diff options
Diffstat (limited to 'vorlesungen/slides/4')
22 files changed, 1288 insertions, 1 deletions
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index ad1081e..5aac429 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -17,6 +17,20 @@ 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/galois/erweiterung.tex \ + ../slides/4/galois/automorphismus.tex \ + ../slides/4/galois/konstruktion.tex \ + ../slides/4/galois/wuerfel.tex \ + ../slides/4/galois/winkeldreiteilung.tex \ + ../slides/4/galois/quadratur.tex \ + ../slides/4/galois/radikale.tex \ + ../slides/4/galois/aufloesbarkeit.tex \ + ../slides/4/galois/sn.tex \ ../slides/4/chapter.tex + diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index a10712a..0691e39 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -16,3 +16,16 @@ \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} +\folie{4/galois/erweiterung.tex} +\folie{4/galois/automorphismus.tex} +\folie{4/galois/konstruktion.tex} +\folie{4/galois/wuerfel.tex} +\folie{4/galois/winkeldreiteilung.tex} +\folie{4/galois/quadratur.tex} +\folie{4/galois/radikale.tex} +\folie{4/galois/aufloesbarkeit.tex} +\folie{4/galois/sn.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/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex new file mode 100644 index 0000000..ef5902b --- /dev/null +++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex @@ -0,0 +1,120 @@ +% +% aufloesbarkeit.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Auflösbarkeit} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Radikalerweiterung} +Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$ +einer Radikalerweiterung +\[ +\Bbbk \subset \Bbbk(\alpha) +\] +sind festgelegt durch Wahl von $f(\alpha)$. + +\begin{itemize} +\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt +\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein +\end{itemize} + +\end{block}} +\uncover<8->{% +\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$} +$\mathbb{Q}\subset \Bbbk$, +$n$ verschiedene Nullstellen $\mathbb{C}$: +\[ +\uncover<9->{ +\operatorname{Gal}(\Bbbk/\mathbb{Q}) +\cong +S_n} +\uncover<10->{ +\quad +\text{auflösbar?}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{\uncover<5->{Galois-Gruppen}} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{1.2} + +\uncover<2->{ +\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3}); +\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen}; +} + +\node at (0,0) {$\mathbb{Q}$}; +\node at (0,{1*\s}) {$E_1$}; +\node at (0,{2*\s}) {$E_2$}; +\node at (0,{3*\s}) {$E_3$}; +\node at (0,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) {$\Bbbk$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s}); + +\begin{scope}[xshift=0.5cm] +\uncover<7->{ +\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3}); +\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe}; +} +\uncover<5->{ +\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$}; +\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$}; +\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$}; +\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$}; +\node at (1,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$}; +\node at (1,{0.5*\s}) {$\cap\mathstrut$}; +\node at (1,{1.5*\s}) {$\cap\mathstrut$}; +\node at (1,{2.5*\s}) {$\cap\mathstrut$}; +\node at (1,{3.5*\s}) {$\cap\mathstrut$}; +\node at (1,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<6->{ +\begin{scope}[xshift=2.5cm] +\node at (0,{0*\s}) {$G_n$}; +\node at (0,{1*\s}) {$G_{n-1}$}; +\node at (0,{2*\s}) {$G_{n-2}$}; +\node at (0,{3*\s}) {$G_{n-3}$}; +\node at (0,{5*\s}) {$G_0=\{e\}$}; +\node at (0,{0.5*\s}) {$\cap\mathstrut$}; +\node at (0,{1.5*\s}) {$\cap\mathstrut$}; +\node at (0,{2.5*\s}) {$\cap\mathstrut$}; +\node at (0,{3.5*\s}) {$\cap\mathstrut$}; +\node at (0,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<7->{ +\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$}; +\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$}; +\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$}; +\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch}; +} + +\end{scope} +\end{scope} + + + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex new file mode 100644 index 0000000..6051813 --- /dev/null +++ b/vorlesungen/slides/4/galois/automorphismus.tex @@ -0,0 +1,118 @@ +% +% automorphismus.