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treffen zu können ersetzen wir die Elemente durch Nach den o.\,s. Tafeln ist die erste Gruppe, $S_{2}$, kommutativ und die zweite, $S_{3}$, nicht. +%% ******************************************************************************** +%% FILE: body/ska/ska6.tex +%% ******************************************************************************** + +\setcounternach{chapter}{6} +\chapter[Woche 6]{Woche 6} + \label{ska:6} + +%% SKA 6-1 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{1} +\section[Aufgabe 1]{} + \label{ska:6:ex:1} +\let\sectionname\altsectionname + +Betrachte die Verknüpfung $(\ntrlpos,\ast)$, +die vermöge $a\ast b:=a^{b}$ definiert wird. +Wir prüfen das \emph{Assoziativitätsgesetz}. +Seien zu diesem Zwecke erstmals $a,b,c\in\ntrlpos$. +Dann + + \begin{mathe}[mc]{rclclcl} + \eqtag[eq:1:ska:6:ex:1] + a\ast (b\ast c) &= &a\ast b^{c} &= &a^{b^{c}},\\ + (a\ast b)\ast c &= &a^{b}\ast c &= &(a^{b})^{c} &= &a^{bc}.\\ + \end{mathe} + +Darum + + \begin{mathe}[mc]{rcl} + \eqtag[eq:2:ska:6:ex:1] + a\ast (b\ast c)=(a\ast b)\ast c + &\eqcrefoverset{eq:1:ska:6:ex:1}{\Longleftrightarrow} + &a^{b^{c}} = a^{bc}\\ + &\Longleftrightarrow + &a=1\,\text{oder}\,b^{c}=bc\\ + &\Longleftrightarrow + &a=1\,\text{oder}\,b^{c-1}=c.\\ + \end{mathe} + +Diese analytische Untersuchung hilft uns, ein \textbf{Gegenbeispiel} zu finden: +Seien $a:=2$, $b:=3$, $c:=2$. +Dann $a\neq 1$ und $b^{c-1}=3^{1}\neq c$. +Darum laut \eqcref{eq:2:ska:6:ex:1} gilt $a\ast (b\ast c)\neq (a\ast b)\ast c$. + +Also erfüllt $(\ntrlpos,\ast)$ das Assoziativitätsaxiom \uline{nicht}. + +%% SKA 6-2 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{2} +\section[Aufgabe 2]{} + \label{ska:6:ex:2} +\let\sectionname\altsectionname + +Die Multiplikationstabelle für $(R,+,\cdot)$ mit $R=\{0,1\}$ +ist wie folgt + + \hraum + \begin{tabular}[mc]{|C|CC|} + \hline + + &0 &1\\ + \hline + 0 &0 &1\\ + 1 &1 &0\\ + \hline + \end{tabular} + \quad + \begin{tabular}[mc]{|C|CC|} + \hline + \cdot &0 &1\\ + \hline + 0 &0 &0\\ + 1 &0 &1\\ + \hline + \end{tabular} + \hraum + +%% SKA 6-3 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{3} +\section[Aufgabe 3]{} + \label{ska:6:ex:3} +\let\sectionname\altsectionname + +Betrachte $\{0,1,a\}$ mit folgenden Operationen $+$, $\cdot$: + + \hraum + \begin{tabular}[mc]{|C|CCC|} + \hline + + &0 &1 &a\\ + \hline + 0 &0 &1 &a\\ + 1 &1 &a &0\\ + a &a &0 &1\\ + \hline + \end{tabular} + \quad + \begin{tabular}[mc]{|C|CCC|} + \hline + \cdot &0 &1 &a\\ + \hline + 0 &0 &0 &0\\ + 1 &0 &1 &a\\ + a &0 &a &1\\ + \hline + \end{tabular} + \hraum + +Diese Struktur ist offensichtlich isomorph zu + $(\intgr/3\intgr,+,\cdot)$ +mittels ${0\mapsto 0}$, ${1\mapsto 1}$, ${a\mapsto 2}$. +Da $(\intgr/3\intgr,+,\cdot)$ ein Körper ist, +ist $(\{0,1,a\},+,\cdot)$ ebenfalls wegen des Isomorphismus ein Körper. +Insbesondere sind die o.\,s. definierten Operationen assoziativ. + +%% SKA 6-4 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{4} +\section[Aufgabe 4]{} + \label{ska:6:ex:4} +\let\sectionname\altsectionname + +\begin{claim*} + Sei $K$ ein Körper und seien $x,y\in K$ mit $xy=0$. + Dann gilt $x=0$ oder $y=0$. +\end{claim*} + + \begin{proof} + Angenommen, dies sei nicht der Fall. + Dann $x\neq 0$ und $y\neq 0$. + Darum existieren multiplikative Inverse, $x^{-1},y^{-1}\in K$. + Aus $xy=0$ folgt dann + + \begin{mathe}[mc]{rcl} + 1 &= &y^{-1}y\\ + &= &y^{-1}1y\\ + &= &y^{-1}(x^{-1}x)y\\ + &= &(y^{-1}x^{-1})(xy)\\ + &= &(y^{-1}x^{-1})0\\ + &= &0.