From b33c550fbb360a6dbf5a9310d7ab42907619265e Mon Sep 17 00:00:00 2001 From: flavis Date: Mon, 4 May 2020 09:11:27 +0200 Subject: [PATCH] update proseminar skript and ana --- sose2020/ana/lectures | 2 +- sose2020/proseminar/skript.pdf | Bin 145344 -> 150566 bytes sose2020/proseminar/skript.tex | 92 ++++++++++++++++++++++----------- 3 files changed, 63 insertions(+), 31 deletions(-) diff --git a/sose2020/ana/lectures b/sose2020/ana/lectures index 0ee655c..82dc897 160000 --- a/sose2020/ana/lectures +++ b/sose2020/ana/lectures @@ -1 +1 @@ -Subproject commit 0ee655c856d37c6cf66ac432352429d31e4bf214 +Subproject commit 82dc89716804cf1e3b99c702813f0f6884552775 diff --git a/sose2020/proseminar/skript.pdf b/sose2020/proseminar/skript.pdf index c1d5c82830f1bc37c4bfc79b6a6afd8ce0b399ce..7f9722b3b2201401d4dc9d9719420d767a233d8d 100644 GIT binary patch delta 42591 zcmV(}K+wOy?+B)t36LWLF)%Qbfk-HS)m&S1+cp+{_pi_+HDjT;U);$gN&qda`kq?DH3#|2}9CMt`?Km1xM4Fl(P8k;>EA$(-~vDxH!Lmy!z$k 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Es sei $R$ ein Ring. Ein Element $x \in R$ heißt ein Nullteiler, wenn es ein $y \in R, y\neq 0$ @@ -28,6 +34,8 @@ $y \in R, y \neq 0$ existiert.} \begin{definition}[Einheit] Es sei $R$ ein Ring. Ein Element $x \in R$ heißt eine Einheit, wenn es ein $y \in R$ mit $xy = 1$ gibt. + Bezeichnung: $x^{-1} := y$. + Die Einheiten im Ring $\Z/n\Z$ heißen \underline{prime Restklassen modulo n}. \textit{also: wenn ein Inverses bezüglich Multiplikation existiert} \end{definition} @@ -49,15 +57,15 @@ $y \in R, y \neq 0$ existiert.} Insbesondere gibt es für jedes $x \in R^{\times }$ genau ein $y \in R^{\times }$ mit $xy = 1$. Dieses Element bezeichnen wir mit $x^{-1}$ und nennen es das (multiplikativ) Inverse zu $x$. + + Die Gruppe $(\Z / n \Z)^{\times}$ wird als \underline{prime Restklassengruppe modulo n} bezeichnet. \item Ist $x \in R^{\times }$, dann ist $x$ kein Nullteiler. \label{lemma:einheitengruppe:nullteiler} \item Falls $R$ endlich ist, dann gilt auch die Umkehrung von (b): Ist $x \in R$ kein Nullteiler, dann ist $x$ eine Einheit. \end{enumerate} \textit{Insbesondere gilt im Fall $R = \Z / n \Z$ , $n \in \N$, für $\overline{x} \in R$, dass $\overline{x}$ - genau dann eine Einheit ist, wenn $\overline{x}$ kein Nullteiler ist.} Die Einheiten im - Ring $\Z / n \Z$ nennt man prime Restklassen modulo n, die Gruppe $(\Z / n \Z)^{\times }$ - dementsprechend die Gruppe der primen Restklassen modulo n. + genau dann eine Einheit ist, wenn $\overline{x}$ kein Nullteiler ist.} \end{lemma} \begin{proof} @@ -127,8 +135,9 @@ $y \in R, y \neq 0$ existiert.} insbesondere ist $\Z / n \Z$ nicht nullteilerfrei. \end{proof} -\textit{Anmerkung: Den erweiterten euklidischen Algorithmus verwenden wir noch weiter um die Einheiten -im Ring $\Z / n \Z$ zu charakterisieren.} +\textit{Anmerkung: Letzte Woche haben wir bereits Kongruenzrelationen in $\Z$ kennengelernt, heute: +lernen Kongruenzen zu lösen. Frage: Wann hat eine Kongruenz eine Lösung in $\Z$?} +\vspace{5mm} \begin{lemma} Es seien $a$, $b \in \Z$, $n \in \N$. Dann sind äquivalent: @@ -158,6 +167,22 @@ im Ring $\Z / n \Z$ zu charakterisieren.} \] und damit die Behauptung zeigt. \end{proof} +\begin{bsp} + %\begin{enumerate}[(a)] + %\item Die Kongruenz $15x \equiv 7$ $(\text{mod } 21)$ hat wegen $ggT(15,21) = 3 \nmid 7$ keine + % Lösung. + %\item + Die Kongruenz $15x \equiv 6$ $(\text{mod } 21)$ hat wegen $\text{ggT}(15,21) = 3 \mid 6$ eine Lösung. + Der erweiterte Euklidische Algorithmus ergibt + \[ + \text{ggT}(15, 21) = 3 = 3 \cdot 15 + (-2) \cdot 21 + .