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%
% phi.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{$\mathbb{Q}(\varphi)=\mathbb{Q}[X]/(X^2-X-1)$}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Der Ring $\mathbb{Z}(\varphi)$}
$\mathbb{Z}(\varphi)$ als Teilrung:
{\color{blue}
\[
R=\{a+b\varphi\;|\; a,b\in\mathbb{Z}\}
\]}%
\uncover<2->{$\varphi\not\in\mathbb{Q}$}\uncover<3->{
$\Rightarrow$
$1$ und $\varphi$ sind inkommensurabel}\uncover<4->{
$\Rightarrow$
$R$ dicht in $\mathbb{R}$}
\end{block}
\uncover<5->{%
\begin{block}{Algebraische Konstruktion}
\uncover<8->{%
Das Polynom $X^2-X-1$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$}
\[
\uncover<8->{\mathbb{Q}[X]/(X^2-X-1)
=}
{\color{red}\{a+b\varphi\;|\;a,b\in\mathbb{Z}\}}
\]\uncover<7->{%
mit der Rechenregel: $X^2=X+1$}
\end{block}}
\uncover<9->{%
\begin{block}{Körper}
$\mathbb{Q}(\varphi) = \mathbb{Q}[X]/(X^2+X+1)$
\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick,scale=0.92]
\begin{scope}
\pgfmathparse{(sqrt(5)-1))/2}
\xdef\gphi{\pgfmathresult}
\clip (-3.2,-3.2) rectangle (3.2,3.2);
\foreach \x in {-10,...,10}{
\pgfmathparse{int(\x/\gphi)-10}
\xdef\s{\pgfmathresult}
\pgfmathparse{int(\x/\gphi)+10}
\xdef\t{\pgfmathresult}
\foreach \y in {\s,...,\t}{
\uncover<4->{
\fill[color=blue] ({\x-\y*\gphi},0)
circle[radius=0.05];
}
\uncover<6->{
\draw[color=blue,line width=0.1pt]
({\x-\y*\gphi-3.2},3.2)
--
({\x-\y*\gphi+3.2},-3.2);
}
}
}
\end{scope}
\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
\uncover<5->{
\draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}];
\foreach \x in {-3,...,3}{
\foreach \y in {-5,...,5}{
\fill[color=red]
({\x},{\y*\gphi}) circle[radius=0.08];
}
}
}
\end{tikzpicture}
\end{center}
\end{column}
\end{columns}
\end{frame}
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