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