Merge pull request #1 from AndreasFMueller/master

update

48 files changed, 2321 insertions, 0 deletions

diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc
new file mode 100644
index 0000000..442bd15
--- /dev/null
+++ b/vorlesungen/slides/3/Makefile.inc
@@ -0,0 +1,37 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter3 =								\
+	../slides/3/motivation.tex					\
+	../slides/3/inverse.tex						\
+	../slides/3/polynome.tex					\
+	../slides/3/division.tex					\
+	../slides/3/division2.tex					\
+	../slides/3/ringstruktur.tex					\
+	../slides/3/teilbarkeit.tex					\
+	../slides/3/ideal.tex						\
+	../slides/3/nichthauptideal.tex					\
+	../slides/3/nichthauptideal2.tex				\
+	../slides/3/idealverband.tex					\
+	../slides/3/maximalideal.tex					\
+	../slides/3/quotientenring.tex					\
+	../slides/3/faktorisierung.tex					\
+	../slides/3/faktorzerlegung.tex					\
+	../slides/3/einsetzen.tex					\
+	../slides/3/maximalergrad.tex					\
+	../slides/3/minimalbeispiel.tex					\
+	../slides/3/fibonacci.tex					\
+	../slides/3/minimalpolynom.tex					\
+	../slides/3/drehmatrix.tex					\
+	../slides/3/drehfaktorisierung.tex				\
+	../slides/3/operatoren.tex					\
+	../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/adjalgebra.tex b/vorlesungen/slides/3/adjalgebra.tex
new file mode 100644
index 0000000..e65b621
--- /dev/null
+++ b/vorlesungen/slides/3/adjalgebra.tex
@@ -0,0 +1,43 @@
+%
+% adjalgebra.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Adjunktion einer Nullstelle, abstrakt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom.
+
+\uncover<2->{%
+\begin{block}{Existenz}
+Es gibt ein ``Objekt'' $\alpha$ mit
+\(
+m(\alpha) = 0
+\)
+\end{block}}
+
+\uncover<3->{%
+\begin{block}{Körpererweiterung}
+Der kleinste Körper, der $\Bbbk$ und $\alpha$ enthält ist
+\[
+\Bbbk(\alpha)
+=
+\left
+\{ p(\alpha)
+\;\left|\;
+\begin{minipage}{8cm}\raggedright
+$p\in\Bbbk[X]$ ein Polynom vom Grad
+$\deg p<\deg m$
+\end{minipage}
+\right.
+\right\}
+\]
+\uncover<4->{Das Polynom $m$ definiert, wie mit $\alpha$ gerechnet werden
+muss:
+\[
+\alpha^n = -m_0-m_1\alpha-m_2\alpha^2 - \dots - m_{n-1}\alpha^{n-1}
+\]}
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/adjunktion.tex b/vorlesungen/slides/3/adjunktion.tex
new file mode 100644
index 0000000..a974a76
--- /dev/null
+++ b/vorlesungen/slides/3/adjunktion.tex
@@ -0,0 +1,35 @@
+%
+% adjunktion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Adjunktion einer Nullstelle von $m(X)$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom.
+\uncover<2->{%
+\[
+X^n = -m_{n-1}X^{n-1} - \dots - m_1X - m_0
+\]
+}%
+\uncover<3->{%
+Nullstelle $W$ als Operator betrachten:
+\[
+W = \begin{pmatrix}
+     0&     0&     0&\dots &     0&   -m_0\\
+     1&     0&     0&\dots &     0&   -m_1\\
+     0&     1&     0&\dots &     0&   -m_2\\
+     0&     0&     1&\dots &     0&   -m_3\\
+\vdots&\vdots&\vdots&\ddots&\vdots& \vdots\\
+     0&     0&     0&\dots &     1&-m_{n-1}
+\end{pmatrix}
+\]}
+\uncover<4->{%
+Man kann nachrechnen, dass immer $m(W)=0$.
+}
+\medskip
+
+\uncover<5->{$\Rightarrow \Bbbk(W) = \{p(W)\;|\;p\in\Bbbk[X], \deg p<\deg m\}$
+ist ein Körper, in dem $m(X)$ faktorisiert werden kann $m(X) = (X-W)q(X)$.}
+\end{frame}
diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex
new file mode 100644
index 0000000..3fbc3fd
--- /dev/null
+++ b/vorlesungen/slides/3/chapter.tex
@@ -0,0 +1,33 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{3/motivation.tex}
+\folie{3/inverse.tex}
+\folie{3/polynome.tex}
+\folie{3/division.tex}
+\folie{3/division2.tex}
+\folie{3/ringstruktur.tex}
+\folie{3/teilbarkeit.tex}
+\folie{3/ideal.tex}
+\folie{3/nichthauptideal.tex}
+\folie{3/nichthauptideal2.tex}
+\folie{3/maximalideal.tex}
+\folie{3/idealverband.tex}
+\folie{3/quotientenring.tex}
+\folie{3/faktorisierung.tex}
+\folie{3/faktorzerlegung.tex}
+\folie{3/einsetzen.tex}
+\folie{3/maximalergrad.tex}
+\folie{3/minimalbeispiel.tex}
+\folie{3/fibonacci.tex}
+\folie{3/minimalpolynom.tex}
+\folie{3/drehmatrix.tex}
+\folie{3/drehfaktorisierung.tex}
+\folie{3/operatoren.tex}
+\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/division.tex b/vorlesungen/slides/3/division.tex
new file mode 100644
index 0000000..94df27b
--- /dev/null
+++ b/vorlesungen/slides/3/division.tex
@@ -0,0 +1,32 @@
+%
+% division.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynomdivision}
+\begin{block}{Aufgabe}
+Finde Quotient und Rest für
+$a= X^4- X^3-7X^2+ X+6\in\mathbb{Z}[X]$
+und
+$b= X^2+X+1\in\mathbb{Z}[X]$
+\end{block}
+\uncover<2->{%
+\begin{block}{Lösung}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcrcrcrcr}
+\llap{$($}X^4&-& X^3&-&7X^2&+& X&+&6\rlap{$)$}&\;\mathstrut:\mathstrut&(X^2&+&X&+&1)&=&\uncover<3->{X^2}&\uncover<7->{-&2X}&\uncover<11->{-&6}=q\\
+\uncover<4->{\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}}& &  & & & &   & & & & & &   & &  & & \\ 
+   &\uncover<5->{-&2X^3&-&8X^2}&\uncover<6->{+& X}& & & &   & & & & & &   & &  & & \\
+   &\uncover<8->{\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}}& & & &   & & & & & &   & &  & & \\ 
+   & &    &\uncover<9->{-&6X^2&+&3X}&\uncover<10->{+&6}& &   & & & & & &   & &  & & \\
+   & &    &\uncover<12->{\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}}& &   & & & & & &   & &  & & \\ 
+   & &    & &    & &\uncover<13->{9X&+&12\rlap{$\mathstrut=r$}}& &   & & & & & &   & &  & &
+\end{array}
+\]
+\uncover<14->{
+Funktioniert weil $b$ normiert ist!
