new slides

1 files changed, 94 insertions, 0 deletions

diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
new file mode 100644
index 0000000..54b941b
--- /dev/null
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -0,0 +1,94 @@
+%
+% winkeldreiteilung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Winkeldreiteilung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.43\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{5}
+\def\a{25}
+
+\uncover<3->{
+	\draw[line width=0.7pt] (\r,0) arc (0:90:\r);
+}
+
+\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
+\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
+
+\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
+
+\uncover<2->{
+	\fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
+	\draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
+	\node[color=red] at ({0.5*\a}:{0.7*\r})
+		{$\displaystyle\frac{\alpha}{3}$};
+}
+
+\uncover<3->{
+	\fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
+	\draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
+
+	\fill[color=red] ({\a}:\r) circle[radius=0.05];
+	\draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
+
+	\draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
+	\draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
+}
+
+
+\uncover<4->{
+\node at ({0.5*\r},-0.5) [below] {$\displaystyle
+\cos{\color{blue}\alpha}
+=
+4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
+$};
+}
+
+\uncover<5->{
+	\node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
+	\node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.53\textwidth}
+\begin{block}{Aufgabe}
+Teile einen Winkel in drei gleiche Teile
+\end{block}
+\vspace{-2pt}
+\uncover<6->{%
+\begin{block}{Algebraisierte Aufgabe}
+Konstruiere $x$ aus $a$ derart, dass
+\[
+p(x)
+=
+x^3-\frac34 x -a = 0
+\]
+\uncover<7->{%
+$a=0$:}
+\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
+\end{block}}
+\vspace{-2pt}
+\uncover<10->{%
+\begin{proof}[Unmöglichkeitsbeweis]
+\begin{itemize}
+\item<11->
+$a\ne 0$ $\Rightarrow$  $p(x)$ irreduzibel
+\item<12->
+$p(x)$ definiert eine Körpererweiterung vom Grad $3$
+\item<13->
+Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}