From 23491ce7858277d2915cbd7f8babcf3772868416 Mon Sep 17 00:00:00 2001 From: flavis Date: Thu, 7 May 2020 22:20:02 +0200 Subject: [PATCH] finish la2 --- lecture.cls | 4 + sose2020/la/uebungen/la2.pdf | Bin 0 -> 115928 bytes sose2020/la/uebungen/la2.tex | 253 +++++++++++++++++++++++++++++++++++ 3 files changed, 257 insertions(+) create mode 100644 sose2020/la/uebungen/la2.pdf create mode 100644 sose2020/la/uebungen/la2.tex diff --git a/lecture.cls b/lecture.cls index 147516b..a685c74 100644 --- a/lecture.cls +++ b/lecture.cls @@ -23,6 +23,7 @@ \RequirePackage{forloop} \RequirePackage{totcount} \RequirePackage{calc} +\RequirePackage{wasysym} \usetikzlibrary{quotes, angles} @@ -145,3 +146,6 @@ \makeatletter \renewcommand\d[1]{\ensuremath{% \;\mathrm{d}#1\@ifnextchar\d{\!}{}}} \makeatother + +% contradiction +\newcommand{\contr}{\text{\Large\lightning}} diff --git a/sose2020/la/uebungen/la2.pdf b/sose2020/la/uebungen/la2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..94bf126e0542b5a5f706e1c8ae5d6264c3cbf317 GIT binary patch literal 115928 zcma&NQ*dru*DV~|wzJ~oiEZ1qZQJIGZQD*(Y};6|ZTu~v}&V| zHb-ltcX9<0F*-(iRv7ZRrQtOgHX;C#y^$3R4-X83jH#Wuiv`ciE!_G)y~1_nFxWpnQ_)avyTMQ43s z>uu5A{*5CtqE0@Jk^KvJ}FY|;E|{#5RiUFkaX8OGA)>*
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,,$\subseteq$'': Sei $r \in I (J + K)$. Dann ex. $a_1, \ldots, a_n \in I$, + $b_1, \ldots, b_n \in (J + K)$ und + $\forall b_i: \exists c_i \in J, d_i \in K$, s.d. + \[ + r = \sum_{i=1}^{n} a_i b_i = \sum_{i=1}^{n} a_i (c_i + d_i) + = \sum_{i=1}^{n} (a_i c_i + a_i d_i) + = \underbrace{\sum_{i=1}^{n} a_i c_i}_{ \in IJ} + + \underbrace{\sum_{i=1}^{n} a_i c_i}_{\in IK } \in IJ + IK + .\] + ,,$\supseteq$'': Wegen $J, K$ Ideale gilt $0 \in J$, $0 \in K$, d.h. + $IJ = I (J + (0)) \subseteq I (J + K)$ und $IK \subseteq I ((0) + K) \subseteq I (J + K)$. + Da $I (J + K)$ Ideal, folgt $IJ + IK \subseteq I (J + K)$. + \end{proof} + \item Beh.: $(I \cap J)(I + J) \subseteq IJ \subseteq I \cap J$. + \begin{proof} + Sei $r \in (I \cap J)(I + J)$. Dann ex. + $a_1, \ldots, a_n \in I \cap J$, $c_1, \ldots, c_n \in I$ und + $d_1, \ldots, d_n \in J$ s.d. + \[ + r = \sum_{i=1}^{n} a_i (c_i + d_i) + = \sum_{i=1}^{n} (a_i c_i + c_i d_i + a_i d_i - c_i d_i) + = \underbrace{\sum_{i=1}^{n} c_i ( a_i + d_i)}_{\in IJ} + + \underbrace{\sum_{i=1}^{n} d_i (a_i - c_i)}_{\in IJ} \in IJ + .