From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001 From: Joshua Baer Date: Mon, 12 Apr 2021 21:51:55 +0200 Subject: update to current state of book --- vorlesungen/slides/4/galois/winkeldreiteilung.tex | 188 +++++++++++----------- 1 file changed, 94 insertions(+), 94 deletions(-) (limited to 'vorlesungen/slides/4/galois/winkeldreiteilung.tex') 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} -- cgit v1.2.1