ExampleOfTrace 2006-05-30 02:09:06 2006-09-24 22:39:47 Example trace % 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 $\omega$ be a complex root of unity different than 1. Then $\omega$ and $\omega^2$ are the conjugate roots of the minimal polynomial $x^2 + x +1$. Since $\mathbb{Q}(\omega)$ is the splitting field of $x^2 + x +1$, it is Galois over $\mathbb{Q}$. Moreover the Galois group $Gal(\mathbb{Q}(\omega)/\mathbb{Q}))$ is formed by the identity and the automorphism $g(\omega) = \omega^2$ The elements of $\mathbb{Q}(\omega)$ have the form $a + b\omega$, $a,b\in \mathbb{Q}$. Then we obtain \[N_{\mathbb{Q}(\omega)}^{\mathbb{Q}}(a + b\omega) = (a +b\omega)(a +b\omega^2) = a^2 - ab + b^2, Tr_{\mathbb{Q}(\omega)}^{\mathbb{Q}}(a + b\omega) = (a +b\omega) + (a + b\omega^2) = 2a - b \]