add new slides

6 files changed, 228 insertions, 2 deletions

diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc
index f2edc80..a70f73b 100644
--- a/vorlesungen/slides/3/Makefile.inc
+++ b/vorlesungen/slides/3/Makefile.inc
@@ -13,6 +13,8 @@ chapter3 =								\
 	../slides/3/ringstruktur.tex					\
 	../slides/3/teilbarkeit.tex					\
 	../slides/3/ideal.tex						\
+	../slides/3/idealverband.tex					\
+	../slides/3/maximalideal.tex					\
 	../slides/3/quotientenring.tex					\
 	../slides/3/faktorisierung.tex					\
 	../slides/3/faktorzerlegung.tex					\
@@ -26,5 +28,6 @@ chapter3 =								\
 	../slides/3/operatoren.tex					\
 	../slides/3/adjunktion.tex					\
 	../slides/3/adjalgebra.tex					\
+	../slides/3/wurzel2.tex						\
 	../slides/3/chapter.tex
 
diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex
index deec12e..ea2718d 100644
--- a/vorlesungen/slides/3/chapter.tex
+++ b/vorlesungen/slides/3/chapter.tex
@@ -11,6 +11,8 @@
 \folie{3/ringstruktur.tex}
 \folie{3/teilbarkeit.tex}
 \folie{3/ideal.tex}
+\folie{3/maximalideal.tex}
+\folie{3/idealverband.tex}
 \folie{3/quotientenring.tex}
 \folie{3/faktorisierung.tex}
 \folie{3/faktorzerlegung.tex}
@@ -24,3 +26,4 @@
 \folie{3/operatoren.tex}
 \folie{3/adjunktion.tex}
 \folie{3/adjalgebra.tex}
+\folie{3/wurzel2.tex}
diff --git a/vorlesungen/slides/3/idealverband.tex b/vorlesungen/slides/3/idealverband.tex
new file mode 100644
index 0000000..3434868
--- /dev/null
+++ b/vorlesungen/slides/3/idealverband.tex
@@ -0,0 +1,78 @@
+%
+% idealverband.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Idealverband}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {$\mathbb{Z}$};
+
+\uncover<2->{
+\node at (-6,-2) {$2\mathbb{Z}$};
+\node at (-2,-2) {$3\mathbb{Z}$};
+\node at (2,-2) {$5\mathbb{Z}$};
+\node at (6,-2) {$7\mathbb{Z}$};
+\node at (7,-2) {$\dots$};
+}
+
+\uncover<3->{
+\node at (-4,-4) {$6\mathbb{Z}$};
+\node at (-2,-4) {$10\mathbb{Z}$};
+\node at (0,-4) {$15\mathbb{Z}$};
+\node at (2,-4) {$21\mathbb{Z}$};
+\node at (4,-4) {$35\mathbb{Z}$};
+\node at (6,-4) {$\dots$};
+}
+
+\uncover<4->{
+\node at (-2,-6) {$30\mathbb{Z}$};
+\node at (0,-6) {$70\mathbb{Z}$};
+\node at (2,-6) {$105\mathbb{Z}$};
+}
+
+\uncover<5->{
+	\node at (-5,-6) {$\dots$};
+	\node at (5,-6) {$\dots$};
+}
+
+\uncover<2->{
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-6,-2);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-2,-2);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (2,-2);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (6,-2);
+}
+
+\uncover<3->{
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-4,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-2,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (-4,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (0,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (2,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (-2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (0,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (4,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (4,-4);
+}
+
+\uncover<4->{
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-4,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (0,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (-2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (4,-4);
+
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (0,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (2,-4);
+\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (4,-4);
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
diff --git a/vorlesungen/slides/3/maximalideal.tex b/vorlesungen/slides/3/maximalideal.tex
new file mode 100644
index 0000000..21a945a
--- /dev/null
+++ b/vorlesungen/slides/3/maximalideal.tex
@@ -0,0 +1,64 @@
+%
+% maximalideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Maximale Ideale}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Teilbarkeit}
+$a|b$
+\uncover<2->{$\Rightarrow$
+$b\in aR$}
+\uncover<3->{$\Rightarrow$
+$bR\subset aR$}
+\end{block}
+\uncover<4->{%
+\begin{block}{Nicht mehr teilbar}
+$a\in R$ nicht faktorisierbar
+\\
+\uncover<5->{$\Rightarrow$
+\\
+es gibt kein Ideal zwischen $aR$ und $R$}
+\\
+\uncover<6->{$\Leftrightarrow$
+\\
+$J$ ein Ideal
+$aR \subset J \subset R$, dann ist
+$J=aR$ oder $J=R$}
+\end{block}}
+\uncover<7->{
+\begin{block}{maximales Ideal}
+$I\subset R$ heisst maximal, wenn für jedes Ideal $J$
+mit $I\subset J\subset R$ gilt
+$I=J$ oder $J=R$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{
+\begin{block}{Beispiele}
+\begin{itemize}
+\item Primzahlen $p$ erzeugen maximale Ideale in $\mathbb{Z}$
+\item<9-> Irreduzible Polynome erzeugen maximale Ideale in $\Bbbk[X]$
+\end{itemize}
+\end{block}}
+\uncover<10->{%
+\begin{block}{Körper}
+$M\subset R$ ein maximales Ideal, dann ist
+$R/M$ ein Körper
+\end{block}}
+\uncover<11->{%
+\begin{block}{Beispiel}
+\begin{itemize}
+\item
+$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$
+\item<12->
+$m$ ein irreduzibles Polynom:
+$\Bbbk[X]/ (m)$ ist ein Körper
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex
new file mode 100644
index 0000000..48cc210
--- /dev/null
+++ b/vorlesungen/slides/3/wurzel2.tex
@@ -0,0 +1,79 @@
+%
+% wurzel2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\mathbb{Z}(\sqrt{2})\only<7->{ = \mathbb{Z}[X]/(X^2-2)}$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Der Ring $\mathbb{Z}(\sqrt{2})$}
+$\mathbb{Z}(\sqrt{2})$ als Teilring:
+{\color{blue}
+\[
+R=\{ a+b\sqrt{2}\;|\; a,b\in\mathbb{Z} \} \subset \mathbb{R}
+\]}%
+\uncover<2->{$\sqrt{2}\not\in\mathbb{Q}$}\uncover<3->{
+$\Rightarrow$
+$1$ und $\sqrt{2}$ sind inkommensurabel}\uncover<4->{
+$\Rightarrow$
+$R$ dicht in $\mathbb{R}$}
+\end{block}
+\uncover<5->{%
+\begin{block}{Algebraische Konstruktion}
+\uncover<8->{%
+Das Polynom $X^2-2$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$}
+\[
+\uncover<8->{\mathbb{Z}[X]/(X^2-2)
+=}
+{\color{red}\{a+bX\;|\;a,b\in\mathbb{Z}\}}
+\]\uncover<7->{%
+mit Rechenregel: $X^2=2$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+\begin{scope}
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\foreach \x in {-10,...,10}{
+	\pgfmathparse{int(\x/sqrt(2))-5}
+	\xdef\s{\pgfmathresult}
+	\pgfmathparse{int(\x/sqrt(2))+5}
+	\xdef\t{\pgfmathresult}
+	\foreach \y in {\s,...,\t}{
+		\uncover<4->{
+			\fill[color=blue] ({\x-\y*sqrt(2)},0)
+				circle[radius=0.05];
+		}
+		\uncover<6->{
+			\draw[color=blue,line width=0.1pt] 
+				({\x-\y*sqrt(2)-3.2},3.2) 
+				--
+				({\x-\y*sqrt(2)+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 {-2,...,2}{
+			\fill[color=red]
+				({\x},{\y*sqrt(2)}) circle[radius=0.08];
+		}
+	}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 2423a3f..9402625 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -3,5 +3,4 @@
 %
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\folie{3/ideal.tex}
-\folie{3/quotientenring.tex}
+\folie{3/wurzel2.tex}