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# 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.

Synonym:

Chebotarev density theorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

11R37*no label found*11R44

*no label found*11R45

*no label found*

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