diff options
| -rw-r--r-- | f8.tex | 144 |
1 files changed, 144 insertions, 0 deletions
@@ -0,0 +1,144 @@ +%% This is a LaTeX document. Hey, Emacs, -*- latex -*- , get it? +\documentclass[12pt,a4paper]{article} +\usepackage[francais]{babel} +\usepackage[latin1]{inputenc} +\usepackage{times} +% A tribute to the worthy AMS: +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsthm} +% +\usepackage{mathrsfs} +\usepackage{wasysym} +\usepackage{url} +% +\theoremstyle{definition} +\newtheorem{comcnt}{Tout}[subsection] +\newcommand\thingy{% +\refstepcounter{comcnt}\smallbreak\noindent\textbf{\thecomcnt.} } +\newtheorem{defn}[comcnt]{Définition} +\newtheorem{prop}[comcnt]{Proposition} +\newtheorem{lem}[comcnt]{Lemme} +\newtheorem{thm}[comcnt]{Théorème} +\newtheorem{cor}[comcnt]{Corollaire} +\newtheorem{rmk}[comcnt]{Remarque} +\newtheorem{exmps}[comcnt]{Exemples} +\newcommand{\limp}{\mathrel{\Rightarrow}} +\newcommand{\liff}{\mathrel{\Longleftrightarrow}} +\newcommand{\pgcd}{\operatorname{pgcd}} +\newcommand{\ppcm}{\operatorname{ppcm}} +\newcommand{\signe}{\operatorname{signe}} +\newcommand{\tee}{\mathbin{\top}} +\newcommand{\Frob}{\operatorname{Fr}} +\renewcommand{\qedsymbol}{\smiley} +% +% +% +\begin{document} + +\pagestyle{empty} + +Exemple de $\mathbb{F}_8$ vu comme $\mathbb{F}_2[t]/(f)$ avec $f = t^3 ++ t + 1$ : + +Représentation par des polynômes de degré $<3$ en $t$ : + +{\footnotesize +\begin{center} +$ +\begin{array}{r|cccccccc} + +&\{0\}&\{1\}&\{2\}&\{3\}&\{4\}&\{5\}&\{6\}&\{7\}\\ +\hline + 0=\{0\}&\{0\}&\{1\}&\{2\}&\{3\}&\{4\}&\{5\}&\{6\}&\{7\}\\ + 1=\{1\}&\{1\}&\{0\}&\{3\}&\{2\}&\{5\}&\{4\}&\{7\}&\{6\}\\ + \bar t=\{2\}&\{2\}&\{3\}&\{0\}&\{1\}&\{6\}&\{7\}&\{4\}&\{5\}\\ + \bar t+1=\{3\}&\{3\}&\{2\}&\{1\}&\{0\}&\{7\}&\{6\}&\{5\}&\{4\}\\ + \bar t^2=\{4\}&\{4\}&\{5\}&\{6\}&\{7\}&\{0\}&\{1\}&\{2\}&\{3\}\\ + \bar t^2+1=\{5\}&\{5\}&\{4\}&\{7\}&\{6\}&\{1\}&\{0\}&\{3\}&\{2\}\\ + \bar t^2+\bar t=\{6\}&\{6\}&\{7\}&\{4\}&\{5\}&\{2\}&\{3\}&\{0\}&\{1\}\\ +\bar t^2+\bar t+1=\{7\}&\{7\}&\{6\}&\{5\}&\{4\}&\{3\}&\{2\}&\{1\}&\{0\}\\ +\end{array} +$ +\end{center} +} + +{\footnotesize +\begin{center} +$ +\begin{array}{r|cccccccc} + \times&\{0\}&\{1\}&\{2\}&\{3\}&\{4\}&\{5\}&\{6\}&\{7\}\\ +\hline + 0=\{0\}&\{0\}&\{0\}&\{0\}&\{0\}&\{0\}&\{0\}&\{0\}&\{0\}\\ + 