# abelian number field

###### Definition 1.

An abelian number field is a number field^{} $K$ such that $K\mathrm{/}\mathrm{Q}$ is an abelian extension^{}, i.e. $K\mathrm{/}\mathrm{Q}$ is Galois and $\mathrm{Gal}\mathit{}\mathrm{(}K\mathrm{/}\mathrm{Q}\mathrm{)}$ is an abelian group^{}.

The abelian number fields are classified by the Kronecker-Weber Theorem^{}.

###### Definition 2.

A cyclic number field is an (abelian) number field $K$ such that $K\mathrm{/}\mathrm{Q}$ is a Galois extension^{} and $\mathrm{Gal}\mathit{}\mathrm{(}K\mathrm{/}\mathrm{Q}\mathrm{)}$ is a finite cyclic group^{} (therefore abelian).

Title | abelian number field |
---|---|

Canonical name | AbelianNumberField |

Date of creation | 2013-03-22 16:01:24 |

Last modified on | 2013-03-22 16:01:24 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 11-00 |

Related topic | GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ |

Defines | cyclic number field |