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{4pt} +\setlength{\belowdisplayskip}{4pt} +\frametitle{Galois-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{3.0} +\begin{scope}[xshift=-1.5cm] +\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut}; +\node at (0,{\s}) {$G$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0); +\node at (0,{-0.5*\s}) [left] {$[F:E]$}; +\node at (0,{0.5*\s}) [left] {$[G:F]$}; +\node at (0,0) {$F$}; +\node at (0,{-\s}) {$E$}; +\end{scope} +\uncover<3->{ +\begin{scope}[xshift=1.8cm] +\node at (0,{\s+0.1}) [above] {Gruppe\strut}; +\fill (0,{-\s}) circle[radius=0.06]; +\fill (0,0) circle[radius=0.06]; +\fill (0,{\s}) circle[radius=0.06]; +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=100,in=-100] (0,{\s}); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=80,in=-80] (0,0); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,0) to[out=80,in=-80] (0,{\s}); +\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$}; +\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$}; +\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$}; +\end{scope} +\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s}); +\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$}; +} +\uncover<4->{ +\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s}); +\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<2->{% +\begin{block}{Automorphismus} +\vspace{-10pt} +\[ +\operatorname{Aut}(F) += +\left\{ +f\colon F\to F +\left| +\begin{aligned} +f(x+y)&=f(x)+f(y)\\ +f(xy)&=f(x)f(y) +\end{aligned} +\right. +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Galois-Gruppe} +Automorphismen, die $E$ festlassen +\[ +{\color{red} +\operatorname{Gal}(F/E) +} += +\left\{ +\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Fixkörper} +$H\subset \operatorname{Aut}(F)$: +\begin{align*} +{\color{blue}F^H} +&= +\{x\in F\;|\; hx = x\forall h\in H\} +=\operatorname{Fix}(H) +\end{align*} +\end{block}} +\vspace{-13pt} +\uncover<5->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<6-> +\( +\operatorname{Gal}(\mathbb{C}/\mathbb{R}) += +\{ +\operatorname{id}_{\mathbb{C}}, +\operatorname{conj}\colon z\mapsto\overline{z} +\} +\) +\item<7-> +\( +\mathbb{C}^{\operatorname{conj}} += +\mathbb{R} +\) +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex new file mode 100644 index 0000000..6909849 --- /dev/null +++ b/vorlesungen/slides/4/galois/erweiterung.tex @@ -0,0 +1,65 @@ +% +% erweiterung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Körpererweiterungen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Körpererweiterung} +$E,F$ Körper: $E\subset F$ +\end{block} +\uncover<6->{% +\begin{block}{Vektorraum} +$F$ ist ein Vektorraum über $E$ +\end{block}} +\uncover<7->{% +\begin{block}{Endliche Körpererweiterung} +$\dim_E F < \infty$ +\end{block}} +\uncover<8->{% +\begin{block}{Adjunktion eines $\alpha$} +$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und +$\alpha$ enthält. +\end{block}} +\uncover<9->{% +\begin{block}{Algebraische Erweiterung} +$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von +$m(X)\in\Bbbk[X]$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiele} +\begin{enumerate} +\item<3-> +$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$ +\item<4-> +$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$ +\item<5-> +$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$ +\end{enumerate} +\end{block}} +\uncover<7->{% +\begin{block}{Grad} +$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls +\[ +\dim_E F = n =: [F:E] +\] +\uncover<8->{% +Gleichbedeutend: $\deg m(X) = n$} +\uncover<10->{% +\[ +E\subset F\subset G +\Rightarrow +[G:E] = [G:F]\cdot [F:E] +\] +(in unseren Fällen)} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile new file mode 100644 index 0000000..444944e --- /dev/null +++ b/vorlesungen/slides/4/galois/images/Makefile @@ -0,0 +1,12 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: wuerfel2.png wuerfel.png + +wuerfel.png: wuerfel.pov common.inc + povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov + +wuerfel2.png: wuerfel2.pov common.inc + povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc new file mode 100644 index 0000000..6cfcabe --- /dev/null +++ b/vorlesungen/slides/4/galois/images/common.inc @@ -0,0 +1,89 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" +#include "textures.inc" +#include "stones.