Merge branch 'master' of https://github.com/AndreasFMueller/SeminarMatrizen

7 files changed, 320 insertions, 1 deletions

diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index ad1081e..6616f56 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -17,6 +17,10 @@ chapter4 =								\
 	../slides/4/euklidpoly.tex					\
 	../slides/4/polynomefp.tex					\
 	../slides/4/schieberegister.tex					\
+	../slides/4/charakteristik.tex					\
+	../slides/4/char2.tex						\
+	../slides/4/frobenius.tex					\
+	../slides/4/qundr.tex						\
 	../slides/4/alpha.tex						\
 	../slides/4/chapter.tex
 
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index a10712a..6872018 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -16,3 +16,7 @@
 \folie{4/polynomefp.tex}
 \folie{4/alpha.tex}
 \folie{4/schieberegister.tex}
+\folie{4/charakteristik.tex}
+\folie{4/char2.tex}
+\folie{4/frobenius.tex}
+\folie{4/qundr.tex}
diff --git a/vorlesungen/slides/4/char2.tex b/vorlesungen/slides/4/char2.tex
new file mode 100644
index 0000000..2b5709a
--- /dev/null
+++ b/vorlesungen/slides/4/char2.tex
@@ -0,0 +1,48 @@
+%
+% char2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Charakteristik 2}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Plus und Minus}
+\[
+x+x = 2x = 0
+\uncover<2->{\Rightarrow
+-x=x}
+\]
+\end{block}
+\uncover<3->{%
+\begin{block}{Quadrieren}
+In $\mathbb{F}_2$ ist $2=0$, d.h
+\[
+(x+y)^2
+=
+x^2 + 2xy + y^2
+\uncover<4->{=
+x^2 + y^2}
+\]
+für alle $x,y\in\Bbbk$
+\end{block}}
+\uncover<6->{%
+\begin{block}{Frobenius-Automorphismus}
+\[
+(x+y)^{2^n} = x^{2^n}+y^{2^n}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Pascal-Dreieck}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/30-endlichekoerper/images/binomial2.pdf}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/charakteristik.tex b/vorlesungen/slides/4/charakteristik.tex
new file mode 100644
index 0000000..a0d6d3e
--- /dev/null
+++ b/vorlesungen/slides/4/charakteristik.tex
@@ -0,0 +1,71 @@
+%
+% charakteristisk.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Primkörper und Charakteristik}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Primkörper}
+$1\in\Bbbk$
+\begin{enumerate}
+\item<2->
+$n\cdot 1\ne 0\;\forall n\in\mathbb{N}$\uncover<3->{:
+$\Rightarrow$
+$\mathbb{Z}\subset \Bbbk$}
+\uncover<4->{%
+$\Rightarrow$
+$\mathbb{Q}\subset \Bbbk$}
+\item<5->
+$\{n\mathbb{Z}\;|\;
+\text{$n\cdot 1 = 0$ in $\Bbbk$}\}
+=
+p\mathbb{Z}$
+\uncover<6->{
+$\Rightarrow$
+$\mathbb{F}_p\subset \Bbbk$}
+\end{enumerate}
+\end{block}
+\uncover<7->{%
+\begin{block}{Primkörper}
+Der Primkörper $\operatorname{Prim}(\Bbbk)$
+eines Körpers $\Bbbk$ ist der kleinste in $\Bbbk$
+enthaltene Körper
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Charakteristik}
+\vspace{-10pt}
+\[
+\operatorname{char}(\Bbbk)
+=
+\begin{cases}
+\uncover<9->{p&\qquad \operatorname{Prim}(\Bbbk) = \mathbb{F}_p}\\
+\uncover<10->{0&\qquad \operatorname{Prim}(\Bbbk) = \mathbb{Q}}
+\end{cases}
+\]
+\vspace{-10pt}
+\end{block}}