\] Also folgt $(I \cap J)(I + J) \subseteq IJ$. + + Sei nun $r \in IJ$. Dann ex. $a_1, \ldots, a_n \in I$ und $b_1, \ldots, b_n \in J$ mit + \[ + r = \sum_{i=1}^{n} a_i b_i + .\] Wegen $a_i \in I$ und $I$ Ideal, ist $a_i b_i \in I$ und wegen + $b_i \in J$ und $J$ Ideal, ist $a_i b_i \in J$, also $r \in I \cap J$. + Damit folgt $IJ \subseteq I \cap J$ und die Behauptung. + \end{proof} + \item Beh.: Ist $I + J = (1)$, dann ist $I \cap J = IJ$. + \begin{proof} + Es sei $I + J = (1)$. Mit (b) folgt bereits $IJ \subseteq I \cap J$. + + Außerdem folgt mit (b): $(I \cap J)(I + J) = (I \cap J)(1) \subseteq IJ$. + Bleibt zu zeigen: $I \cap J \subseteq (I \cap J)(1)$. Sei dazu $r \in I \cap J$. Dann + ist $r = r \cdot 1 \in (I \cap J)(1)$. Damit folgt + $I \cap J \subseteq IJ$ und die Behauptung. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Beh.: Er füttert und badet die Python vom Mittwoch dieser Woche in sieben Tagen am selben Tag. + \begin{proof} + Zunächst ist die Fütterung durch $\Z / 4 \Z$ und das Baden durch $\Z / 7 \Z$ modelliert. + Weiter sind $4 \Z$ und $7 \Z$ wegen $2 \cdot 4 - 7 = 1$ relativ prim. + + Außerdem ist $4 \Z \cap 7 \Z \supseteq 28 \Z$, denn: $28 = 4 \cdot 7 \in 4 \Z \cap 7 \Z$. + Weiter ist $4 \Z \cap 7 \Z \subseteq 28 \Z$, denn: $\forall r \in 28\Z$ gilt + $4 \mid r$ und $7 \mid r \implies r \in 4 \Z \cap 7 \Z$. + + Mit dem chinesischen Restsatz folgt damit: + \[ + \Z / 28 \Z \stackrel{\sim }{=} \Z / 4 \Z \times \Z / 7 \Z + .\] Startzeitpunkt Mittwoch ist $(\overline{1}, \overline{0}) \in \Z / 4 \Z \times \Z / 7 \Z$. + Nach dem chinesischen Restsatz existiert auch die Abbildung + \[ + \varphi \colon \Z \to \Z / 4 \Z \times \Z / 7 \Z + .\] Es ist $\varphi(21) = (\overline{1}, \overline{0})$, da $21 \equiv 1$ $(\text{mod } 4)$ + und $21 \equiv 0$ $(\text{mod } 7)$. Also ist in 7 Tagen, ausgehend von Mittwoch dieser Woche: + \[ + \varphi(21 + 7) = \varphi(28) = (\overline{0}, \overline{0}) + .\] + Also wird die Python an diesem Tag sowohl gebadet, als auch gefüttert, und danach + alle 28 Tage wieder. + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Sei $\Z[\sqrt{-3}] := \{a + b \sqrt{-3} \mid a, b \in \Z\} \subset \mathbb{C}$ und + $\delta \colon \Z[\sqrt{-3} ] \to \N_0$, $a + b \sqrt{-3} \mapsto a^2 + 3b^2$. + \begin{enumerate}[(a)] + \item Beh.: $\delta(1) = 1$ und $\delta (x\cdot y) = \delta (x) \cdot \delta (y)$ + $\forall x, y \in \Z[\sqrt{-3}] $ + \begin{proof} + Es gilt $\delta (1) = 