4$.

Going deeper

For a student

The theorem generalises Dirichlet's theorem on arithmetic progressions 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 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.