From a791903ec7acf9a6d8dd0e780e01204b1188ff7d Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Mon, 22 Mar 2021 20:19:15 +0100
Subject: new slides

---
 vorlesungen/slides/4/frobenius.tex | 54 ++++++++++++++++++++++++++++++++++++++
 1 file changed, 54 insertions(+)
 create mode 100644 vorlesungen/slides/4/frobenius.tex

(limited to 'vorlesungen/slides/4/frobenius.tex')

diff --git a/vorlesungen/slides/4/frobenius.tex b/vorlesungen/slides/4/frobenius.tex
new file mode 100644
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+++ 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}
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