abelian extension AbelianExtension 2002-11-15 02:15:48 2006-04-30 18:09:24 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $K$ be a Galois extension of $F$. The extension is said to be an {\em abelian extension} if the Galois group $\textrm{Gal$(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 \PMlinkname{primitive nth root of unity}{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.