+}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/division2.tex b/vorlesungen/slides/3/division2.tex
new file mode 100644
index 0000000..0602598
--- /dev/null
+++ b/vorlesungen/slides/3/division2.tex
@@ -0,0 +1,34 @@
+%
+% division2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Division in $\Bbbk[X]$}
+\vspace{-5pt}
+\begin{block}{Aufgabe}
+Finde Quotienten und Rest der Polynome
+$a(X) = X^4-X^3-7X^2+X+6$
+und
+$b(X) = 2X^2+X+1$
+\end{block}
+\uncover<2->{%
+\begin{block}{Lösung}
+\vspace{-15pt}
+\[
+\arraycolsep=1.4pt
+\renewcommand{\arraystretch}{1.2}
+\begin{array}{rcrcrcrcrcrcrcrcrcrcr}
+\llap{$($}X^4&-&       X^3&-&         7X^2&+&          X&+&           6\rlap{$)$}&\mathstrut\;:\mathstrut&(2X^2&+&X&+&1)&=&\uncover<3->{\frac12X^2}&\uncover<7->{-&\frac34X}&\uncover<11->{-\frac{27}{8}} = q\\
+\uncover<4->{\llap{$-($}X^4&+&\frac12X^3&+&   \frac12X^2\rlap{$)$}}& &           & &            & &    & & & & & &          & &        &             \\
+   &\uncover<5->{-&\frac32X^3&-&\frac{15}2X^2}&\uncover<6->{+&          X}& &            & &    & & & & & &          & &        &             \\
+   &\uncover<8->{\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}}& &            & &    & & & & & &          & &        &             \\
+   & &          &\uncover<9->{-&\frac{27}4X^2&+&\frac{ 7}4X}&\uncover<10->{+&           6}& &    & & & & & &          & &        &             \\
+   & &          &\uncover<12->{\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}}& &    & & & & & &          & &        &             \\
+   & &          & &             & &\uncover<13->{\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}}& &    & & & & & &          & &        &             \\
+\end{array}
+\]
+Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/drehfaktorisierung.tex b/vorlesungen/slides/3/drehfaktorisierung.tex
new file mode 100644
index 0000000..64418d5
--- /dev/null
+++ b/vorlesungen/slides/3/drehfaktorisierung.tex
@@ -0,0 +1,75 @@
+%
+% drehfaktorisierung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{4pt}
+\setlength{\belowdisplayskip}{4pt}
+\frametitle{Faktorisierung von $X^2+X+1$}
+\vspace{-3pt}
+$X^2+X+1$ kann faktorisiert werden, wenn man $i\sqrt{3}$
+hinzufügt:
+\uncover<2->{%
+\[
+\biggl(X+\frac12+\frac{i\sqrt{3}}2\biggr)
+\biggl(X+\frac12-\frac{i\sqrt{3}}2\biggr)
+=
+X^2+X+\frac14
++
+\frac34
+\uncover<3->{=
+X^2+X+1}
+\]}
+\vspace{-10pt}
+\uncover<4->{%
+\begin{block}{Was ist $i\sqrt{3}$?}
+Matrix mit Minimalpolynom $X^2+3$:
+\[
+W=\begin{pmatrix}0&-3\\1&0\end{pmatrix}
+\uncover<5->{%
+\qquad\Rightarrow\qquad
+W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I}
+\uncover<6->{%
+\qquad\Rightarrow\qquad
+W^2+3I=0}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{block}{Faktorisierung von $X^2+X+1$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<8->{B_\pm
+&=
+-\frac12I\pm\frac12W}
+&
+&\uncover<10->{\Rightarrow
+&
+(X+B_+)(X+B_-)}
+&\uncover<11->{=
+(X+\frac12I+\frac12W)
+(X+\frac12I-\frac12W)}
+\\
+&\uncover<9->{=
+\smash{
+{\textstyle\begin{pmatrix}-\frac12&-\frac32\\\frac12&-\frac12\end{pmatrix}}
+}}
+&
+&
+&
+&\uncover<12->{=
+X^2+X + \frac14I - \frac14W^2}
+\\
+&
+&
+&%\Rightarrow
+&
+&\uncover<13->{=
+X^2+X + \frac14I + \frac34I}
+\uncover<14->{=
+X^2+X+I}
+\end{align*}
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/drehmatrix.tex b/vorlesungen/slides/3/drehmatrix.tex
new file mode 100644
index 0000000..9e5eb65
--- /dev/null
+++ b/vorlesungen/slides/3/drehmatrix.tex
@@ -0,0 +1,66 @@
+%
+% drehmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Analyse einer Drehung um $120^\circ$}
+$D$ eine Drehung des $\mathbb{R}^3$ um $120^\circ$
+\begin{enumerate}
+\item<2->
+Drehwinkel = $120^\circ\quad\Rightarrow\quad D^3 = I$
+\uncover<3->{
+$\quad\Rightarrow\quad \chi_D(X)=X^3-1$
+}
+\item<4->
+$m_D(X)=X^3-1$
+\item<5->
+$m_D$ ist nicht irreduzibel, weil $m_D(1)=0$:
+$
+m_D(X) = (X-1)(X^2+X+1)
+$
+\item<6->
+Welche Matrix hat $X^2+X+1$ als Minimalpolynom?
