# abelian extension

Let $K$ be a Galois extension of $F$. The extension is said to be an 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 AbelianExtension 2013-03-22 13:09:28 2013-03-22 13:09:28 scanez (1021) scanez (1021) 5 scanez (1021) Definition msc 12F10 KroneckerWeberTheorem KummerTheory