67 lines
1.6 KiB
TeX
67 lines
1.6 KiB
TeX
|
\documentclass[12pt,a4paper,german]{article}
|
||
|
|
\usepackage{url}
|
||
|
|
%\usepackage{graphics}
|
||
|
|
\usepackage{times}
|
||
|
|
\usepackage[T1]{fontenc}
|
||
|
|
\usepackage{ngerman}
|
||
|
|
\usepackage{float}
|
||
|
|
\usepackage{diagbox}
|
||
|
|
\usepackage[utf8]{inputenc}
|
||
|
|
\usepackage{geometry}
|
||
|
|
\usepackage{amsfonts}
|
||
|
|
\usepackage{amsmath}
|
||
|
|
\usepackage{cancel}
|
||
|
|
\usepackage{wasysym}
|
||
|
|
\usepackage{csquotes}
|
||
|
|
\usepackage{graphicx}
|
||
|
|
\usepackage{epsfig}
|
||
|
|
\usepackage{paralist}
|
||
|
|
\usepackage{tikz}
|
||
|
|
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
|
||
|
|
|
||
|
|
%%%%%%%%%% Fill out the the definitions %%%%%%%%%
|
||
|
|
\def \name {Valentin Brandl} %
|
||
|
|
\def \matrikel {108018274494} %
|
||
|
|
\def \pname {Marvin Herrmann} %
|
||
|
|
\def \pmatrikel {108018265436} %
|
||
|
|
\def \gruppe {2 (Mi 10-12 Andre)}
|
||
|
|
\def \qname {Pascal Brackmann}
|
||
|
|
\def \qmatrikel {108017113834} %
|
||
|
|
\def \uebung {11} %
|
||
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
|
||
|
|
% DO NOT MODIFY THIS HEADER
|
||
|
|
\newcommand{\hwsol}{
|
||
|
|
\vspace*{-2cm}
|
||
|
|
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
|
||
|
|
\noindent \pmatrikel \quad \pname \\
|
||
|
|
\noindent \qmatrikel \quad \qname \\
|
||
|
|
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
|
||
|
|
}
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
%Import header
|
||
|
|
\hwsol
|
||
|
|
|
||
|
|
\section*{Aufgabe 11.2}
|
||
|
|
|
||
|
|
Sei $p$ prim.
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
f(x) = x^p - x \in \mathbb{F}_p[x]
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
Nullstellen berechnen:
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
x^p - 1 &\equiv 0 &\mod p \\
|
||
|
|
x (x^{p-1}-1) &\equiv 0 &\mod p \\
|
||
|
|
\Rightarrow (x \equiv 0 \mod p) \lor (x^{p-1} \equiv 1 \mod p)
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
Gilt wegen Satz von Fermat $\forall x \in \{1,...,p-1\}$
|
||
|
|
|
||
|
|
$\Rightarrow$ $\forall x \in \mathbb{F}_p$ gilt: $x$ ist Nullstelle von $x^p - x$
|
||
|
|
|
||
|
|
\end{document}
|