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authordavid <david>2008-10-21 17:30:00 (GMT)
committerdavid <david>2008-10-21 17:30:00 (GMT)
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parent19b95aef46a4f132f6145478e3792e69eb74a3be (diff)
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Tables of the finite field with 8 elements.
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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}