+\uncover<7->{%
+\[
+\arraycolsep=1.4pt
+W
+=
+\biggl(\begin{array}{cc}
+-\frac12          & -\frac{\sqrt{3}}2 \\
+ \frac{\sqrt{3}}2 & -\frac12
+\end{array}\biggr)
+\quad\Rightarrow\quad
+W^2+W+I
+=
+\biggl(\begin{array}{cc}
+-\frac12          & -\frac{\sqrt{3}}2 \\
+ \frac{\sqrt{3}}2 & -\frac12
+\end{array}\biggr)
++
+\biggl(\begin{array}{cc}
+-\frac12          & \frac{\sqrt{3}}2 \\
+ -\frac{\sqrt{3}}2 & -\frac12
+\end{array}\biggr)
++
+\biggl(\begin{array}{cc}
+1&0\\0&1
+\end{array}\biggr)
+=0
+\]}
+\item<8-> In einer geeigneten Basis hat $D$ die Form
+\[
+D=\begin{pmatrix}
+1&0&0\\
+0&-\frac12 & -\frac{\sqrt{3}}2 \\
+0&\frac{\sqrt{3}}2 & -\frac12
+\end{pmatrix}
+\uncover<9->{=
+\begin{pmatrix}
+1&0&0\\
+0&\cos 120^\circ & -\sin 120^\circ\\
+0&\sin 120^\circ & \cos 120^\circ
+\end{pmatrix}}
+\]
+\end{enumerate}
+\end{frame}
diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex
new file mode 100644
index 0000000..7f54abb
--- /dev/null
+++ b/vorlesungen/slides/3/einsetzen.tex
@@ -0,0 +1,54 @@
+%
+% einsetzen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Matrix in ein Polynom einsetzen}
+\vspace{-10pt}
+\[
+\begin{array}{rcrcrcrcrcrcr}
+p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\
+\uncover<2->{\bigg\downarrow\hspace*{4pt}}  & &
+\uncover<3->{\bigg\downarrow\hspace*{4pt}}  & &
+\uncover<4->{\bigg\downarrow\hspace*{10pt}} & &     & &
+\uncover<5->{\bigg\downarrow\hspace*{4pt}}  & &
+\uncover<6->{\bigg\downarrow\hspace*{2pt}}  & &
+\uncover<7->{\bigg\downarrow\hspace*{0pt}}  \\
+\uncover<2->{p(A)}&\uncover<3->{=&a_nA^n}&\uncover<4->{+&a_{n-1}A^{n-1}}&\uncover<5->{+&\dots&+&a_2A^2}&\uncover<6->{+&a_1A}&\uncover<7->{+&a_0         I}
+\end{array}
+\]
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Nilpotente Matrizen}
+$p(X) = (X-a)^n$
+\[
+\uncover<9->{p(A) = 0}
+\uncover<10->{
+\quad\Rightarrow\quad
+\text{$A-aI$ ist nilpotent}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Eigenwerte}
+$p(X) = (X-\lambda_1)(X-\lambda_2)$,\\
+$A$ eine $2\times 2$-Matrix
+\[
+\uncover<12->{p(A)=0}
+\uncover<13->{\quad\Rightarrow\quad
+\left\{
+\begin{aligned}
+&\text{$A-\lambda_1I$ ist singulär}\\
+&\text{$A-\lambda_2I$ ist singulär}
+\end{aligned}
+\right.}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/faktorisierung.tex b/vorlesungen/slides/3/faktorisierung.tex
new file mode 100644
index 0000000..b4ea1d5
--- /dev/null
+++ b/vorlesungen/slides/3/faktorisierung.tex
@@ -0,0 +1,47 @@
+%
+% faktorisierung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Faktorisierung}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Primzahlen\strut}
+Eine Zahl $p\in\mathbb{Z}$, $p>1$ heisst Primzahl, wenn sie nicht als Produkt
+$p=ab$ mit $a,b\in\mathbb{Z},a>1, b>1$ geschrieben werden kann.
+\begin{align*}
+\uncover<2->{p&=7}
+\\
+\uncover<3->{2021 &= 43 \cdot 47}
+\\
+\uncover<4->{2048 &= 2^{11}}
+\\
+\uncover<5->{4095667&=2021\cdot 2027}
+\\
+\uncover<6->{p&=43, 47, 1291, 2017, 2027}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Irreduzible Polynome in $\mathbb{Q}[X]$}
+Ein Polynome $p\in\mathbb{Q}[X]$, $\deg p>0$ wenn es nicht als Produkt
+$p=ab$ mit $a,b\in\mathbb{Q}[X]$, $\deg a>0$, $\deg b>0$ geschrieben
+werden kann.
+\begin{align*}
+\uncover<8->{p&=X-9}
+\\
+\uncover<9->{X^2-1&= (X+1)(X-1)}
+\\
+\uncover<10->{X^2-2&\text{\; irreduzibel}}
+\\
+\uncover<11->{X^2-2&=(X-\sqrt{2})(X+\sqrt{2})}
+\end{align*}
+\uncover<12->{%
+aber: $X\pm\sqrt{2}\not\in\mathbb{Q}[X]$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/faktorzerlegung.tex b/vorlesungen/slides/3/faktorzerlegung.tex
new file mode 100644
index 0000000..eb44cf3
--- /dev/null
+++ b/vorlesungen/slides/3/faktorzerlegung.tex
@@ -0,0 +1,62 @@
+%
+% faktorzerlegung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Faktorzerlegung}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{in $\mathbb{Z}$}
+Jede Zahl kann eindeutig in Primfaktoren zerlegt werden:
+\[
+z = p_1^{n_1}\cdot p_2^{n_2} \cdot\dots\cdot p_k^{n_k}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{in $\mathbb{Q}[X]$}
+Jedes Polynom $p\in\mathbb{Q}[X]$
+kann eindeutig faktorisiert werden in irreduzible, normierte Polynome
+\[
+p
+=
+a_n
+p_1^{n_1}
+\cdot
+p_2^{n_2}
+\cdot
+\dots
+\cdot
+p_k^{n_k}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Polynomfaktorisierung hängt vom Koeffizientenring ab}
+Ist $X^2-2$ irreduzibel?
+\vspace{-5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{in $\mathbb{Q}[X]$}
+\[
+X^2-2\quad\text{ist irreduzibel}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{in $\mathbb{R}[X]$}
+\[
+X^2-2 = (X-\sqrt{2})(X+\sqrt{2})
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex
new file mode 100644
index 0000000..3d01020
--- /dev/null
+++ b/vorlesungen/slides/3/fibonacci.tex
@@ -0,0 +1,71 @@
+%
+% fibonacci.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+
+\begin{frame}[t]
+\frametitle{Fibonacci}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{block}{Fibonacci-Rekursion}
+$x_i$ Fibonacci-Zahlen\uncover<2->{, d.~h.~$x_{n+1\mathstrut}=x_{n\mathstrut}+x_{n-1\mathstrut}$}
+\[
+\uncover<3->{
+v_n
+=
+\begin{pmatrix}
+x_{n+1}\\
+x_n
+\end{pmatrix}}
+\uncover<4->{
+\quad\Rightarrow\quad
+v_n = 
+\underbrace{
+\begin{pmatrix}
+1&1\\
+1&0
+\end{pmatrix}
+}_{\displaystyle=\Phi}
+v_{n-1}}
+\uncover<5->{
+\quad\Rightarrow\quad
+v_n
+=
+\Phi^n
+v_0}\uncover<6->{,
+\;
+v_0 = \begin{pmatrix} 1\\0\end{pmatrix}}
+\]
+\end{block}
+\vspace{-5pt}
+\uncover<7->{%
+\begin{block}{Rekursionsformel für $\Phi$}
+\vspace{-12pt}
+\begin{align*}
+v_{n}&=v_{n-1}+v_{n-2}
+&&\uncover<8->{\Rightarrow&
+\Phi^n v_0 &= \Phi^{n-1} v_0 + \Phi^{n-2}v_0}
+&&\uncover<9->{\Rightarrow&