\] Damit folgt \textit{wie im Beweis durch Multiplikation mit 2} + \[ + 6 = 6 \cdot 15 + (-4) \cdot 21 \equiv 15 \cdot 6 \quad (\text{mod } 21) + \] d.h. $x = 6$ ist eine Lösung der Kongruenz. + %\end{enumerate} +\end{bsp} + \begin{korrolar} Es sei $a \in \Z, n \in \N$. Dann sind äquivalent \begin{enumerate}[(i)] @@ -172,25 +197,18 @@ im Ring $\Z / n \Z$ zu charakterisieren.} $\overline{a} \cdot \overline{x} = \overline{1}$ in $\Z / n \Z$, welche genau dann lösbar ist, wenn $\overline{x}$ eine Einheit in $\Z / n \Z$ ist. - (i) $\iff$ (iii): $\text{ggT}(a,n) = 1 \iff \text{ggT}(a,n) \mid 1$. Folgt mit $b = 1$ aus + (i) $\iff$ (iii): $\text{ggT}(a,n) = 1 \iff \text{ggT}(a,n) \mid 1$. + \textit{Anmerkung: Hinrichtung klar, Rückrichtung: $\text{ggT}(a,n) \mid 1$ heißt, + 1 ist ein Vielfaches des $\text{ggT}(a,n)$, d.h. + der $\text{ggT}(a,n)$ ist bereits 1}. Folgt mit $b = 1$ aus \ref{lemma:kongruenz}. + + \textit{Hier wird klar, warum die Einheiten in $\Z / n \Z$ prime Restklassen heißen: + teilerfremd wird auch als relativ prim bezeichnet.} \end{proof} -\begin{bsp} - %\begin{enumerate}[(a)] - %\item Die Kongruenz $15x \equiv 7$ $(\text{mod } 21)$ hat wegen $ggT(15,21) = 3 \nmid 7$ keine - % Lösung. - %\item - Die Kongruenz $15x \equiv 6$ $(\text{mod } 21)$ hat wegen $\text{ggT}(15,21) = 3 \mid 6$ eine Lösung. - Der erweiterte Euklidische Algorithmus ergibt - \[ - \text{ggT}(15, 21) = 3 = 3 \cdot 15 + (-2) \cdot 21 - .\] Damit folgt \textit{wie im Beweis durch Multiplikation mit 2} - \[ - 6 = 6 \cdot 15 + (-4) \cdot 21 \equiv 15 \cdot 6 \quad (\text{mod } 21) - \] d.h. $x = 6$ ist eine Lösung der Kongruenz. - %\end{enumerate} -\end{bsp} +\textit{Frage: Wie können große Potenzen modulo n vereinfacht werden? Dazu wird Satz v. Euler-Fermat +helfen, dafür brauchen wir aber noch ein paar Definitionen} \begin{definition}[Ordnung] Es sei $G$ eine endliche Gruppe. Die Ordnung von $G$ (Notation: $|G|$) ist definiert als die Anzahl @@ -265,13 +283,27 @@ Notation für Gruppen: Verknüpfung multiplikativ und neutrales Element ist $1$. \end{proof} \begin{bsp} - Es ist $3^{19} \equiv 10$ $(\text{mod } 17)$, denn $\overline{3} \in (\Z / 17 \Z)^{\times}$ und - \[ - 3^{16} = 3^{\varphi(17)} \qquad\qquad \stackrel{\text{Satz v. Euler-Fermat}}{\equiv} \qquad \qquad 1 \quad (\text{mod } 17) - .\] Damit folgt - \[ - 3^{19} = 3^{3} \cdot 3^{16} = 27 \cdot 1 \equiv 10 \quad (\text{mod } 17) - .\] + \begin{enumerate}[(1)] + \item Es ist $3^{19} \equiv 10$ $(\text{mod } 17)$, denn $\overline{3} \in (\Z / 17 \Z)^{\times}$ und + \[ + 3^{16} = 3^{\varphi(17)} \qquad\qquad \stackrel{\text{Satz v. Euler-Fermat}}{\equiv} \qquad \qquad 1 \quad (\text{mod } 17) + .\] Damit folgt + \[ + 3^{19} = 3^{3} \cdot 3^{16} \equiv 27 \cdot 1 \equiv 10 \quad (\text{mod } 17) + .\] + \item Was ist die letzte Dezimalstelle von $7^{222}$? Also welche Zahl ist + $7^{222}$ kongruent modulo 10? + + Zunächst $\varphi(10) = 4$. Und $\text{ggT}(7,10) = 1$. Dann folgt + \[ + 7^{4} = 7^{\varphi(10)} \qquad\qquad \stackrel{\text{Satz v. Euler-Fermat}}{\equiv} + \qquad \qquad 1 \quad (\text{mod } 10) + .\] Dann teile $222$ durch $4$ mit Rest. Damit + \[ + 7^{222} = 7^{4 \cdot 55 + 2} = (7^{4})^{55} \cdot 7^{2} \equiv 1^{55} \cdot 7^{2} + \equiv 49 \equiv 9 \quad (\text{mod } 10) + .\] + \end{enumerate} \end{bsp} \begin{korrolar}[Kleiner Satz von Fermat]