diff options
Diffstat (limited to 'vorlesungen/slides/3')
| -rw-r--r-- | vorlesungen/slides/3/Makefile.inc | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/3/chapter.tex | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/3/multiplikation.tex | 180 | ||||
| -rw-r--r-- | vorlesungen/slides/3/phi.tex | 85 | ||||
| -rw-r--r-- | vorlesungen/slides/3/wurzel2.tex | 4 |
5 files changed, 273 insertions, 0 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} |