+\Phi^{n-2}(\Phi^2-\Phi-I)v_0&=0}
+\\
+\end{align*}
+\vspace{-22pt}%
+
+\uncover<10->{$\Phi$ ist $\chi_\Phi(X)=m_\Phi(X) = X^2-X-1$, irreduzibel}
+\end{block}}
+
+\uncover<11->{%
+\begin{block}{Faktorisierung}
+\vspace{-12pt}
+\[
+(X-\Phi)(X-(I-\Phi))
+\uncover<12->{=
+X^2-X +\Phi(I-\Phi)}
+\uncover<13->{=
+X^2-X -(\underbrace{\Phi^2-\Phi}_{\displaystyle=I})
+}
+\]
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/ideal.tex b/vorlesungen/slides/3/ideal.tex
new file mode 100644
index 0000000..f7f432e
--- /dev/null
+++ b/vorlesungen/slides/3/ideal.tex
@@ -0,0 +1,63 @@
+%
+% ideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ideal}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Voraussetzungen}
+$R$ ein Ring, $r\in R$
+\end{block}
+\uncover<2->{%
+\begin{block}{Vielfache\uncover<4->{ = Hauptideal}}
+Die Menge aller Elemente, die durch $r$ teilbar sind\uncover<3->{:
+\[
+(r)=rR
+\]}
+\uncover<4->{heisst {\em Hauptideal}}
+\end{block}}
+\uncover<5->{%
+\begin{block}{Ideal}
+$I\subset R$ mit
+\(RI\subset I\), \(I+I\subset I\)
+\end{block}}
+\uncover<6->{%
+\begin{block}{Hauptidealring}
+Jedes Ideal von $R$ ist ein Hauptideal
+\\
+\uncover<7->{{\usebeamercolor[fg]{title}Beispiele:}
+$\mathbb{Z}$,
+$\Bbbk[X]$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Grösster gemeinsamer Teiler}
+$a,b\in R$
+\begin{align*}
+\uncover<9->{(a) + (b)
+&= aR + bR}
+\intertext{\uncover<10->{ist eine Ideal }\uncover<11->{$\Rightarrow$ ein Hauptideal}}
+&\uncover<12->{= cR}\uncover<13->{ = \operatorname{ggT}(a,b)R}
+\end{align*}
+\uncover<14->{Existenz des $\operatorname{ggT}(a,b)$ ist eine
+gemeinsame Eigenschaft}
+\end{block}}
+\uncover<15->{%
+\begin{block}{Allgemein}
+\begin{itemize}
+\item<16->
+Alle euklidischen Ringe sind Hauptidealringe
+\item<17->
+Alle solchen Ringe verwenden den gleichen Algorithmus
+für $\operatorname{ggT}(a,b)$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
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/images/Makefile b/vorlesungen/slides/3/images/Makefile
new file mode 100644
index 0000000..e338fcf
--- /dev/null
+++ b/vorlesungen/slides/3/images/Makefile
@@ -0,0 +1,55 @@
+#
+# Makefile -- build images
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	hauptideal.jpg nichthauptideal.jpg hauptideal2.jpg hauptidealX.jpg \
+	hauptidealR.jpg hauptidealXR.jpg ring.jpg
+
+ring.png:	ring.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Oring.png ring.pov
+ring.jpg:	ring.png
+	convert ring.png -density 300 -units PixelsPerInch ring.jpg
+
+hauptideal.png:	hauptideal.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Ohauptideal.png hauptideal.pov
+hauptideal.jpg:	hauptideal.png
+	convert hauptideal.png -density 300 -units PixelsPerInch \
+		hauptideal.jpg
+
+hauptidealR.png:	hauptidealR.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealR.png hauptidealR.pov
+hauptidealR.jpg:	hauptidealR.png
+	convert hauptidealR.png -density 300 -units PixelsPerInch \
+		hauptidealR.jpg
+
+hauptideal2.png:	hauptideal2.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Ohauptideal2.png hauptideal2.pov
+hauptideal2.jpg:	hauptideal2.png
+	convert hauptideal2.png -density 300 -units PixelsPerInch \
+		hauptideal2.jpg
+
+hauptidealX.png:	hauptidealX.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealX.png hauptidealX.pov
+hauptidealX.jpg:	hauptidealX.png
+	convert hauptidealX.png -density 300 -units PixelsPerInch \
+		hauptidealX.jpg
+
+hauptidealXR.png:	hauptidealXR.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealXR.png hauptidealXR.pov
+hauptidealXR.jpg:	hauptidealXR.png
+	convert hauptidealXR.png -density 300 -units PixelsPerInch \
+		hauptidealXR.jpg
+
+nichthauptideal.png:	nichthauptideal.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Onichthauptideal.png nichthauptideal.pov
+nichthauptideal.jpg:	nichthauptideal.png
+	convert nichthauptideal.png -density 300 -units PixelsPerInch \
+		nichthauptideal.jpg
+
+ideal:	ideal.pov ideal.ini common.inc
+	rm -rf ideal
+	mkdir ideal
+	povray +A0.1 +W1920 +H1080 -Oideal/i.png ideal.ini
+
+
diff --git a/vorlesungen/slides/3/images/common.inc b/vorlesungen/slides/3/images/common.inc
new file mode 100644
index 0000000..36c4e6b
--- /dev/null
+++ b/vorlesungen/slides/3/images/common.inc
@@ -0,0 +1,277 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.40;
+#declare O = <0, 0, 0>;
+#declare at = 0.10;
+
+#declare Xunten = -10;
+#declare Xoben = 10;
+#declare Yunten = -8;
+#declare Yoben = 8;
+#declare Zunten = 0;
+#declare Zoben = 20;
+
+#declare phi0 = 2 * pi * 290 / 360;
+
+camera {
+	location <60 * cos(2*pi*T+phi0), 15, 60 * sin(2*pi*T+phi0) + 10>
+	look_at <0, -2, 10>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-14, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+light_source {
+	<41, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+arrow(< -12.0,   0.0,  0  >, < 12.0,  0.0,  0.0 >, at, Gray)
+arrow(<   0.0,   0.0, -2.0>, <  0.0,  0.0, 22.0 >, at, Gray)
+arrow(<   0.0, -10.0,  0  >, <  0.0, 10.0,  0.0 >, at, Gray)
+
+#macro kasten()
+	box { <-10.5,-8.5,-0.5>, <10.5,8.5,20.5> }
+#end
+
+#declare gruen = rgb<0.2,0.4,0.2>;
+#declare blau  = rgb<0.0,0.4,0.8>;
+#declare rot   = rgb<1.0,0.4,0.0>;
+
+#declare r = 0.4;
+
+#macro Zring()
+	union {
+		#declare X = Xunten;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X, Y, Z>, r }
+
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 1;
+		#end
+		pigment {
+			color rot
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Hauptideal()
+	union {
+		#declare A = Xunten;
+		#while (A <= Xoben + 0.5)
+			#declare B = Zunten;
+			#while (B <= Zoben + 0.5)
+				#declare Y = A + B;
+				#if ((Y >= Yunten - 0.5) & (Y <= Yoben + 0.5))
+					sphere { <A, Y, B>, r }
+				#end
+				#declare B = B + 1;
+			#end
+			#declare A = A + 1;
+		#end
+		pigment {
+			color blau
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro HauptidealR()
+	intersection {
+		kasten()
+		#declare n = vnormalize(< 1, -2, 1 >);
+		plane { n, 0.1 }
+		plane { -n, 0.1 }
+		pigment {
+			color blau
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Ideal2()
+	union {
+		#declare X = Xunten;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X, Y, Z>, r }
+					#declare Z = Z + 2;
+				#end
+				#declare Y = Y + 2;
+			#end
+			#declare X = X + 2;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro IdealX()
+	union {
+		#declare Y = Yunten;
+		#while (Y <= Yoben + 0.5)
+			#declare Z = Zunten;
+			#while (Z <= Zoben + 0.5)
+				sphere { <0, Y, Z>, r }
+				#declare Z = Z + 1;
+			#end
+			#declare Y = Y + 1;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro IdealXR()
+	intersection {
+		kasten()
+		plane { <1,0,0>, 0.1 }
+		plane { <-1,0,0>, 0.1 }
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Nichthauptideal()
+	union {
+		#declare X = Xunten/2;
+		#while (X <= Xoben/2 + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = 0;
+				#while (Z <= Zoben + 0.5)
+					sphere { <2*X,Y,Z>, r }
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 1;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro NichthauptidealKomplement()
+	union {
+		#declare X = Xunten + 1;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X,Y,Z>, r }
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 2;
+		#end
+		pigment {
+			color rot
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+
+
+
+
+
+
diff --git a/vorlesungen/slides/3/images/hauptideal.jpg b/vorlesungen/slides/3/images/hauptideal.jpg
new file mode 100644
index 0000000..769f53c
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal.jpg
diff --git a/vorlesungen/slides/3/images/hauptideal.pov b/vorlesungen/slides/3/images/hauptideal.pov
new file mode 100644
index 0000000..a934e57
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Hauptideal()
+
diff --git a/vorlesungen/slides/3/images/hauptideal2.jpg b/vorlesungen/slides/3/images/hauptideal2.jpg
new file mode 100644
index 0000000..51823f3
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal2.jpg
diff --git a/vorlesungen/slides/3/images/hauptideal2.pov b/vorlesungen/slides/3/images/hauptideal2.pov
new file mode 100644
index 0000000..9da5a1a
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal2.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Ideal2()
+
diff --git a/vorlesungen/slides/3/images/hauptidealR.jpg b/vorlesungen/slides/3/images/hauptidealR.jpg
new file mode 100644
index 0000000..fae5840
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealR.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealR.pov b/vorlesungen/slides/3/images/hauptidealR.pov
new file mode 100644
index 0000000..330e523
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealR.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealR.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+HauptidealR()
+
diff --git a/vorlesungen/slides/3/images/hauptidealX.jpg b/vorlesungen/slides/3/images/hauptidealX.jpg
new file mode 100644
index 0000000..f9b4540
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealX.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealX.pov b/vorlesungen/slides/3/images/hauptidealX.pov
new file mode 100644
index 0000000..d0045f9
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealX.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealX.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+IdealX()
+
diff --git a/vorlesungen/slides/3/images/hauptidealXR.jpg b/vorlesungen/slides/3/images/hauptidealXR.jpg
new file mode 100644
index 0000000..d8906c8
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealXR.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealXR.pov b/vorlesungen/slides/3/images/hauptidealXR.pov
new file mode 100644
index 0000000..5daa3e6
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealXR.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealXR.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+IdealXR()
+
diff --git a/vorlesungen/slides/3/images/ideal.ini b/vorlesungen/slides/3/images/ideal.ini
new file mode 100644
index 0000000..66aa191
--- /dev/null
+++ b/vorlesungen/slides/3/images/ideal.ini
@@ -0,0 +1,7 @@
+Input_File_Name=ideal.pov
+Initial_Frame=0
+Final_Frame=2500
+Initial_Clock=0
+Final_Clock=5
+Cyclic_Animation=off
+Pause_when_Done=off
diff --git a/vorlesungen/slides/3/images/ideal.pov b/vorlesungen/slides/3/images/ideal.pov
new file mode 100644
index 0000000..88afaf7
--- /dev/null
+++ b/vorlesungen/slides/3/images/ideal.pov
@@ -0,0 +1,26 @@
+//
+// ideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = clock;
+#include "common.inc"
+
+#if (T < 1)
+Zring()
+#else
+    #if (T < 2)
+	Hauptideal()
+    #else
+	#if (T < 3)
+	    Ideal2()
+	#else
+	    #if (T < 4)
+		IdealX()
+	    #else
+		Nichthauptideal()
+		NichthauptidealKomplement()
+	    #end
+	#end
+    #end
+#end
diff --git a/vorlesungen/slides/3/images/nichthauptideal.jpg b/vorlesungen/slides/3/images/nichthauptideal.jpg
new file mode 100644
index 0000000..55858d0
--- /dev/null
+++ b/vorlesungen/slides/3/images/nichthauptideal.jpg
diff --git a/vorlesungen/slides/3/images/nichthauptideal.pov b/vorlesungen/slides/3/images/nichthauptideal.pov
new file mode 100644
index 0000000..72a6330
--- /dev/null
+++ b/vorlesungen/slides/3/images/nichthauptideal.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Nichthauptideal()
+NichthauptidealKomplement()
