diff options
Diffstat (limited to 'vorlesungen/slides/3')
| -rw-r--r-- | vorlesungen/slides/3/Makefile.inc | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/3/chapter.tex | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/3/idealverband.tex | 78 | ||||
| -rw-r--r-- | vorlesungen/slides/3/maximalideal.tex | 64 | ||||
| -rw-r--r-- | vorlesungen/slides/3/wurzel2.tex | 79 |
5 files changed, 227 insertions, 0 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} |