Tchebotarev density theorem

Let $L/K$ be any finite Galois extension of number fields with Galois group $G$ . For any conjugacy class $C\\subset G$ , the subset of prime ideals $\\mathfrak{p}\\subset K$ which are unramified in $L$ and satisfy the property

[L/K,\\mathfrak{P}]\\in C\\ \\text{for any prime }\\ \\mathfrak{P}\\subset L\\ \\text{%containing }\\ \\mathfrak{p}

has analytic density $\\frac{|C|}{|G|}$ , where $[L/K,\\mathfrak{P}]$ denotes the Artin symbol at $\\mathfrak{P}$ .

Note that the conjugacy class of $[L/K,\\mathfrak{P}]$ is independent of the choice of prime $\\mathfrak{P}$ lying over $\\mathfrak{p}$ , since any two such choices of primes are related by a Galois automorphism and their corresponding Artin symbols are conjugate by this same automorphism.