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.133; +#declare O = <0, 0, 0>; +#declare E = <1, 1, 1>; +#declare a = pow(2, 1/3); +#declare at = 0.02; + +camera { + location <3, 2, 12> + look_at E * (a / 2) * 0.93 + right x * imagescale + up y * imagescale +} + +light_source { + <11, 20, 16> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro wuerfelgitter(A, AT) + cylinder { O, <A, 0, 0>, AT } + cylinder { O, <0, A, 0>, AT } + cylinder { O, <0, 0, A>, AT } + cylinder { <A, 0, 0>, <A, A, 0>, AT } + cylinder { <A, 0, 0>, <A, 0, A>, AT } + cylinder { <0, A, 0>, <A, A, 0>, AT } + cylinder { <0, A, 0>, <0, A, A>, AT } + cylinder { <0, 0, A>, <A, 0, A>, AT } + cylinder { <0, 0, A>, <0, A, A>, AT } + cylinder { <A, A, 0>, <A, A, A>, AT } + cylinder { <A, 0, A>, <A, A, A>, AT } + cylinder { <0, A, A>, <A, A, A>, AT } + sphere { <0, 0, 0>, AT } + sphere { <A, 0, 0>, AT } + sphere { <0, A, 0>, AT } + sphere { <0, 0, A>, AT } + sphere { <A, A, 0>, AT } + sphere { <A, 0, A>, AT } + sphere { <0, A, A>, AT } + sphere { <A, A, A>, AT } +#end + +#macro wuerfel() + union { + box { O, E } + wuerfelgitter(1, 0.5*at) + texture { + T_Grnt24 + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro wuerfel2() + union { + wuerfelgitter(a, at) + pigment { + color rgb<0.8,0.4,0.4> + } + finish { + specular 0.9 + metallic + } + } +#end diff --git a/vorlesungen/slides/4/galois/images/wuerfel.png b/vorlesungen/slides/4/galois/images/wuerfel.png Binary files differnew file mode 100644 index 0000000..ff6fc14 --- /dev/null +++ b/vorlesungen/slides/4/galois/images/wuerfel.png diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov new file mode 100644 index 0000000..a5db465 --- /dev/null +++ b/vorlesungen/slides/4/galois/images/wuerfel.pov @@ -0,0 +1,9 @@ +// +// wuerfel.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +wuerfel() + diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.png b/vorlesungen/slides/4/galois/images/wuerfel2.png Binary files differnew file mode 100644 index 0000000..68919cc --- /dev/null +++ b/vorlesungen/slides/4/galois/images/wuerfel2.png diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov new file mode 100644 index 0000000..ac32b2f --- /dev/null +++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov @@ -0,0 +1,9 @@ +// +// wuerfel.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +wuerfel() +wuerfel2() diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex new file mode 100644 index 0000000..094b570 --- /dev/null +++ b/vorlesungen/slides/4/galois/konstruktion.tex @@ -0,0 +1,147 @@ +% +% konstruktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Konstruktion mit Zirkel und Lineal} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Strahlensatz} +\uncover<6->{% +Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$ +kann mit Zirkel und Lineal konstruiert werden.} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Kreis--Gerade} +Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$ +\uncover<13->{% +$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden} +\end{block}} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{0.5} +\def\t{0.45} + +\coordinate (A) at (0,0); +\coordinate (B) at ({10*\t},0); + +\uncover<2->{ + \draw (0,0) -- (30:{10.5*\s}); +} + +\uncover<3->{ + \foreach \x in {0,...,10}{ + \fill (30:{\x*\s}) circle[radius=0.03]; + } + \foreach \x in {0,1,2,3,4,7,8,9}{ + \node at (30:{\x*\s}) [above] {\tiny $\x$}; + } + \node at (30:{10*\s}) [above right] {$q=10$}; +} + +\uncover<4->{ + \foreach \x in {1,...,10}{ + \fill (0:{\x*\t}) circle[radius=0.03]; + \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t}); + } +} + +\draw (A) -- (0:{10.5*\t}); +\node at (A) [below left] {$A$}; +\node at (B) [below right] {$B$}; +\fill (A) circle[radius=0.05]; +\fill (B) circle[radius=0.05]; + +\uncover<5->{ + \node at (30:{6*\s}) [above left] {$p=6$}; + \draw[line width=0.2pt] (0,0) -- (0,-0.4); + \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4); + \draw[<->] (0,-0.3) -- ({6*\t},-0.3); + \node at ({3*\t},-0.4) [below] + {$\displaystyle\frac{p}{q}\cdot\overline{AB}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +%\foreach \x in {8,...,14}{ +% \only<\x>{\node at (4,4) {$\x$};} +%} + +\def\r{4} +\def\a{50} + +\coordinate (A) at ({\r*cos(\a)},0); + +\uncover<10->{ + \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle; + \fill[color=gray] + (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle; +} + +\draw[->] (0,0) -- (95:\r); +\node at (95:{0.5*\r}) [left] {$c$}; + +\begin{scope} + \clip (-1,-0.3) rectangle (4.5,4.1); + \uncover<10->{ + \draw (-1,0) -- (5,0); + \draw[->] (0,0) -- (\r,0); + \draw (0,0) circle[radius=\r]; + \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5); + } +\end{scope} + +\uncover<11->{ + \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle; +} + +\uncover<9->{ + \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle; + \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07]; + \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1); +} + +\uncover<11->{ + \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r); + \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$}; +} + +\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0); +\fill[color=blue] (0,0) circle[radius=0.04]; +\fill[color=blue] (A) circle[radius=0.04]; +\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$}; + +\uncover<12->{ + \fill[color=white,opacity=0.8] + ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25}) + rectangle + ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25}); + + \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right] + {$b=\sqrt{c^2-a^2}$}; + \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r); + \fill[color=red] (\a:\r) circle[radius=0.05]; + \fill[color=red] (A) circle[radius=0.05]; +} + +\end{tikzpicture} +\end{center}} +\end{column} +\end{columns} +\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:} +Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$} +\end{frame} diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex new file mode 100644 index 0000000..f5763b9 --- /dev/null +++ b/vorlesungen/slides/4/galois/quadratur.tex @@ -0,0 +1,66 @@ +% +% quadratur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Quadratur des Kreises} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{center} +\uncover<2->{% +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.8} +\pgfmathparse{sqrt(3.14159)*\r/2} +\xdef\s{\pgfmathresult} + +\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s); +\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r]; + +\uncover<3->{ + \draw[->,color=red] (0,0) -- (50:\r); + \fill[color=red] (0,0) circle[radius=0.04]; + \node[color=red] at (50:{0.5*\r}) [below right] {$r$}; +} + +\uncover<4->{ + \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7}); + \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7}); + \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6}); + \node[color=blue] at (0,{-\s-0.6}) [below] {$l$}; +} + +\uncover<5->{ + \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2} + \;\Rightarrow\; + {\color{blue}l}={\color{red}\sqrt{\pi}r}$}; +} + +\end{tikzpicture}} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Aufgabe} +Konstruiere ein zu einem Kreis flächengleiches Quadrat +\end{block} +\uncover<6->{% +\begin{block}{Modifizierte Aufgabe} +Konstruiere eine Strecke, deren Länge Lösung der Gleichung +$x^2-\pi=0$ ist. +\end{block}} +\uncover<7->{% +\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch] +\begin{itemize} +\item<8-> Lösung in einem Erweiterungskörper +\item<9-> Lösung ist Nullstelle eines Polynoms +\item<10-> Lösung ist algebraisch +\item<11-> $\pi$ ist {\bf nicht} algebraisch +\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})} +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex new file mode 100644 index 0000000..e9e4ce8 --- /dev/null +++ b/vorlesungen/slides/4/galois/radikale.tex @@ -0,0 +1,69 @@ +% +% radikale.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lösung durch Radikale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Problemstellung} +Finde Nullstellen eines Polynomes +\[ +p(X) += +a_nX^n + a_{n-1}X^{n-1} ++\dots+ +a_1X+a_0 +\] +$p\in\mathbb{Q}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Radikale} +Geschachtelte Wurzelausdrücke +\[ +\sqrt[3]{ +-\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} ++ +\sqrt[3]{ +-\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} +\] +\uncover<3->{(Lösung von $x^3+px+q=0$)} +\end{block}} +\uncover<4->{% +\begin{block}{Lösbar durch Radikale} +Nullstelle von $p(X)$ ist ein Radikal +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Algebraische Formulierung} +Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$, +finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart, +dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{: +$\Bbbk = \mathbb{Q}[X]/(p)$} +\end{block}} +\uncover<7->{% +\begin{block}{Radikalerweiterung} +Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften +\begin{itemize} +\item<8-> $\alpha$ ist eine Einheitswurzel +\item<9-> $\alpha^k\in\Bbbk$ +\end{itemize} +\end{block}} +\vspace{-5pt} +\uncover<10->{% +\begin{block}{Lösbar durch Radikale} +Radikalerweiterungen +\[ +\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex new file mode 100644 index 0000000..1cae3fa --- /dev/null +++ b/vorlesungen/slides/4/galois/sn.tex @@ -0,0 +1,87 @@ +% +% sn.