+\uncover<11->{%
+\begin{block}{Vektorraum}
+$\Bbbk$ ist ein Vektorraum über $\operatorname{Prim}(\Bbbk)$
+durch Einschränkung der Multiplikation auf $\operatorname{Prim}(\Bbbk)$
+(Körperstruktur vergessen)
+\end{block}}
+\uncover<12->{%
+\begin{block}{Endliche Körper}
+\begin{itemize}
+\item<13->
+Endliche Körper haben immer Charakteristik $p\ne 0$
+\item<14->
+$\Bbbk$ ist eine endlichdimensionaler $\mathbb{F}_p$-Vektorraum
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex
index be5b3ca..c63afec 100644
--- a/vorlesungen/slides/4/euklidmatrix.tex
+++ b/vorlesungen/slides/4/euklidmatrix.tex
@@ -18,7 +18,7 @@ a_k = b_kq_k + r_k
 \;\Rightarrow\;
 \left\{
 \begin{aligned}
-a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\
+a_{k+1} &= b_k = \phantom{a_k-q_k}b_k \\
 b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k 
 \end{aligned}
 \right.}
diff --git a/vorlesungen/slides/4/frobenius.tex b/vorlesungen/slides/4/frobenius.tex
new file mode 100644
index 0000000..56fd78f
--- /dev/null
+++ b/vorlesungen/slides/4/frobenius.tex
@@ -0,0 +1,54 @@
+%
+% frobenius.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Frobenius-Automorphismus}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+$\operatorname{Prim}(\Bbbk) = \mathbb{F}_p$
+\uncover<2->{%
+\begin{block}{Binomial-Koeffizienten}
+\vspace{-10pt}
+\begin{align*}
+\binom{p}{k}
+&=
+\frac{
+{\color{red}p}\cdot(p-1)\cdot(p-2)\cdot\dots\cdot (p-k+1)
+}{
+1\cdot2\cdot3\cdot\dots\cdot k
+}
+\intertext{{\color{red}$p$} wird nicht gekürzt wegen}
+\uncover<3->{1&\not\equiv 0 \mod p}\\
+\uncover<3->{2&\not\equiv 0 \mod p}\\
+\uncover<3->{ &\phantom{a}\vdots}\\
+\uncover<3->{k&\not\equiv 0 \mod p}
+\end{align*}
+\vspace{-10pt}
+\end{block}}
+\vspace{-5pt}
+\uncover<4->{%
+\begin{block}{Frobenius-Authomorphismus}
+\vspace{-10pt}
+\begin{align*}
+\uncover<5->{(x+y)^{p\phantom{\mathstrut^n}}
+&=
+x^{p\phantom{\mathstrut}^n}+y^{p\phantom{mathstrut^n}}}
+\\
+\uncover<6->{(x+y)^{p^n} &= x^{p^n}+y^{p^n}}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Pascal-Dreieck}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/30-endlichekoerper/images/binomial5.pdf}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/qundr.tex b/vorlesungen/slides/4/qundr.tex
new file mode 100644
index 0000000..a6f89bd
--- /dev/null
+++ b/vorlesungen/slides/4/qundr.tex
@@ -0,0 +1,138 @@
+%
+% qundr.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\definecolor{darkblue}{rgb}{0,0,0.8}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (ll) at (-6,-3.6);
+\coordinate (lr) at (6,-3.6);
+\coordinate (ur) at (6,3.6);
+\coordinate (ul) at (-6,3.6);
+
+\def\d{0.6}
+\def\D{0.5}
+
+\coordinate (q) at (0,{-2.25+\d});
+\coordinate (r) at (-1.5,{\d+\D});
+\coordinate (a) at (1.5,{\d-\D});
+\coordinate (c) at (0,{2.25+\d});
+
+\coordinate (m1) at ($0.5*(q)+0.5*(r)$);
+\coordinate (m2) at ($0.5*(q)+0.5*(a)$);
+\coordinate (m3) at ($0.5*(c)+0.5*(r)$);
+\coordinate (m4) at ($0.5*(c)+0.5*(a)$);
+
+\def\t{1.5}
+\coordinate (M1) at ($(m1)+\t*(m1)-\t*(m4)$);