1^2 = 1$. + + Seien $x, y \in \Z[\sqrt{-3} ]$ mit $x = a + \sqrt{-3} b$ und $y = c + \sqrt{-3}d$. + Dann folgt: + \begin{align*} + \delta (x \cdot y) &= \delta (ac - 3bd + \sqrt{-3} (bc + ad)) \\ + &= a^2c^2 - 6bdac + 9b^2d^2 + 3(b^2c^2 + 2bdac + a^2d^2) \\ + &= a^2 c^2 + 9b^2d^2 + 3b^2c^2 + 3 a^2d^2 \\ + &= (a^2 + 3b^2) (c^2 + 3d^2) \\ + &= \delta (a + \sqrt{-3} b) \cdot \delta (c + \sqrt{-3} d) \\ + &= \delta (x) \cdot \delta (y) + .\end{align*} + \end{proof} + \item Beh.: $\Z[\sqrt{-3}]^{\times } = \{ x \in \Z[\sqrt{-3}] \mid \delta (x) = 1\} = \{\pm 1\} $ + \begin{proof} + Zunächst gilt: $\{\pm 1\} \subseteq \{ x \in \Z[\sqrt{-3} ] \mid \delta (x) = 1\} $, denn + $\delta (1) = 1 = \delta (-1)$. + + Sei nun $x \in \Z[-3]$ mit $\delta (x) = 1$. Dann ex. $a, b \in \Z$ mit + $x = a + \sqrt{-3} b$. Damit folgt + \[ + \delta (x) = \underbrace{a^2}_{\ge 0} + \underbrace{3b^2}_{\ge 0} = 1 + .\] Ang. $b \neq 0$, dann ist $b^2 \ge 1 \implies 3b^2 \ge 3 \implies \delta (x) \ge 3$ + $\contr$. Also ist $b = 0 \implies a^2 = 1 \implies x \in \{\pm 1\}$. + Damit gilt $\{x \in \Z[\sqrt{-3} ] \mid \delta (x) = 1\} = \{\pm 1\} $. + + Sei nun $x \in \Z[\sqrt{-3}]^{\times }$. Dann ex. $y \in \Z[\sqrt{-3}]$ mit + $xy = 1 \implies \delta (xy) = \delta (x) \delta (y) = 1$. Wegen + $\delta (x), \delta (y) \in \N_0 \implies \delta(x) = 1$. + Also + $\Z[\sqrt{-3}]^{\times} \subseteq \{x \in \Z[\sqrt{-3} ] \mid \delta (x) = 1\} = \{\pm 1\} $. + + Offensichtlich ist $1 \cdot 1 = (-1) \cdot (-1) = 1$, also + $\{\pm 1\} \subseteq \Z[\sqrt{-3}]^{\times }$. Damit folgt die Beh. + \end{proof} + \item Beh.: $\delta^{-1}(2) = \emptyset$. + \begin{proof} + Ang. es ex. ein $x \in \Z[\sqrt{-3} ]$ mit $\delta (x) = 2$. Dann ex. $a, b \in \Z$ + mit $x = a + \sqrt{-3} b$. Dann folgt $\delta (x) = a^2 + 3b^2 = 2$. + + Ang.: $b \neq 0 \implies b^2 \ge 1 \implies 3b^2 \ge 3$. + Wegen $a^2 \ge 0 \implies \delta (x) \ge 3$ $\contr$. Also $b = 0$. + + Damit folgt $a^2 = 2$, aber $a \in \Z$ $\contr$. + \end{proof} + + Beh.: $2$ und $1 \pm \sqrt{-3} $ sind irreduzibel. + \begin{proof} + Es gilt zunächst $\delta (2) = \delta (1 \pm \sqrt{-3}) = 4$. + + Seien $x, y \in R$ mit $xy = 2$. Dann folgt + $\delta (x) \cdot \delta (y) = \delta (xy) = \delta (2) = 4$. Da aber + $\delta (a) \in \N_0$ $\forall a \in \Z[\sqrt{-3}]$, folgt, dass $\delta (2) = 4$ nur + als $\delta (2) = 2 \cdot 2$ oder $\delta (2) = 4 \cdot 1$ darstellbar ist. + + Wegen $\delta^{-1}(2) = \emptyset$, folgt, dass entweder $\delta (x) = 1$ oder $\delta (y) = 1$ + ist. Wegen (b) folgt damit, dass $x \in \Z[\sqrt{-3} ]^{\times }$ oder + $y \in \Z[\sqrt{-3}]^{\times }$. + + Für $1 \pm \sqrt{-3} $ analog. + \end{proof} + + Beh.: $2$ ist kein Primelement in $\Z[\sqrt{-3} ]$. + \begin{proof} + Es gilt $2 \mid 4 = (1 + \sqrt{-3})(1 - \sqrt{-3})$. Aber + $2 \nmid (1 \pm \sqrt{-3})$, da $1 \pm \sqrt{-3} $ irreduzibel. Also + ist $2$ kein Primelement. + \end{proof} + \item Beh.: $\text{GGT}(4, 2+2\sqrt{-3}) = \emptyset$. + \begin{proof} + Zunächst ist $\delta (4) = \delta (2 + 2 \sqrt{-3}) = 16$. Sei $b \in \Z[\sqrt{-3}]$ + Teiler von $a \in \Z[\sqrt{-3}]$ mit $\delta(a) = 16$. Dann ex. + $c \in \Z[\sqrt{-3}]$ s.d. $a = bc$. Dann gilt + $\delta(a) = \delta (bc) = \delta (b) \cdot \delta (c) = 16$. Wegen + $\delta (b) \in \N_0 \implies \delta (b) \in \{1, 2, 4, 8, 16\}$. Wegen + $\delta^{-1}(2) = \emptyset$ und $2 \cdot 8 = 16 \implies \delta(b) \in \{1, 4, 16\} $. + + Es lässt sich nachrechnen, dass + \begin{align*} + \delta^{-1}(1) &= \{\pm 1 \} \\ + \delta^{-1}(4) &= \{\pm 2, \pm (1 \pm \sqrt{-3}) \} \\ + \delta ^{-1}(16) &= \{\pm 4, \pm (2 \pm \sqrt{-3}) \} + .\end{align*} + Damit folgt, dass alle möglichen gemeinsamen Teiler + gegeben sind durch: + \[ + T = \{\pm 1, \pm 2, \pm (1 \pm \sqrt{-3}), \pm 4, \pm (2 \pm \sqrt{-3})\} + .\] + Es gilt weiter $4 \nmid (2 \pm 2\sqrt{-3})$, da $\forall a, b \in \Z$ gilt: + \[ + 4 (a + \sqrt{-3} b) = \underbrace{4a}_{\neq 2} + 4 b \sqrt{-3} \neq 2 \pm 2 \sqrt{-3} + .\] Da $-4$ bzw. $- (2 \pm \sqrt{-3})$ assoziierte zu $4$ bzw. $2 \pm \sqrt{-3} $ sind, + folgt $\pm 4 \nmid \pm (2 \pm 2 \sqrt{-3})$. + + Außerdem gilt $(2 \pm 2 \sqrt{-3}) \nmid 4$, da, ang. es ex. $a, b \in \Z$ mit: + \[ + 4 = (2 \pm 2 \sqrt{-3} ) (a + \sqrt{-3} b) + = \underbrace{2a \mp 6b}_{=4} + \sqrt{-3} (\underbrace{2b \pm 2a}_{= 0}) + \implies 2a = \mp 2b \implies = \mp \underbrace{8b}_{\in \Z} = 4 \quad \contr + .\] Analog folgt wieder $\pm (2 \pm 2 \sqrt{-3}) \nmid \pm 4$. + Damit folgt $\pm 4$, $\pm (2 \pm 2 \sqrt{-3}) \not\in \text{GGT}(4, 2 + 2 \sqrt{-3})$. + + Da $\pm 2, \pm (1 \pm \sqrt{-3}) \not\in \Z[\sqrt{-3}]^{\times }$, folgt + $\pm 2 \nmid \pm 1, \pm (1 \pm \sqrt{-3}) \nmid \pm 1$. Wegen + $\pm 2$ und $\pm (1 \pm \sqrt{-3})$ irreduzibel, folgt + $\pm 2 \nmid \pm (1 \pm \sqrt{-3})$ und $\pm (1 \pm \sqrt{-3}) \nmid \pm 2$. + + Damit folgt die Behauptung. + \end{proof} + \item Beh.: $\Z[\sqrt{-3}]$ ist nicht faktoriell. + \begin{proof} + Es ist $2$ und $1 \pm \sqrt{-3}$ irreduzibel. Damit folgt + \[ + 4 = 2 \cdot 2 = (1 + \sqrt{-3})(1 - \sqrt{-3}) + .