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
Canonical name	TchebotarevDensityTheorem
Date of creation	2013-03-22 12:46:49
Last modified on	2013-03-22 12:46:49
Owner	djao (24)
Last modified by	djao (24)
Numerical id	5
Author	djao (24)
Entry type	Theorem
Classification	msc 11R37
Classification	msc 11R44
Classification	msc 11R45
Synonym	Chebotarev density theorem