5 files changed, 179 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index ad1081e..88ae3bb 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -17,6 +17,9 @@ 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/alpha.tex						\
 	../slides/4/chapter.tex
 
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index a10712a..8c9a30b 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -16,3 +16,6 @@
 \folie{4/polynomefp.tex}
 \folie{4/alpha.tex}
 \folie{4/schieberegister.tex}
+\folie{4/charakteristik.tex}
+\folie{4/char2.tex}
+\folie{4/frobenius.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/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}