diff --git a/vorlesungen/slides/3/images/ring.jpg b/vorlesungen/slides/3/images/ring.jpg
new file mode 100644
index 0000000..27721b1
--- /dev/null
+++ b/vorlesungen/slides/3/images/ring.jpg
diff --git a/vorlesungen/slides/3/images/ring.pov b/vorlesungen/slides/3/images/ring.pov
new file mode 100644
index 0000000..f854335
--- /dev/null
+++ b/vorlesungen/slides/3/images/ring.pov
@@ -0,0 +1,10 @@
+//
+// ring.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Zring()
+
diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex
new file mode 100644
index 0000000..4ad22d2
--- /dev/null
+++ b/vorlesungen/slides/3/inverse.tex
@@ -0,0 +1,89 @@
+%
+% inverse.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Inverse Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.24\textwidth}
+\begin{block}{Imaginäre Einheit}
+\vspace{-15pt}
+\begin{align*}
+J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix}
+\\
+0&=
+J^2 + I
+\\
+0&=
+J+J^{-1}
+\\
+J^{-1}&=-J
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.25\textwidth}
+\uncover<2->{%
+\begin{block}{Wurzel $\sqrt{2}$}
+\vspace{-15pt}
+\begin{align*}
+W&=\begin{pmatrix}0&2\\1&0\end{pmatrix}
+\\
+0 &= X^2-2
+\\
+0 &= W-2W^{-1}
+\\
+W^{-1}&=\frac12 W
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.41\textwidth}
+\uncover<3->{%
+\begin{block}{Drehmatrix}
+\vspace{-15pt}
+\begin{align*}
+D&=\begin{pmatrix}
+\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\
+\sin \frac{\pi}{1291} &  \cos\frac{\pi}{1291}
+\end{pmatrix}
+\\
+0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\\
+0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{D-2\cos\frac{\pi\mathstrut}{1291\mathstrut}+D^{-1}}
+\\
+D^{-1}
+&= \only<-3>{-D^{1290}\phantom{+\frac{\mathstrut}{\mathstrut}}}%
+\only<4->{-D+2I\cos\frac{\pi\mathstrut}{1291\mathstrut}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-25pt}
+\uncover<5->{
+\begin{block}{3D-Beispiel}
+$p(x) = -x^3-5x^2+5x+1$
+\[
+A=
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}
+\quad\Rightarrow\quad
+A^{-1}
+=
+A^2+5A-5I
+=
+\begin{pmatrix*}[r]
+-1& 1&-1\\
+ 7&-9&10\\
+ 3&-4& 5
+\end{pmatrix*}
+\]
+\end{block}}
+\vspace{-10pt}
+
+\end{frame}
diff --git a/vorlesungen/slides/3/maximalergrad.tex b/vorlesungen/slides/3/maximalergrad.tex
new file mode 100644
index 0000000..d33ddc0
--- /dev/null
+++ b/vorlesungen/slides/3/maximalergrad.tex
@@ -0,0 +1,72 @@
+%
+% maximalergrad.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Jede Matrix hat eine Polynomrelation}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-5pt}
+\begin{block}{Dimension des Matrizenrings}
+Der Ring $M_{n}(\Bbbk)$ ist ein $n^2$-dimensionaler Vektorraum mit
+Basis
+{\tiny
+\begin{align*}
+&\uncover<2->{\begin{pmatrix}
+1&0&\dots&0\\
+0&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<3->{\begin{pmatrix}
+0&1&\dots&0\\
+0&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<4->{\dots}
+&
+&\uncover<5->{\begin{pmatrix}
+0&0&\dots&1\\
+0&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+\\
+&\uncover<6->{\begin{pmatrix}
+0&0&\dots&0\\
+1&0&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<7->{\begin{pmatrix}
+0&0&\dots&0\\
+0&1&\dots&0\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+&
+&\uncover<8->{\dots}
+&
+&\uncover<9->{\begin{pmatrix}
+0&0&\dots&0\\
+0&0&\dots&1\\
+\vdots&\vdots&\ddots&\vdots\\
+\end{pmatrix}}
+\end{align*}}
+\end{block}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Potenzen von $A$}
+Die $n^2+1$ Matrizen $I,A,A^2,\dots,A^{n^2-1},A^{n^2}$ müssen linear abhängig
+sein:
+\[
+\uncover<11->{
+a_0I+a_1A+a_2A^2+\dots+a_{n^2-1}A^{n^2-1}+a_{n^2}A^{n^2} = 0
+}
+\]
+\uncover<12->{d.~h.~$p(X) = a_0+a_1X+a_2X^2+\dots +a_{n^2-1}X^{n^2-1}+a_{n^2}A^{n^2}\in\Bbbk[X]$ ist ein Polynom mit $p(A)=0$.}
+\end{block}}
+\uncover<13->{%
+$\Rightarrow$ $A$ über die Eigenschaften (Faktorisierung) von $p$ studieren
+}
+\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/minimalbeispiel.tex b/vorlesungen/slides/3/minimalbeispiel.tex
new file mode 100644
index 0000000..f94cf8d
--- /dev/null
+++ b/vorlesungen/slides/3/minimalbeispiel.tex
@@ -0,0 +1,36 @@
+%
+% minimalbeispiel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Beispiel für $p(A)=0$}
+\begin{block}{Potenzen einer $2\times 2$-Matrix $A$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-10pt}
+\[
+I  ={\tiny\begin{pmatrix}  1 &  0 \\  0 &  1 \end{pmatrix}},\quad
+A  ={\tiny\begin{pmatrix}  3 &  2 \\ -1 & -2 \end{pmatrix}},\quad
+\uncover<2->{A^2={\tiny\begin{pmatrix}  7 &  2 \\ -1 &  2 \end{pmatrix}}}
+\uncover<3->{,\quad A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}}}
+\uncover<4->{,\quad A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 &  2 \end{pmatrix}}}
+\]
+\end{block}
+\vspace{-5pt}
+\uncover<5->{%
+\begin{block}{linear abhängig}
+Bereits die ersten $3$ sind linear abhängig:
+\[
+-4I - A + A^2
+=
+-4\begin{pmatrix}  1 &  0 \\  0 &  1 \end{pmatrix}
+-\begin{pmatrix}  3 &  2 \\ -1 & -2 \end{pmatrix}
++\begin{pmatrix}  7 &  2 \\ -1 &  2 \end{pmatrix}
+= 
+\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
+\]
+\uncover<6->{$p(X) = X^2 - X - 4 \in \mathbb{Q}[X]$ hat die Eigenschaft
+$p(A)=0$}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/minimalpolynom.tex b/vorlesungen/slides/3/minimalpolynom.tex
new file mode 100644
index 0000000..2b36a65
--- /dev/null
+++ b/vorlesungen/slides/3/minimalpolynom.tex
@@ -0,0 +1,30 @@
+%
+% minimalpolynom.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Minimalpolynom}
+\begin{block}{Definition}
+Zu jeder $n\times n$-Matrix $A$
+gibt es ein Polynom $m_A(X)\in\Bbbk[X]$ minimalen Grades $\deg m_A\le n^2$
+derart, dass $m_A(A)=0$.
