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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}