From a791903ec7acf9a6d8dd0e780e01204b1188ff7d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 22 Mar 2021 20:19:15 +0100 Subject: new slides --- vorlesungen/slides/4/Makefile.inc | 3 ++ vorlesungen/slides/4/chapter.tex | 3 ++ vorlesungen/slides/4/char2.tex | 48 ++++++++++++++++++++++ vorlesungen/slides/4/charakteristik.tex | 71 +++++++++++++++++++++++++++++++++ vorlesungen/slides/4/frobenius.tex | 54 +++++++++++++++++++++++++ vorlesungen/slides/test.tex | 2 + 6 files changed, 181 insertions(+) create mode 100644 vorlesungen/slides/4/char2.tex create mode 100644 vorlesungen/slides/4/charakteristik.tex create mode 100644 vorlesungen/slides/4/frobenius.tex (limited to 'vorlesungen/slides') 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} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index e4b9ad7..466256b 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -15,3 +15,5 @@ % XXX Motiviation für *-Operation %\folie{5/normal.tex} +\folie{4/char2.tex} +\folie{4/frobenius.tex} -- cgit v1.2.1