+\end{block}
+\uncover<2->{%
+\begin{block}{Strategie}
+Das Minimalpolynom ist eine ``Invariante'' der Matrix $A$
+\end{block}}
+\uncover<3->{%
+\begin{block}{Satz von Cayley-Hamilton}
+Für jede $n\times n$-Matrix $A\in M_n(\Bbbk)$ gilt $\chi_A(A)=0$
+\uncover<4->{%
+\[
+\Downarrow
+\]
+Das Minimalpolynom $m_A\in \Bbbk[X]$ ist ein Teiler
+des charakteristischen Polynoms $\chi_A\in \Bbbk[X]$}
+\\
+\uncover<5->{$\Rightarrow $
+Faktorzerlegung on $\chi_A(X)$ ermitteln!}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex
new file mode 100644
index 0000000..048e6a2
--- /dev/null
+++ b/vorlesungen/slides/3/motivation.tex
@@ -0,0 +1,108 @@
+%
+% motivation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Motivation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.24\textwidth}
+\begin{block}{Imaginäre Einheit}
+\vspace{-15pt}
+\begin{align*}
+J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix}
+\\
+p(X) &= X^2 + 1
+\\
+p(J) &= J^2 + I = 0
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.25\textwidth}
+\uncover<2->{%
+\begin{block}{Wurzel $\sqrt{2}$}
+\vspace{-15pt}
+\begin{align*}
+W&=\begin{pmatrix}0&2\\1&0\end{pmatrix}
+\\
+p(X) &= X^2-2
+\\
+p(W) &= W^2-2I=0
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.41\textwidth}
+\uncover<3->{%
+\begin{block}{Drehmatrix}
+\vspace{-15pt}
+\begin{align*}
+D&=\begin{pmatrix}
+\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\
+\sin \frac{\pi}{1291} &  \cos\frac{\pi}{1291}
+\end{pmatrix}
+\\
+p(X)&=
+\ifthenelse{\boolean{presentation}}{\only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{X^2-2X\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\\
+p(D) &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{}
+\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-20pt}
+\uncover<5->{
+\begin{block}{3D-Beispiel}
+$p(x) = -x^3-5x^2+5x+1$
+\[
+\ifthenelse{\boolean{presentation}}{
+\only<5-8>{
+A=
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}}
+\only<6-8>{
+\quad\Rightarrow\quad}}{}
+\uncover<6->{
+-
+\only<-9>{A^3}\only<10->{
+\begin{pmatrix*}[r]
+-169&-35&35\\
+-185&-39&40\\
+ -45&-10&11
+\end{pmatrix*}}
+-5
+\only<-8>{A^2}\only<9->{
+\begin{pmatrix*}[r]
+29&6&-6\\
+32&6&-5\\
+ 8&1& 0
+\end{pmatrix*}}
++5
+\only<-7>{A}\only<8->{
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}}
++
+\only<-6>{I}\only<7->{
+\begin{pmatrix*}[r]
+1&0&0\\
+0&1&0\\
+0&0&1
+\end{pmatrix*}}
+}
+\uncover<11->{=0}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}$\Rightarrow$
+Rechenregeln von Matrizen können durch Polynome ausgedrückt werden}
+}
+\end{frame}
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/nichthauptideal.tex b/vorlesungen/slides/3/nichthauptideal.tex
new file mode 100644
index 0000000..46074b9
--- /dev/null
+++ b/vorlesungen/slides/3/nichthauptideal.tex
@@ -0,0 +1,78 @@
+%
+% nichthauptideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Nicht-Hauptideal in $\mathbb{Z}[X]$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Hauptideal\uncover<2->{ = ``Gerade''}}
+\vspace{-10pt}
+\begin{align*}
+\langle X+1\rangle&=(X+1) = {\color{red}(X+1)\cdot\mathbb{Z}[X]}
+\end{align*}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+\draw[->] (-6.3,0) -- (6.8,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-6.2) -- (0,6.6) coordinate[label={right:$\mathbb{Z}X$}];
+\foreach \x in {-6,...,6}{
+	\fill[color=red] (\x,\x) circle[radius=0.12];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Ideal mit zwei Erzeugenden}
+\vspace{-10pt}
+\begin{align*}
+\uncover<6->{
+{\color{darkgreen}
+\langle 2,X\rangle
+}
+&=}
+\uncover<5->{
+{\color{red}2\cdot\mathbb{Z}[X]}
+}
+\uncover<6->{+}
+\uncover<4->{
+{\color{blue}X\cdot\mathbb{Z}[X]}
+}
+\end{align*}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+\draw[->] (-6.3,0) -- (6.9,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-6.2) -- (0,7.0) coordinate[label={right:$\mathbb{Z}X$}];
+\uncover<6->{
+	\foreach \x in {-6,-4,...,6}{
+		\foreach \y in {-6,...,6}{
+			\fill[color=darkgreen] (\x,\y) circle[radius=0.20];
+		}
+	}
+}
+\uncover<5->{
+	\foreach \x in {-6,-4,...,6}{
+		\foreach \y in {-6,-4,...,6}{
+			\fill[color=red] (\x,\y) circle[radius=0.16];
+		}
+	}
+}
+\uncover<4->{
+	\foreach \y in {-6,...,6}{
+		\fill[color=blue] (0,\y) circle[radius=0.12];
+	}
+}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex
new file mode 100644
index 0000000..e1424ff
--- /dev/null
+++ b/vorlesungen/slides/3/nichthauptideal2.tex
@@ -0,0 +1,95 @@
+%
+% nichthauptideal2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Das Ideal $\langle 2,X\rangle \subset \mathbb{Z}[X]$}
+\vspace{-12pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\c{\clip (-2.8,-2.0) rectangle (2.8,2.0);}
+
+\def\labels{
+	\fill[color=white,opacity=0.5] (1.5,-0.1) circle[radius=0.2];
+	\node at (1.5,-0.1) {$1$};
+	\fill[color=white,opacity=0.5] (-0.9,1.7) circle[radius=0.2];
+	\node at (-0.9,1.7) {$X$};
+	\fill[color=white,opacity=0.5] (0.8,0.7) circle[radius=0.2];
+	\node at (0.8,0.7) {$X^2$};
+}
+
+\only<-3>{
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/ring.jpg}};
+	\end{scope}
+	\node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$};
+\end{scope}
+}
+
+\uncover<2->{
+\begin{scope}[xshift=-3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptideal.jpg}};
+	\end{scope}
+	\node[color=blue] at (-0.2,-1.2) {$(X+1)\cdot\mathbb{Z}[X]$};
+	\labels
+\end{scope}
+}
+
+\uncover<3->{
+\begin{scope}[xshift=-3.0cm,yshift=-1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptideal2.jpg}};
+	\end{scope}
+	\node[color=darkgreen] at (-3.0,-0.8) {$2\cdot\mathbb{Z}[X]$};
+\end{scope}
+
+\begin{scope}[xshift=3.0cm,yshift=-1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptidealX.jpg}};
+	\end{scope}
+	\node[color=darkgreen] at (2.5,-0.8) {$X\cdot\mathbb{Z}[X]$};
+	\labels
+\end{scope}
+}
+
+\uncover<4->{
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/nichthauptideal.jpg}};
+	\end{scope}
+	\node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$};
+	\node[color=darkgreen] at (1.9,-0.4) [right] {$\langle 2,X\rangle$};
+\end{scope}
+}
+
+\draw[color=gray!50] (-6.6,0) -- (6.4,0);
+\draw[color=gray!50] (0,-3.8) -- (0,3.8);
+
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\fill[color=white,opacity=0.5] (1.5,-0.6) circle[radius=0.2];
