Trace2 2002-02-07 13:18:45 2005-04-03 23:11:33 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 trace} $\operatorname{Tr}_F^K(x)$ of $x$ is defined to be the sum 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, $$ \operatorname{Tr}_F^K(x) := \sum_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) $$ The trace of $x$ is always an element of $F$, since any element of $\operatorname{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes $\operatorname{Tr}_F^K(x)$. The name ``trace'' derives from the fact that, when $K/F$ is finite, the trace of $x$ is simply the trace of the linear transformation $T: K \longrightarrow K$ of vector spaces over $F$ defined by $T(v) := xv$.