\\ + \end{mathe} + + Das ist ein Widerspruch! + \end{proof} + +\textbf{Bemerkung.} +Wir haben hier den Satz $\forall{a\in K:~}a\cdot 0=0$ (siehe \cite[Satz 4.2.2]{sinn2020}) +in Anspruch genommen haben. + +%% SKA 6-5 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{5} +\section[Aufgabe 5]{} + \label{ska:6:ex:5} +\let\sectionname\altsectionname + +Sei $R=\intgr/6\intgr$ versehen mit Addition und Multiplikation modulo $6$.\\ +Beachte: das additive Neutralelement ist die Äquivalenzklasse $[0]$.\\ +Für $x=[2]$ und $y=[3]$ gilt $x,y\neq [0]$ und aber $xy=[2\cdot 3]=[6]=[0]$. + +\textbf{Bemerkung.} Wir nennen solche Elemente \emph{Nullteiler}. + +%% SKA 6-6 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{6} +\section[Aufgabe 6]{} + \label{ska:6:ex:6} +\let\sectionname\altsectionname + +\begin{enumerate}{\bfseries (a)} + %% SKA 6-7(a) + \item + \begin{mathe}[tc]{ll} + &z_{1}:=\dfrac{2\imageinh}{5-3\imageinh} + = \dfrac{2\imageinh\cdot(5+3\imageinh)}{5^{2}+3^{2}} + = \dfrac{-6 + 10\imageinh}{34} + = -\frac{3}{17} + \imageinh\frac{5}{17}\\ + \Longrightarrow + &\ReTeil(z_{1}) = -\frac{3}{17}, + \quad + \ImTeil(z_{1}) = \frac{5}{17}.\\ + \\ + &z_{2}:=\dfrac{3 - 2\imageinh}{1 + 2\imageinh} + = \dfrac{(3 - 2\imageinh)\cdot(1-2\imageinh)}{1^{2}+2^{2}} + = \dfrac{(3+(-2)(-2)\imageinh^{2})) + (-6+-2)\imageinh}{5} + = \dfrac{(3-4) + -8\imageinh}{5} + = -\frac{1}{5} + \imageinh\frac{8}{5}\\ + \Longrightarrow + &\ReTeil(z_{2}) = -\frac{1}{5}, + \quad + \ImTeil(z_{2}) = \frac{8}{5}.\\ + \end{mathe} + + %% SKA 6-7(b) + \item\voritemise + + \begin{algorithm}[\rtab][\rtab] + Gaußverfahren angewandt auf $(A|\mathbf{b})$: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +1+\imageinh &1 &0\\ +-2 &2-\imageinh &1\\ +\end{matrix}\\ + \end{mathe} + + Wende die Zeilentransformation + ${Z_{1}\leftsquigarrow (1-\imageinh)\cdot Z_{1}}$ + an: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +2 &1-\imageinh &0\\ +-2 &2-\imageinh &1\\ +\end{matrix}\\ + \end{mathe} + + Wende die Zeilentransformation + ${Z_{2}\leftsquigarrow Z_{2}+Z_{1}}$ + an: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +2 &1-\imageinh &0\\ +0 &3-2\imageinh &1\\ +\end{matrix}\\ + \end{mathe} + + Aus der Stufenform erschließt sich + + \begin{mathe}[bc]{rclclcl} + y &= &\frac{1}{3-2\imageinh} + &= &\frac{3+2\imageinh}{3^{2}+2^{2}} + &= &\frac{3+2\imageinh}{13}\\ + x &= &\frac{1}{2}(-(1-\imageinh)y) + &= &-\frac{1}{2}\frac{3-\imageinh}{13} + &= &\frac{-3+\imageinh}{26}.\\ + \end{mathe} + \end{algorithm} +\end{enumerate} + +%% SKA 6-7 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{7} +\section[Aufgabe 7]{} + \label{ska:6:ex:7} +\let\sectionname\altsectionname + +\begin{enumerate}{\bfseries (a)} + %% SKA 6-7(a) + \item + Es gilt $4\cdot 7\equiv -1\cdot 2\equiv -2\equiv 3$ modulo $5$. + %% SKA 6-7(b) + \item + Es gilt $2^{100031}\equiv (-1)^{\text{ungerade Zahl}}\equiv -1\equiv 2$ modulo 3.