Norm 2002-02-07 13:20:40 2003-10-06 17:58:18 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 $K/F$ be a Galois extension, and let $x \in K$. The {\em norm} $\operatorname{N}_F^K(x)$ of $x$ is defined to be the product of all the elements of the orbit of $x$ under the group action of the Galois group $\operatorname{Gal}(K/F)$ on $K$; taken with multiplicities if $K/F$ is a finite extension. In the case where $K/F$ is a finite extension, the norm of $x$ can be defined to be the determinant of the linear transformation $[x]: K \to K$ given by $[x](k) := xk$, where $K$ is regarded as a vector space over $F$. This definition does not require that $K/F$ be Galois, or even that $K$ be a field---for instance, it remains valid when $K$ is a division ring (although $F$ does have to be a field, in order for determinant to be defined). Of course, for finite Galois extensions $K/F$, this definition agrees with the previous one, and moreover the formula $$ \operatorname{N}_F^K(x) := \prod_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) $$ holds. The norm of $x$ is always an element of $F$, since any element of $\operatorname{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes $\operatorname{N}_F^K(x)$.