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-rw-r--r--vorlesungen/slides/4/Makefile.inc1
-rw-r--r--vorlesungen/slides/4/chapter.tex1
-rw-r--r--vorlesungen/slides/4/qundr.tex138
3 files changed, 140 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index 88ae3bb..6616f56 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -20,6 +20,7 @@ chapter4 = \
../slides/4/charakteristik.tex \
../slides/4/char2.tex \
../slides/4/frobenius.tex \
+ ../slides/4/qundr.tex \
../slides/4/alpha.tex \
../slides/4/chapter.tex
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index 8c9a30b..6872018 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -19,3 +19,4 @@
\folie{4/charakteristik.tex}
\folie{4/char2.tex}
\folie{4/frobenius.tex}
+\folie{4/qundr.tex}
diff --git a/vorlesungen/slides/4/qundr.tex b/vorlesungen/slides/4/qundr.tex
new file mode 100644
index 0000000..a6f89bd
--- /dev/null
+++ b/vorlesungen/slides/4/qundr.tex
@@ -0,0 +1,138 @@
+%
+% qundr.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\definecolor{darkblue}{rgb}{0,0,0.8}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (ll) at (-6,-3.6);
+\coordinate (lr) at (6,-3.6);
+\coordinate (ur) at (6,3.6);
+\coordinate (ul) at (-6,3.6);
+
+\def\d{0.6}
+\def\D{0.5}
+
+\coordinate (q) at (0,{-2.25+\d});
+\coordinate (r) at (-1.5,{\d+\D});
+\coordinate (a) at (1.5,{\d-\D});
+\coordinate (c) at (0,{2.25+\d});
+
+\coordinate (m1) at ($0.5*(q)+0.5*(r)$);
+\coordinate (m2) at ($0.5*(q)+0.5*(a)$);
+\coordinate (m3) at ($0.5*(c)+0.5*(r)$);
+\coordinate (m4) at ($0.5*(c)+0.5*(a)$);
+
+\def\t{1.5}
+\coordinate (M1) at ($(m1)+\t*(m1)-\t*(m4)$);
+\coordinate (M2) at ($(m2)+\t*(m2)-\t*(m3)$);
+\coordinate (M4) at ($(m4)+\t*(m4)-\t*(m1)$);
+\coordinate (M3) at ($(m3)+\t*(m3)-\t*(m2)$);
+
+\begin{scope}
+\clip (ll) rectangle (ur);
+
+\uncover<3->{
+ \fill[color=blue!30]
+ ($0.9*(m1)+0.1*(M1)+(-6,0)$) -- ($0.9*(m1)+0.1*(M1)$)
+ -- (M4) -- (ul) -- cycle;
+}
+
+\uncover<4->{
+ \fill[color=red!60,opacity=0.5]
+ ($0.9*(m2)+0.1*(M2)$) -- ($0.9*(m2)+0.1*(M2)+(6,0)$)
+ -- (ur) -- (M3) -- cycle;
+}
+
+\uncover<2->{
+ \fill[color=darkgreen!60,opacity=0.5]
+ ($1.09*(m3)-0.09*(M3)$) -- ($1.09*(m3)-0.09*(M3)+(-6,0)$)
+ -- (ll) -- (M2) -- cycle;
+}
+
+\uncover<6->{
+ \fill[color=gray,opacity=0.5]
+ ({6-0.1},{\d+0.22}) rectangle ({6-2.4},{\d+0.62});
+ \node[color=yellow] at (6,\d) [above left] {überabzählbar\strut};
+
+ \fill[color=gray,opacity=0.5]
+ ({-6+0.1},{\d-0.15}) rectangle ({-6+1.75},{\d-0.55});
+ \node[color=yellow] at (-6,\d) [below right] {abzählbar\strut};
+
+ \draw[color=yellow,line width=2pt] (-7,\d) -- (7,\d);
+}
+
+\end{scope}
+
+\node at (q) {$\mathbb{Q}$\strut};
+\node at ($(q)+(0,-0.2)$) [below] {Primkörper};
+
+\uncover<3->{
+ \node at (r) {$\mathbb{R}$\strut};
+ \node at (r) [left] {$\text{reelle Zahlen}=\mathstrut$};
+ \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (r);
+ \node at ($0.5*(q)+0.5*(r)$)
+ [below,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$};
+ \node[color=blue] at (ul)
+ [above right] {topologische Vervollständigung};
+}
+
+\uncover<4->{
+ \node at (a) {$\mathbb{A}$\strut};
+ \node at (a) [right] {$\mathstrut = \text{algebraische Zahlen}$};
+ \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (a);
+ \node at ($0.5*(q)+0.5*(a)$)
+ [below,rotate={atan((2.25-\D)/1.5)}] {index $\infty$};
+ \node[color=red] at (ur)
+ [above left] {algebraische Vervollständigung};
+}
+
+\uncover<5->{
+ \node at (c) {$\mathbb{C}$\strut};
+ \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (r) -- (c);
+ \draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (a) -- (c);
+ \node at ($(c)+(0,0.2)$) [above] {komplexe Zahlen};
+ \node at ($0.5*(r)+0.5*(c)$)
+ [above,rotate={atan((2.25-\D)/1.5)}] {index 2};
+ \node at ($0.5*(a)+0.5*(c)$)
+ [above,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$};
+}
+
+\uncover<3->{
+ \node[color=darkblue] at (ul) [below right]
+ {\begin{minipage}{0.3\textwidth}\raggedright
+ Grenzwerte von Cauchy-Folgen in $\mathbb{Q}$ hinzufügen
+ \end{minipage}};
+}
+
+\uncover<4->{
+ \node[color=darkred] at (ur) [below left]
+ {\begin{minipage}{0.3\textwidth}\raggedleft
+ Nullstellen von Polynomen in $\mathbb{Q}[X]$ hinzufügen
+ \end{minipage}};
+}
+
+\uncover<2->{
+ \node[color=darkgreen] at (ll) [above right]
+ {\begin{minipage}{0.4\textwidth}\raggedright
+ \begin{block}{Archimedische Eigenschaft}
+ Für $a>b >0$ gibt es $n\in\mathbb{N}$ mit
+ $n\cdot b > a$
+ \end{block}
+ \end{minipage}};
+
+ \node[color=darkgreen] at (ll) [below right]
+ {geordneter Körper, nötig für die Definition von Cauchy-Folgen};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup