diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/4/galois/winkeldreiteilung.tex | 188 |
1 files changed, 94 insertions, 94 deletions
diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex index 54b941b..28c07fe 100644 --- a/vorlesungen/slides/4/galois/winkeldreiteilung.tex +++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex @@ -1,94 +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} +%
+% 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}
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