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%% 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}
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