GaloisClosure 2002-11-16 02:44:12 2006-10-15 17:13:26 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 an extension field of $F$. A \emph{Galois closure} of $K/F$ is a field $L \supseteq K$ that is a Galois extension of $F$ and is minimal in that respect, i.e. no proper subfield of $L$ containing $K$ is normal over $F$.