+	\node at (1.5,-0.6) {$1$};
+	\fill[color=white,opacity=0.5] (-0.4,1.7) circle[radius=0.2];
+	\node at (-0.4,1.7) {$X$};
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex
new file mode 100644
index 0000000..d646353
--- /dev/null
+++ b/vorlesungen/slides/3/operatoren.tex
@@ -0,0 +1,51 @@
+%
+% operatoren.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{$X$ als Operator}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Polynome}
+$a(X)=a_0+a_1X+\dots+a_nX^n$
+\uncover<2->{%
+\[
+a(X)
+=
+\begin{pmatrix}
+a_0\\a_1\\a_2\\a_3\\\vdots\\a_n
+\end{pmatrix}
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<3->{%
+\begin{block}{Multiplikation mit $X$}
+\strut
+\[
+\begin{pmatrix}
+1\\0\\0\\0\\\vdots\\0
+\end{pmatrix}
+\uncover<4->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\1\\0\\0\\\vdots\\0
+\end{pmatrix}}
+\uncover<5->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\0\\1\\0\\\vdots\\0
+\end{pmatrix}}
+\uncover<6->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\0\\0\\1\\\vdots\\0
+\end{pmatrix}}
+\uncover<7->{\overset{\cdot X}{\mapsto}\dots}
+\uncover<8->{\overset{\cdot X}{\mapsto}
+\begin{pmatrix}
+0\\0\\0\\0\\\vdots\\1
+\end{pmatrix}}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
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/polynome.tex b/vorlesungen/slides/3/polynome.tex
new file mode 100644
index 0000000..d7179a0
--- /dev/null
+++ b/vorlesungen/slides/3/polynome.tex
@@ -0,0 +1,29 @@
+%
+% polynome.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynome}
+$R$ ein Ring, z.~B.~$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$
+
+\begin{definition}
+Polynome in $X$ mit Koeffizienten in $R$:
+\[
+R[X]
+=
+\{
+a(X)\;|\;
+a(X) = a_nX^n+a_{n-1}X^{n-1} + \dots a_2X^2+a_1X + a_0, a_k\in R
+\}
+\]
+\end{definition}
+
+\begin{itemize}
+\item<2-> {\em Grad} des Polynoms: $\deg a(X) = \deg a = n$
+\item<3-> $\deg 0 = -\infty$
+\item<4-> {\em normiertes Polynom}: $a_n=1$
+\end{itemize}
+
+
+\end{frame}
diff --git a/vorlesungen/slides/3/quotientenring.tex b/vorlesungen/slides/3/quotientenring.tex
new file mode 100644
index 0000000..4aa9e43
--- /dev/null
+++ b/vorlesungen/slides/3/quotientenring.tex
@@ -0,0 +1,59 @@
+%
+% Quotientenring.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quotientenring}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Quotientenring}
+$I\subset R$ ein Ideal
+\\
+\uncover<2->{
+$R/I$ hat eine Ringstruktur:
+\begin{align*}
+\uncover<3->{\pi(s)&=s+I}
+\\
+\uncover<4->{\pi(s)\pi(r)&= (s+I)(r+I)}\\
+            &\uncover<5->{= sr +\underbrace{sI + rI}_{\subset RI\subset I} + II = sr+I}
+\\
+\uncover<6->{\pi(s)+\pi(r)&= (s+I)+(r+I)}\\
+	     &\uncover<7->{=s+r+I=\pi(s+r)}
+\end{align*}}
+\end{block}
+\vspace{-15pt}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Beispiele}
+\begin{itemize}
+\item
+$\mathbb{Z}/(n)=\mathbb{Z}/n\mathbb{Z}$,
+$\mathbb{F}_p=\mathbb{Z}/(p)=\mathbb{Z}/p\mathbb{Z}$
+\item<8->
+$p\in\Bbbk[X]$
+$\Rightarrow$
+$\Bbbk[X]/(p)$ ist ein Ring
+\end{itemize}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Algebraideal}
+$I\subset A$
+\begin{itemize}
+\item<10->
+$I$ ein Unterraum von $A$ als Vektorraum
+\item<11->
+$I$ ein Ideal von $A$ als Ring
+\end{itemize}
+\end{block}}
+\uncover<12->{%
+\begin{block}{Quotientenalgebra}
+$A/I$ ist eine Algebra
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/ringstruktur.tex b/vorlesungen/slides/3/ringstruktur.tex
new file mode 100644
index 0000000..d653300
--- /dev/null
+++ b/vorlesungen/slides/3/ringstruktur.tex
@@ -0,0 +1,50 @@
+%
+% ringstruktur.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Ringstruktur}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.46\textwidth}
+\begin{block}{Ring}
+Menge $R$ mit zwei zweistelligen Verknüfpungen $+$ und $\cdot$
+mit
+\begin{enumerate}
+\item<3->
+$R$ ist abelsche Gruppe bezüglich $+$
+\item<5->
+$R\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$
+\item<7->
+Für alle $a,b,c\in R$
+\begin{align*}
+a(b+c) &= ab+ac
+\\
+(a+b)c &= ac+bc
+\end{align*}
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<2->{%
+\begin{block}{Polynomring}
+$R$ ein Ring, $R[X]$ ``erbt'' Addition und Multiplikation mit
+\begin{enumerate}
+\item<4->
+$R[X]$ ist abelsche Gruppe bezüglich $+$
+\item<6->
+$R[X]\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$
+\item<8->
+Für alle $a,b,c\in R[X]$
+\begin{align*}
+a(b+c) &= ab+ac
+\\
+(a+b)c &= ac+bc
+\end{align*}
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/teilbarkeit.tex b/vorlesungen/slides/3/teilbarkeit.tex
new file mode 100644
index 0000000..a5ea9b9
--- /dev/null
+++ b/vorlesungen/slides/3/teilbarkeit.tex
@@ -0,0 +1,47 @@
+%
+% teilbarkeit.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Teilen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Teilen in $\mathbb{Z}$}
+Zu zwei Zahlen $a,b\in \mathbb{Z}$, \only<3->{{\color<3-4>{red}$a>b$}, }gibt es
+immer \only<3->{{\color<3-4>{red}genau}} ein Paar $q,r\in\mathbb{Z}$ derart, dass
+\begin{align*}
+a&=bq+r
+\\
+\uncover<3->{{\color<3-4>{red}r}&{\color<3-4>{red}< b}}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Teilen in $\mathbb{Q}[X]$}
+Zu zwei Polynomen $a,b\in\mathbb{Q}[X]$, \only<4->{{\color<4>{red}$\deg a > \deg b$},}
+gibt es 
+immer \only<4->{{\color<4>{red}bis auf eine Einheit genau }}%
+ein Paar $q,r\in\mathbb{Q}[X]$ derart, dass
+\begin{align*}
+a&=bq+r
+\\
+\uncover<4->{{\color<4>{red}\deg r}&{\color<4>{red}< \deg b}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<5->{%
+\begin{block}{Allgemein: euklidischer Ring}
+Nullteilerfreier Ring $R$ mit einer Funktion
+$d\colon R\setminus{0}\to\mathbb{N}$ mit
+\begin{itemize}
+\item Für $x,y\in R$ gilt $d(xy) \ge d(x)$.
+\item Für $x,y\in R$ gibt es $q,r\in R$ derart
+$x=qy +r$ mit $d(y)>d(r)$
+\end{itemize}
+Euklidische Ringe haben ähnliche Eigenschaften wie Polynomringe
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/3/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex
new file mode 100644
index 0000000..d20bfc4
--- /dev/null
+++ b/vorlesungen/slides/3/wurzel2.tex
@@ -0,0 +1,83 @@
+%
+% 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}}
+\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}
+\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}