# Tchebotarev density theorem

Let $L/K$ be any finite Galois extension^{} of number fields^{} with Galois group^{} $G$. For any conjugacy class^{} $C\beta \x8a\x82G$, the subset of prime ideals $\mathrm{\pi \x9d\x94\xad}\beta \x8a\x82K$ which are unramified in $L$ and satisfy the property

$$[L/K,\mathrm{\pi \x9d\x94\x93}]\beta \x88\x88C\beta \x81\u2019\text{for any prime\Beta}\beta \x81\u2019\mathrm{\pi \x9d\x94\x93}\beta \x8a\x82L\beta \x81\u2019\text{containing\Beta}\beta \x81\u2019\mathrm{\pi \x9d\x94\xad}$$ |

has analytic density $\frac{|C|}{|G|}$, where $[L/K,\mathrm{\pi \x9d\x94\x93}]$ denotes the Artin symbol^{} at $\mathrm{\pi \x9d\x94\x93}$.

Note that the conjugacy class of $[L/K,\mathrm{\pi \x9d\x94\x93}]$ is independent of the choice of prime $\mathrm{\pi \x9d\x94\x93}$ lying over $\mathrm{\pi \x9d\x94\xad}$, 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 |