# 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.

Title Tchebotarev density theorem TchebotarevDensityTheorem 2013-03-22 12:46:49 2013-03-22 12:46:49 djao (24) djao (24) 5 djao (24) Theorem msc 11R37 msc 11R44 msc 11R45 Chebotarev density theorem