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nichtauflösbarkeit von $S_n$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Die symmetrische Gruppe $S_n$} +Permutationen auf $n$ Elementen +\[ +\sigma += +\begin{pmatrix} +1&2&3&\dots&n\\ +\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n) +\end{pmatrix} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Signum} +$t(\sigma)=\mathstrut$ Anzahl Transpositionen +\[ +\operatorname{sgn}(\sigma) += +(-1)^{t(\sigma)} += +\begin{cases} +\phantom{-}1&\text{$t(\sigma)$ gerade} +\\ +-1&\text{$t(\sigma)$ ungerade} +\end{cases} +\] +Homomorphismus! +\end{block}} +\uncover<3->{% +\begin{block}{Die alternierende Gruppe $A_n$} +\vspace{-12pt} +\[ +A_n = \ker \operatorname{sgn} += +\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Normale Untergruppe} +\begin{itemize} +\item +$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$ +\item +$G/N$ ist wohldefiniert +\end{itemize} +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Einfache Gruppe} +$G$ einfach $\Leftrightarrow$ +\[ +H\triangleleft G +\; +\Rightarrow +\; +\text{$H=\{e\}$ oder $H=G$} +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$} +\begin{enumerate} +\item<7-> Zeigen, dass $A_5$ einfach ist +\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach +\end{enumerate} +\uncover<9->{% +$\Rightarrow$ i.~A.~keine Lösung der +einer Polynomgleichung vom Grad $\ge 5$ durch Radikale +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex new file mode 100644 index 0000000..54b941b --- /dev/null +++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex @@ -0,0 +1,94 @@ +% +% winkeldreiteilung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Winkeldreiteilung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.43\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{5} +\def\a{25} + +\uncover<3->{ + \draw[line width=0.7pt] (\r,0) arc (0:90:\r); +} + +\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle; +\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$}; + +\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r); + +\uncover<2->{ + \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle; + \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r); + \node[color=red] at ({0.5*\a}:{0.7*\r}) + {$\displaystyle\frac{\alpha}{3}$}; +} + +\uncover<3->{ + \fill[color=blue] ({3*\a}:\r) circle[radius=0.05]; + \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1); + + \fill[color=red] ({\a}:\r) circle[radius=0.05]; + \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1); + + \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}]; + \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}]; +} + + +\uncover<4->{ +\node at ({0.5*\r},-0.5) [below] {$\displaystyle +\cos{\color{blue}\alpha} += +4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3} +$}; +} + +\uncover<5->{ + \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$}; + \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.53\textwidth} +\begin{block}{Aufgabe} +Teile einen Winkel in drei gleiche Teile +\end{block} +\vspace{-2pt} +\uncover<6->{% +\begin{block}{Algebraisierte Aufgabe} +Konstruiere $x$ aus $a$ derart, dass +\[ +p(x) += +x^3-\frac34 x -a = 0 +\] +\uncover<7->{% +$a=0$:} +\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$} +\end{block}} +\vspace{-2pt} +\uncover<10->{% +\begin{proof}[Unmöglichkeitsbeweis] +\begin{itemize} +\item<11-> +$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel +\item<12-> +$p(x)$ definiert eine Körpererweiterung vom Grad $3$ +\item<13-> +Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$ +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex new file mode 100644 index 0000000..ada6079 --- /dev/null +++ b/vorlesungen/slides/4/galois/wuerfel.tex @@ -0,0 +1,64 @@ +% +% wuerfel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Würfelverdoppelung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}}; +\uncover<2->{ +\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}}; +} + +\uncover<3->{ + \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25); + \node[color=blue] at (0.75,-2.3) [above] {$a$}; +} + +\uncover<4->{ + \begin{scope}[yshift=0.03cm] + \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19); + \draw[color=red] (2.85,-2.7) -- (2.85,-3.0); + \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9); + \end{scope} + \node[color=red] at (0.36,-2.9) [below] {$b$}; +} + +\uncover<5->{ +\node at (0,-4) {$ + 2{\color{blue}a}^3={\color{red}b}^3 + \uncover<6->{\;\Rightarrow\; + \frac{b}{a} = \sqrt[3]{2}}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Aufgabe} +Konstruiere einen Würfel mit doppeltem Volumen +\end{block} +\uncover<7->{% +\begin{block}{Algebraisierte Aufgabe} +Konstruiere eine Nullstelle von $p(x)=x^3-2$ +\end{block}} +\uncover<8->{% +\begin{proof}[Unmöglichkeitsbeweis] +\begin{itemize} +\item<9-> +$p(x)$ irreduzibel +\item<10-> +$p(x)$ definiert eine Körpererweiterung vom Grad $3$ +\item<11-> +Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar +\qedhere +\end{itemize} +\end{proof}} +\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 |