1=\{1\}&\{0\}&\{1\}&\{2\}&\{3\}&\{4\}&\{5\}&\{6\}&\{7\}\\ + \bar t=\{2\}&\{0\}&\{2\}&\{4\}&\{6\}&\{3\}&\{1\}&\{7\}&\{5\}\\ + \bar t+1=\{3\}&\{0\}&\{3\}&\{6\}&\{5\}&\{7\}&\{4\}&\{1\}&\{2\}\\ + \bar t^2=\{4\}&\{0\}&\{4\}&\{3\}&\{7\}&\{6\}&\{2\}&\{5\}&\{1\}\\ + \bar t^2+1=\{5\}&\{0\}&\{5\}&\{1\}&\{4\}&\{2\}&\{7\}&\{3\}&\{6\}\\ + \bar t^2+\bar t=\{6\}&\{0\}&\{6\}&\{7\}&\{1\}&\{5\}&\{3\}&\{2\}&\{4\}\\ +\bar t^2+\bar t+1=\{7\}&\{0\}&\{7\}&\{5\}&\{2\}&\{1\}&\{6\}&\{4\}&\{3\}\\ +\end{array} +$ +\end{center} +} + +Représentation par des puissances de l'élément primitif $\bar t$ : + +{\footnotesize +\begin{center} +$ +\begin{array}{r|cccccccc} + +&[^\infty]&[^0]&[^1]&[^2]&[^3]&[^4]&[^5]&[^6]\\ +\hline + 0=[^\infty]&[^\infty]&[^0]&[^1]&[^2]&[^3]&[^4]&[^5]&[^6]\\ + 1=[^0]&[^0]&[^\infty]&[^3]&[^6]&[^1]&[^5]&[^4]&[^2]\\ + \bar t=[^1]&[^1]&[^3]&[^\infty]&[^4]&[^0]&[^2]&[^6]&[^5]\\ +\bar t^2=[^2]&[^2]&[^6]&[^4]&[^\infty]&[^5]&[^1]&[^3]&[^0]\\ +\bar t^3=[^3]&[^3]&[^1]&[^0]&[^5]&[^\infty]&[^6]&[^2]&[^4]\\ +\bar t^4=[^4]&[^4]&[^5]&[^2]&[^1]&[^6]&[^\infty]&[^0]&[^3]\\ +\bar t^5=[^5]&[^5]&[^4]&[^6]&[^3]&[^2]&[^0]&[^\infty]&[^1]\\ +\bar t^6=[^6]&[^6]&[^2]&[^5]&[^0]&[^4]&[^3]&[^1]&[^\infty]\\ +\end{array} +$ +\end{center} +} + +{\footnotesize +\begin{center} +$ +\begin{array}{r|cccccccc} + \times&[^\infty]&[^0]&[^1]&[^2]&[^3]&[^4]&[^5]&[^6]\\ +\hline + 0=[^\infty]&[^\infty]&[^\infty]&[^\infty]&[^\infty]&[^\infty]&[^\infty]&[^\infty]&[^\infty]\\ + 1=[^0]&[^\infty]&[^0]&[^1]&[^2]&[^3]&[^4]&[^5]&[^6]\\ + \bar t=[^1]&[^\infty]&[^1]&[^2]&[^3]&[^4]&[^5]&[^6]&[^0]\\ +\bar t^2=[^2]&[^\infty]&[^2]&[^3]&[^4]&[^5]&[^6]&[^0]&[^1]\\ +\bar t^3=[^3]&[^\infty]&[^3]&[^4]&[^5]&[^6]&[^0]&[^1]&[^2]\\ +\bar t^4=[^4]&[^\infty]&[^4]&[^5]&[^6]&[^0]&[^1]&[^2]&[^3]\\ +\bar t^5=[^5]&[^\infty]&[^5]&[^6]&[^0]&[^1]&[^2]&[^3]&[^4]\\ +\bar t^6=[^6]&[^\infty]&[^6]&[^0]&[^1]&[^2]&[^3]&[^4]&[^5]\\ +\end{array} +$ +\end{center} +} + +Correspondance (log discret) : + +\begin{center} +$ +\begin{array}{c|c|c|c|c|c|c|c} +0&1&\bar t&\bar t+1&\bar t^2&\bar t^2+1&\bar t^2+\bar t&\bar t^2+\bar t+1\\ +\hline +\{0\}&\{1\}&\{2\}&\{3\}&\{4\}&\{5\}&\{6\}&\{7\}\\ +\hline +[^\infty]&[^0]&[^1]&[^3]&[^2]&[^6]&[^4]&[^5]\\ +\hline +0&1&\bar t&\bar t^3&\bar t^2&\bar t^6&\bar t^4&\bar t^5 +\end{array} +$ +\end{center} + +% +% +% +\end{document} |