diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/4/Makefile.inc | 4 | ||||
| -rw-r--r-- | vorlesungen/slides/4/chapter.tex | 4 | ||||
| -rw-r--r-- | vorlesungen/slides/4/char2.tex | 48 | ||||
| -rw-r--r-- | vorlesungen/slides/4/charakteristik.tex | 71 | ||||
| -rw-r--r-- | vorlesungen/slides/4/euklidmatrix.tex | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/4/frobenius.tex | 54 | ||||
| -rw-r--r-- | vorlesungen/slides/4/qundr.tex | 138 |
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 |