# abelian extension

Let $K$ be a Galois extension^{} of $F$. The extension^{} is said to be an
abelian extension^{} if the Galois group^{} Gal$(K/F)$ is abelian^{}.

Examples: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ has Galois group $\mathbb{Z}/2\mathbb{Z}$ so $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is an abelian extension.

Let ${\zeta}_{n}$ be a primitive nth root of unity (http://planetmath.org/RootOfUnity). Then $\mathbb{Q}({\zeta}_{n})/\mathbb{Q}$ has Galois group ${(\mathbb{Z}/n\mathbb{Z})}^{*}$ (the group of units of $\mathbb{Z}/n\mathbb{Z}$) so $\mathbb{Q}({\zeta}_{n})/\mathbb{Q}$ is abelian.

Title | abelian extension |
---|---|

Canonical name | AbelianExtension |

Date of creation | 2013-03-22 13:09:28 |

Last modified on | 2013-03-22 13:09:28 |

Owner | scanez (1021) |

Last modified by | scanez (1021) |

Numerical id | 5 |

Author | scanez (1021) |

Entry type | Definition |

Classification | msc 12F10 |

Related topic | KroneckerWeberTheorem |

Related topic | KummerTheory |