\\ + Es gilt $2^{100302}\equiv (-1)^{\text{gerade Zahl}}\equiv 1$ modulo 3.\\ + %% SKA 6-7(c) + \item + Für $\intgr/5\intgr$ ist die Menge an Möglichkeiten klein, + sodass wir per \emph{brute force} das Inverse von $3$ bestimmen können: + + \hraum + \begin{tabular}[mc]{|C|CCCCC|} + \hline + n &0 &1 &2 &3 &4\\ + \hline + 3\cdot n &0 &3 &\fbox{1} &4 &2\\ + \hline + \end{tabular} + \hraum + + Darum gilt $3\cdot 2\equiv 1$ modulo $5$, sodass \fbox{$3^{-1}=2$} in $\intgr/5\intgr$. + + Für den Fall $\intgr/103\intgr$ gibt es deutlich mehr Möglichkeiten, + sodass es ein Versuch durch rohe Gewalt nicht mehr sinnvoll ist. + Darum wenden wir das Lemma von B\'ezout an und lesen das Resultat daraus. + + \textbf{Zur Erinnerung:} + Durch den Euklidischen Algorithmus auf $(a:=103,\,b:=21)$ angewandt + erhalten wir ganze Zahlen, + ${u,v\in\intgr}$, + so dass ${u\cdot 103 + v\cdot 21=\ggT(103,21)}$ gilt. + Da aber ${103\in\mathbb{P}}$ und ${1<21<103}$, gilt ${\ggT(103,21)=1}$, + sodass die o.\,s. Identität + $1\equiv u\cdot 103+v\cdot 21\equiv 0+v\cdot 21$ modulo $103$ + liefert, + was wiederum bedeutet, dass $21^{-1}=v$ in $\intgr/103\intgr$. + + Zunächst führen wird also den Euklidischen Algorithmus auf $a=103$ und $b=21$ aus: + + \begin{longtable}[mc]{|c|c|} + \hline + \hline + Restberechnung (symbolisch) &Restberechnung (Werte)\\ + \hline + \endhead +$a = b\cdot q_{1} + r_{1}$ &$103 = 21\cdot 4 + 19$\\ +$b = r_{1}\cdot q_{2} + r_{2}$ &$21 = 19\cdot 1 + 2$\\ +$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$19 = 2\cdot 9 + \boxed{\mathbf{1}}$\\ +$r_{2} = r_{3}\cdot q_{4} + r_{4}$ &$2 = 1\cdot 2 + 0$\\ + \hline + \hline + \end{longtable} + + Also gilt $\ggT(103, 21)=1$ wie erwartet. + Und jetzt kehren wir die Ausdrücke um, um die Koeffizienten $u$, $v$ zu bestimmen: + + \begin{longtable}[mc]{|c|c|} + \hline + \hline + Rest (symbolisch) &Rest (Werte)\\ + \hline + \endhead +$r_{1} = a - 4\cdot b$ &$19 = 1\cdot a + -4\cdot b$\\ +$r_{2} = b - 1\cdot r_{1}$ &$2 = -1\cdot a + 5\cdot b$\\ +$r_{3} = r_{1} - 9\cdot r_{2}$ &$\boxed{1 = \mathbf{10}\cdot a + \mathbf{-49}\cdot b}$\\ + \hline + \hline + \end{longtable} + + Darum liefert uns das B\'ezout Lemma + ${1=10\cdot a - 49\cdot b=10\cdot 103-\boxed{49}\cdot 21}$. + Also gilt wie oben \fbox{$21^{-1}=-49=54$} in $\intgr/103\intgr$.\\ + (Man prüft dies: $21\cdot 54=1134=103\cdot 11+1\equiv 1$ modulo $103$. + Also ist das Ergebnis richtig!) + + %% SKA 6-7(d) + \item + Wir führen das Gaußverfahren zuerst in $\intgr$ durch, + nur achten wir darauf, niemals mit Vielfachen von $11$ oder $13$ + zu multiplizieren: + + \begin{algorithm}[\rtab][\rtab] + Gaußverfahren angewandt auf $(A|\mathbf{b})$: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +2 &-3 &0\\ +10 &7 &-5\\ +\end{matrix}\\ + \end{mathe} + + Wende die Zeilentransformation + ${Z_{2}\leftsquigarrow Z_{2}-5\cdot Z_{1}}$ + an: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +2 &-3 &0\\ +0 &22 &-5\\ +\end{matrix}\\ + \end{mathe} + + Wir wenden die Zeilentransformation + ${Z_{1}\leftsquigarrow 7\cdot Z_{1}+Z_{2}}$ + an und erhalten das äquivalente System (\textdagger): + + \begin{mathe}[bc]{c} + \begin{matrix}{cc|c} +14 &1 &-5\\ +0 &22 &-5\\ +\end{matrix}\\ + \end{mathe} + \end{algorithm} + + In $\intgr/11\intgr$ ist das System ab (\textdagger) + + \begin{algorithm}[\rtab][\rtab] + LGS modulo 11: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +\mathbf{3} &1 &6\\ +0 &0 &\mathbf{6}\\ +\end{matrix}\\ + \end{mathe} + + $\Longrightarrow$ \fbox{System unlösbar}, da $6\neq 0$ in $\intgr/11\intgr$. + \end{algorithm} + + In $\intgr/13\intgr$ ist das System ab (\textdagger) + + \begin{algorithm}[\rtab][\rtab] + LGS modulo 13: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +1 &1 &8\\ +0 &9 &8\\ +\end{matrix}\\ + \end{mathe} + + Wende die Zeilentransformationen + ${Z_{1}\leftsquigarrow 9\cdot Z_{1}-\cdot Z_{2}}$ + an: + + \begin{mathe}[mc]{c} + \begin{matrix}{cc|c} +\mathbf{9} &0 &-1\\ +0 &\mathbf{9} &8\\ +\end{matrix}\\ + \end{mathe} + + Daraus ergibt sich die Lösung in $\intgr/13\intgr$: + + \begin{mathe}[bc]{rclclcl} + x_{1} &= &9^{-1}\cdot -1 + &= &3\cdot -1 + &= &\boxed{10}\\ + x_{2} &= &9^{-1}\cdot 8 + &= &3\cdot 8 + &= &\boxed{11}.\\ + \end{mathe} + + Also ist \fbox{$\mathbf{x}=\begin{svector}10\\11\\\end{svector}$} + die Lösung des LGS in $\intgr/13\intgr$. + \end{algorithm} +\end{enumerate} + +%% SKA 6-8 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{8} +\section[Aufgabe 8]{} + \label{ska:6:ex:8} +\let\sectionname\altsectionname + +\begin{satz*} + Sei $p\in\mathbb{P}$ eine Primzahl. + Für jedes $m\in\intgr$ mit $p\ndivides m$ + ist die Abbildung + ${M_{m}:\intgr/p\intgr\to\intgr/p\intgr}$, + die vermöge $M_{m}([x])=[m]\cdot[x](=[mx])$ + wohldefiniert und injektiv. + Insbesondere existert ein $[x]\in\intgr/p\intgr$, + so dass $[m]\cdot [x]=[1]$. +\end{satz*} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + Dass diese Abbildung wohldefiniert ist, folgt aus der Wohldefiniertheit + von Modularmultiplikation. + + \uwave{{\bfseries Injektivität:}}\\ + Seien $[x],[x']\in\intgr/p\intgr$. + Angenommen $M_{m}([x])=M_{m}([x'])$.\\ + \textbf{Zu zeigen:} $[x]=[x']$.\\ + Aus $M_{m}([x])=M_{m}([x'])$ + folgt $[mx]=[mx']$ per Konstruktion von der Abbildung $M_{m}$.\\ + Per Definition der Äquivalenzklassen + gilt somit $mx\equiv mx'$ modulo $p$.\\ + Daraus folgt ${p\divides (mx-mx')}$, + also ${p\divides m\cdot(x-x')}$.\\ + Da $p$ prim ist, gilt $p\divides m$ oder $p\divides (x-x')$ + (siehe \cite[Satz 3.4.14]{sinn2020}).\\ + Per Voraussetzung auf $m$ folgt daraus, dass $p\divides(x-x')$.\\ + Daraus folgt $x\equiv x'$ modulo $p$, + und somit $[x]=[x']$ per Definition der Äquivalenzklassen.\\ + Darum ist $M_{m}$ injektiv. + + Da nun $\intgr/p\intgr$ endlich ist, + sind injektive Abbildungen zwischen $\intgr/p\intgr$ und sich selbst + automatisch surjektiv.\\ + Per Surjektivität existiert ein Element $[x]\in\intgr/p\intgr$, + so dass $[1]=M_{m}([x])=[m]\cdot [x]$.\\ + Das heißt, $[m]$ ist invertierbar innerhalb $\intgr/p\intgr$. + \end{proof} + \end{einzug} + +\textbf{Bemerkung.} +Die letzte Aussage in diesem Satz gilt auch allgemeiner: +Sind $n,m\in\intgr$ teilerfremd, dann ist $[m]$ innerhalb $\intgr/n\intgr$ +invertierbar. +Aber wenn $n$ nicht prim ist, können wir das o.\,s. Argument nicht verwenden, +da das Zwischenresultat der Injektivität nicht mehr gilt. +Stattdessen müssen wir schon das Lemma von B\'ezout anwenden. + \setcounternach{part}{3} \part{Quizzes}