\] Damit hat $4$ keine, bis auf Assoziiertheit und Reihenfolge + eindeutige Zerlegung in irreduzible Elemente. Also folgt die Behauptung. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $R$ ein Ring. + Beh.: $R$ ist noethersch $\iff$ Jedes Ideal in $R$ ist endlich erzeugt. + \begin{proof} + ,,$\implies$'': Sei R noethersch und $I \subseteq R$ ein Ideal. + Dann konstruiere induktiv eine Folge von Idealen $(I_{j})_{j \in \N}$. + Dazu sei $a_1 \in I$. Setze $I_1 := (a_1)$. + + Seien $I_1$ bis $I_j$ bereits konstruiert. Dann sei $a_{j+1} \in I$ mit + $a_{j+1} \not\in I_j$. Falls kein $a_{j+1}$ mit solcher Eigenschaft existiert, folgt + mit $I = (a_1, \ldots, a_j)$ die Behauptung. Sonst setze $I_{j+1} := (a_1, \ldots, a_j, a_{j+1})$. + Dafür gilt nach Konstruktion: $I_j \subseteq I_{j+1} \subseteq I$. + + So ensteht eine aufsteigende Kette von Idealen in $R$. Da $R$ noethersch, ex. $n \in \N$ mit + $I_j = I_n$ $\forall j \ge n$. Für $I_n$ gilt damit $I_n = (a_1, \ldots, a_n) = I$, also + ist $I$ endlich erzeugt. + + ,,$\impliedby$'': Sei jedes Ideal in $R$ endlich erzeugt und $I_1 \subseteq I_2 \subseteq \ldots$ + aufsteigende Kette von Idealen. Dann setze + \[ + J := \bigcup_{k\ge 1} I_k + .\] Zunächst gilt $J$ ist Ideal in $R$, da + \begin{enumerate}[label=(I\arabic*)] + \item $0 \in I_k$ $\forall k \ge 1 \implies 0 \in J$ + \item Seien $a, b \in J \implies \exists k, l \in \N$ mit $a \in I_k$ und $b \in I_l$. + O.E. sei $k \ge l$. Dann ist $I_l \subseteq I_k \implies a, b \in I_k$. Da + $I_k$ Ideal, folgt $a + b \in I_k \implies a + b \in J$. + \item Sei $a \in J$, $r \in R \implies \exists k \in \N$ mit $a \in I_k$. + Da $I_k$ Ideal, folgt $r \cdot a \in I_k \implies r \cdot a \in J$. + \end{enumerate} + + Da jedes Ideal in $R$ endlich erzeugt ist, ex. $a_1, \ldots, a_k \in R$ mit + $J = (a_1, \ldots, a_k)$. Also ist $a_1, \ldots, a_k \in J$. Damit ex. $n \in \N$ mit + $a_1, \ldots, a_k \in I_n$. Also ist + $(a_1, \ldots, a_n) \subseteq I_n \subseteq J = (a_1, \ldots, a_n)$. + Dann folgt $I_n = J$, also gilt $\forall k \ge n$: $I_k = I_n$. Damit ist + $R$ noethersch. + \end{proof} +\end{aufgabe} + +\end{document}