+\coordinate (M2) at ($(m2)+\t*(m2)-\t*(m3)$);
+\coordinate (M4) at ($(m4)+\t*(m4)-\t*(m1)$);
+\coordinate (M3) at ($(m3)+\t*(m3)-\t*(m2)$);
+
+\begin{scope}
+\clip (ll) rectangle (ur);
+
+\uncover<3->{
+	\fill[color=blue!30]
+		($0.9*(m1)+0.1*(M1)+(-6,0)$) -- ($0.9*(m1)+0.1*(M1)$)
+		-- (M4) -- (ul) -- cycle;
+}
+
+\uncover<4->{
+	\fill[color=red!60,opacity=0.5]
+		($0.9*(m2)+0.1*(M2)$) -- ($0.9*(m2)+0.1*(M2)+(6,0)$)
+		-- (ur) -- (M3) -- cycle;
+}
+
+\uncover<2->{
+	\fill[color=darkgreen!60,opacity=0.5]
+		($1.09*(m3)-0.09*(M3)$) -- ($1.09*(m3)-0.09*(M3)+(-6,0)$)
+		-- (ll) -- (M2) -- cycle;
+}
+
+\uncover<6->{
+	\fill[color=gray,opacity=0.5]
+		({6-0.1},{\d+0.22}) rectangle ({6-2.4},{\d+0.62});
+	\node[color=yellow] at (6,\d) [above left] {überabzählbar\strut};
+
+	\fill[color=gray,opacity=0.5]
+		({-6+0.1},{\d-0.15}) rectangle ({-6+1.75},{\d-0.55});
+	\node[color=yellow] at (-6,\d) [below right] {abzählbar\strut};
+
+	\draw[color=yellow,line width=2pt] (-7,\d) -- (7,\d);
+}
+
+\end{scope}
+
+\node at (q) {$\mathbb{Q}$\strut};
+\node at ($(q)+(0,-0.2)$) [below] {Primkörper};
+
+\uncover<3->{
+	\node at (r) {$\mathbb{R}$\strut};
+	\node at (r) [left] {$\text{reelle Zahlen}=\mathstrut$};
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (r);
+	\node at ($0.5*(q)+0.5*(r)$)
+		[below,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$};
+	\node[color=blue] at (ul)
+		[above right] {topologische Vervollständigung};
+}
+
+\uncover<4->{
+	\node at (a) {$\mathbb{A}$\strut};
+	\node at (a) [right] {$\mathstrut = \text{algebraische Zahlen}$};
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (a);
+	\node at ($0.5*(q)+0.5*(a)$)
+		[below,rotate={atan((2.25-\D)/1.5)}] {index $\infty$};
+	\node[color=red] at (ur)
+		[above left] {algebraische Vervollständigung};
+}
+
+\uncover<5->{
+	\node at (c) {$\mathbb{C}$\strut};
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (r) -- (c);
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (a) -- (c);
+	\node at ($(c)+(0,0.2)$) [above] {komplexe Zahlen};
+	\node at ($0.5*(r)+0.5*(c)$)
+		[above,rotate={atan((2.25-\D)/1.5)}] {index 2};
+	\node at ($0.5*(a)+0.5*(c)$)
+		[above,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$};
+}
+
+\uncover<3->{
+	\node[color=darkblue] at (ul) [below right]
+	{\begin{minipage}{0.3\textwidth}\raggedright
+	Grenzwerte von Cauchy-Folgen in $\mathbb{Q}$ hinzufügen
+	\end{minipage}};
+}
+
+\uncover<4->{
+	\node[color=darkred] at (ur) [below left]
+	{\begin{minipage}{0.3\textwidth}\raggedleft
+	Nullstellen von Polynomen in $\mathbb{Q}[X]$ hinzufügen
+	\end{minipage}};
+}
+
+\uncover<2->{
+	\node[color=darkgreen] at (ll) [above right]
+	{\begin{minipage}{0.4\textwidth}\raggedright
+	\begin{block}{Archimedische Eigenschaft}
+	Für $a>b >0$ gibt es $n\in\mathbb{N}$ mit
+	$n\cdot b > a$
+	\end{block}
+	\end{minipage}};
+
+	\node[color=darkgreen] at (ll) [below right]
+		{geordneter Körper, nötig für die Definition von Cauchy-Folgen};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup