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-rw-r--r--vorlesungen/slides/3/Makefile.inc2
-rw-r--r--vorlesungen/slides/3/chapter.tex2
-rw-r--r--vorlesungen/slides/3/multiplikation.tex180
-rw-r--r--vorlesungen/slides/3/phi.tex85
-rw-r--r--vorlesungen/slides/3/wurzel2.tex4
-rw-r--r--vorlesungen/slides/test.tex6
6 files changed, 277 insertions, 2 deletions
diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc
index d6ce6b7..442bd15 100644
--- a/vorlesungen/slides/3/Makefile.inc
+++ b/vorlesungen/slides/3/Makefile.inc
@@ -31,5 +31,7 @@ chapter3 = \
../slides/3/adjunktion.tex \
../slides/3/adjalgebra.tex \
../slides/3/wurzel2.tex \
+ ../slides/3/phi.tex \
+ ../slides/3/multiplikation.tex \
../slides/3/chapter.tex
diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex
index 5287ffe..3fbc3fd 100644
--- a/vorlesungen/slides/3/chapter.tex
+++ b/vorlesungen/slides/3/chapter.tex
@@ -29,3 +29,5 @@
\folie{3/adjunktion.tex}
\folie{3/adjalgebra.tex}
\folie{3/wurzel2.tex}
+\folie{3/phi.tex}
+\folie{3/multiplikation.tex}
diff --git a/vorlesungen/slides/3/multiplikation.tex b/vorlesungen/slides/3/multiplikation.tex
new file mode 100644
index 0000000..13f4e03
--- /dev/null
+++ b/vorlesungen/slides/3/multiplikation.tex
@@ -0,0 +1,180 @@
+%
+% multiplikation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\N{21}
+\begin{frame}[t,fragile]
+\frametitle{Multiplikation mit $\alpha$ in $\mathbb{Z}(\alpha)$}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+
+\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\sqrt{2})$};
+\node[color=blue] at (3.5,3.2) [above left] {$\sqrt{2}\mathbb{Z}(\sqrt{2})$};
+
+\pgfmathparse{sqrt(2)}
+\xdef\a{\pgfmathresult}
+\pgfmathparse{-int(3.2/\a)}
+\xdef\ymin{\pgfmathresult}
+\pgfmathparse{int(3.2/\a)}
+\xdef\ymax{\pgfmathresult}
+
+\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\sqrt{2}$}];
+
+\def\punkt#1#2#3{
+ ({(1-(#3))*(#1)+2*(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1))*\a})
+}
+
+\foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.5pt]
+ \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0};
+ \foreach \y in {\ymin,...,\ymax}{
+ \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08];
+ }
+}
+\foreach \y in {\ymin,...,\ymax}{
+ \draw[color=red,line width=0.5pt]
+ \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0};
+}
+
+
+\def\bildnetz#1{
+ \pgfmathparse{(#1-1)/(\N-1)}
+ \xdef\t{\pgfmathresult}
+ \only<#1>{
+ \uncover<2->{
+ \draw[->,color=blue,line width=1.4pt]
+ (0,\a) -- \punkt{0}{1}{\t};
+ \draw[->,color=blue,line width=1.4pt]
+ (1,0) -- \punkt{1}{0}{\t};
+ }
+ \foreach \x in {-3,...,3}{
+ \draw[color=blue,line width=0.5pt]
+ \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t};
+ \foreach \y in {\ymin,...,\ymax}{
+ \fill[color=blue]
+ \punkt{\x}{\y}{\t}
+ circle[radius=0.06];
+ }
+ }
+ \foreach \y in {\ymin,...,\ymax}{
+ \draw[color=blue,line width=0.5pt]
+ \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t};
+ }
+ }
+}
+
+\begin{scope}
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\ifthenelse{\boolean{presentation}}{
+ \foreach \T in {1,...,\N}{
+ \bildnetz{\T}
+ }
+}{
+ \bildnetz{\N}
+}
+\end{scope}
+
+\uncover<\N->{
+\begin{scope}[yshift=-2.5cm]
+\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8);
+\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8);
+\node at (0,0) {$\displaystyle W=\begin{pmatrix}0&2\\1&0\end{pmatrix}$};
+\end{scope}
+}
+
+\node at (0,-3.7) {$\alpha^2 = 2$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+
+\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\varphi)$};
+\node[color=blue] at (3.5,3.2) [above left] {$\varphi\mathbb{Z}(\varphi)$};
+
+\pgfmathparse{(sqrt(5)+1)/2}
+\xdef\a{\pgfmathresult}
+\pgfmathparse{-int(3.3/\a)}
+\xdef\ymin{\pgfmathresult}
+\pgfmathparse{int(3.3/\a)}
+\xdef\ymax{\pgfmathresult}
+\def\punkt#1#2#3{
+ ({(1-(#3))*(#1)+(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1+#2))*\a})
+}
+
+\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\varphi$}];
+
+\foreach \x in {-3,...,3}{
+ \draw[color=red,line width=0.5pt]
+ \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0};
+ \foreach \y in {\ymin,...,\ymax}{
+ \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08];
+ }
+}
+\foreach \y in {\ymin,...,\ymax}{
+ \draw[color=red,line width=0.5pt]
+ \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0};
+}
+
+\def\bildnetz#1{
+ \pgfmathparse{(#1-1)/(\N-1)}
+ \xdef\t{\pgfmathresult}
+ \only<#1>{
+ \uncover<2->{
+ \draw[->,color=blue,line width=1.4pt]
+ (0,\a) -- \punkt{0}{1}{\t};
+ \draw[->,color=blue,line width=1.4pt]
+ (1,0) -- \punkt{1}{0}{\t};
+ }
+ \foreach \x in {-3,...,3}{
+ \draw[color=blue,line width=0.5pt]
+ \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t};
+ \foreach \y in {\ymin,...,\ymax}{
+ \fill[color=blue] \punkt{\x}{\y}{\t}
+ circle[radius=0.06];
+ }
+ }
+ \foreach \y in {\ymin,...,\ymax}{
+ \draw[color=blue,line width=0.5pt]
+ \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t};
+ }
+ }
+}
+
+\begin{scope}
+
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\ifthenelse{\boolean{presentation}}{
+ \foreach \T in {1,...,\N}{
+ \bildnetz{\T}
+ }
+}{
+ \bildnetz{\N}
+}
+\end{scope}
+
+\uncover<\N->{
+\begin{scope}[yshift=-2.5cm]
+\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8);
+\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8);
+\node at (0,0) {$\displaystyle \Phi=\begin{pmatrix}0&1\\1&1\end{pmatrix}$};
+\end{scope}
+}
+
+\node at (0,-3.7) {$\alpha^2 = \alpha + 1$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/phi.tex b/vorlesungen/slides/3/phi.tex
new file mode 100644
index 0000000..ee0814c
--- /dev/null
+++ b/vorlesungen/slides/3/phi.tex
@@ -0,0 +1,85 @@
+%
+% phi.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\mathbb{Q}(\varphi)=\mathbb{Q}[X]/(X^2-X-1)$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Der Ring $\mathbb{Z}(\varphi)$}
+$\mathbb{Z}(\varphi)$ als Teilrung:
+{\color{blue}
+\[
+R=\{a+b\varphi\;|\; a,b\in\mathbb{Z}\}
+\]}%
+\uncover<2->{$\varphi\not\in\mathbb{Q}$}\uncover<3->{
+$\Rightarrow$
+$1$ und $\varphi$ sind inkommensurabel}\uncover<4->{
+$\Rightarrow$
+$R$ dicht in $\mathbb{R}$}
+\end{block}
+\uncover<5->{%
+\begin{block}{Algebraische Konstruktion}
+\uncover<8->{%
+Das Polynom $X^2-X-1$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$}
+\[
+\uncover<8->{\mathbb{Q}[X]/(X^2-X-1)
+=}
+{\color{red}\{a+b\varphi\;|\;a,b\in\mathbb{Z}\}}
+\]\uncover<7->{%
+mit der Rechenregel: $X^2=X+1$}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Körper}
+$\mathbb{Q}(\varphi) = \mathbb{Q}[X]/(X^2+X+1)$
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.92]
+\begin{scope}
+\pgfmathparse{(sqrt(5)-1))/2}
+\xdef\gphi{\pgfmathresult}
+\clip (-3.2,-3.2) rectangle (3.2,3.2);
+\foreach \x in {-10,...,10}{
+ \pgfmathparse{int(\x/\gphi)-10}
+ \xdef\s{\pgfmathresult}
+ \pgfmathparse{int(\x/\gphi)+10}
+ \xdef\t{\pgfmathresult}
+ \foreach \y in {\s,...,\t}{
+ \uncover<4->{
+ \fill[color=blue] ({\x-\y*\gphi},0)
+ circle[radius=0.05];
+ }
+ \uncover<6->{
+ \draw[color=blue,line width=0.1pt]
+ ({\x-\y*\gphi-3.2},3.2)
+ --
+ ({\x-\y*\gphi+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 {-5,...,5}{
+ \fill[color=red]
+ ({\x},{\y*\gphi}) circle[radius=0.08];
+ }
+ }
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex
index 48cc210..d20bfc4 100644
--- a/vorlesungen/slides/3/wurzel2.tex
+++ b/vorlesungen/slides/3/wurzel2.tex
@@ -33,6 +33,10 @@ Das Polynom $X^2-2$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$}
\]\uncover<7->{%
mit Rechenregel: $X^2=2$}
\end{block}}
+\uncover<9->{%
+\begin{block}{Körper}
+$\mathbb{Q}(\sqrt{2}) = \mathbb{Q}[X]/(X^2-2)$
+\end{block}}
\end{column}
\begin{column}{0.48\textwidth}
\begin{center}
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 276b978..c482bee 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -4,5 +4,7 @@
% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
%
%\folie{3/wurzel2.tex}
-\folie{3/nichthauptideal.tex}
-\folie{3/nichthauptideal2.tex}
+%\folie{3/phi.tex}
+%\folie{3/nichthauptideal.tex}
+%\folie{3/nichthauptideal2.tex}
+\folie{3/multiplikation.tex}
generated by cgit v1.2.1 (git 2.18.0) at 2024-11-29 15:18:19 +0100