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R0@wlUj4)(mqVi%e{}0$i<+=a> literal 0 HcmV?d00001 diff --git a/sose2020/la/uebungen/la1.tex b/sose2020/la/uebungen/la1.tex new file mode 100644 index 0000000..60717f8 --- /dev/null +++ b/sose2020/la/uebungen/la1.tex @@ -0,0 +1,209 @@ +\documentclass[uebung]{../../../lecture} + +\begin{document} + +\author{Dominik Daniel, Christian Merten} +\title{Lineare Algebra II: Übungsblatt 1} + +\punkte + +\setcounter{aufgabe}{3} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh $(*)$.: $R$ nullteilerfrei $\implies \forall f, g \in R[t]$ gilt + $\text{deg}(f\cdot g) = \text{deg}(f) + \text{deg}(g)$. + \begin{proof} + Seien $f, g \in R[t]$. Falls $f \neq 0$ und $g\neq 0$ folgt + $\text{e}(f) \neq 0$ und $\text{e}(g) \neq 0$. Wegen $R$ nullteilerfrei + folgt damit $\text{e}(f\cdot g) \neq 0$. Aus der Multiplikation von + Polynomen folgt somit direkt $\text{deg}(f\cdot g) = f + g$. + + Falls $f = 0$ und/oder $g = 0$: O.B.d.A.: $f = 0$. Dann ist $f \cdot g = 0$. Damit + \[ + \text{deg}(f) + \text{deg}(g) = -\infty + \text{deg}(g) = -\infty = \text{deg}(0) = \text{deg}(f\cdot g) + .\] + \end{proof} + Beh.: $R$ nullteilerfrei $\implies R[t]^{\times } = R^{\times }$ + \begin{proof} + ,,$\subset $'': Sei $f \in R[t]^{\times }$. Dann ex. $g \in (R[t])^{\times}$ mit + $f\cdot g = 1$. Dann folgt mit $(*)$: + \[ + 0 = \text{deg}(1) = \text{deg}(f \cdot g) = \text{deg}(f) + \text{deg}(g) + .\] + Wegen $R$ nullteilerfrei folgt $R \neq 0$ und damit $f, g \neq 0$. Damit folgt + $\text{deg}(f) \in \N_0$ und $\text{deg}(g) \in \N_0$. Also ist + \[ + \text{deg}(f) = \text{deg}(g) = 0 + .\] + Damit ex. $a, b \in R$ mit $f = a$, $g = b$ und $ab = 1$. D.h. $f = a \in R^{\times}$. + + ,,$\supset$'': Sei $a \in R^{\times}$. Dann ex. $b \in R^{\times }$ mit $ab = 1$. + Es ist $a,b \in R[t] \implies a \in R[t]^{\times}$. + \end{proof} + \item $\Z / 4 \Z$ ist wegen $\overline{2} \cdot \overline{2} = \overline{4} = \overline{0}$ und + $\overline{2} \neq 0$ nicht nullteilerfrei. Hier ist + $\overline{1} + \overline{2}t \in (\Z / 4 \Z)^{\times }$, wegen + \[ + (\overline{1} + \overline{2}t) (\overline{1} + \overline{2}t) + = \overline{1} + \underbrace{\overline{4}t + \overline{4}t^2}_{= 0} + = \overline{1} + .\] Aber wegen $2 \neq 0$ ist $1 + \overline{2}t \not\in (\Z/4\Z)^{\times }$. + \item Sei $R = \Z / 4 \Z$ und $f = \overline{2}t$. Dann ist wegen + $f(\overline{2}) = \overline{4}t = \overline{0}$: + \[ + \#\{r \in R | f(r) = 0\} = \#\{\overline{0}, \overline{2}\} = 2 > 1 = \text{deg}(f) + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\varphi$ ist surjektiver Ringhomomorphismus. + \begin{proof} + Zunächst ist $\varphi(1) = 1$. Seien $f, g \in \R[t]$ beliebig. Dann gilt + \begin{align*} + \varphi(f+g) &= (f+g)(i) = f(i) + g(i) = \varphi(f) + \varphi(g) \\ + \varphi(f\cdot g) &= (f\cdot g)(i) = f(i) \cdot g(i) = \varphi(f) \cdot \varphi(g) + .\end{align*} + $\implies \varphi$ Ringhomomorphismus. + + Sei $z \in \mathbb{C}$ beliebig, dann ex. $a, b \in \R$ mit $z = a + ib$. Wähle nun + $f := a + bt \in \R[t]$. Damit folgt $\varphi(f) = a + ib = z$. + $\implies \varphi$ surjektiv. + \end{proof} + \item Beh.: $t^2 + 1 \in \text{ker } \varphi$ und $\forall f \in \R[t] \setminus \{0\} $ + mit $\text{deg}(f) < 2$ gilt $\varphi(f) \neq 0$. + \begin{proof} + $\varphi(t^2 + 1) = i^2 + 1 = -1 + 1 = 0 \implies t^2 + 1 \in \text{ker } \varphi$. + + Sei $f \in \R[t]$ mit $\text{deg}(f) < 2$. Dann ex. $a, b \in \R$ mit + $f = a + bt$. Damit folgt $\varphi(f) = a + ib$. Falls $b \neq 0$ dann + ist $ib \not\in \R \implies -a \neq ib \implies \varphi(f) \neq 0$.\\ + Falls $b = 0$, dann ist wegen $f \neq 0$: $\varphi(f) = a \neq 0$. + \end{proof} + \item Beh.: $\text{ker } \varphi = (t^2 + 1)$. + \begin{proof} + ,,$\subset$'': Sei $f \in \text{ker } \varphi$. Teile $f$ durch $t^2 + 1$ mit Rest. + Dann ex. $g, h \in \R[t]$ mit + \[ + f = g \cdot (t^2+1) + h \qquad \text{deg}(h) < \text{deg}(t^2 + 1) = 2 + .\] Wegen $\varphi(f) = 0$ und $\varphi$ Ringhomomorphismus folgt + \[ + \varphi(f) = \varphi(g) \varphi(i^2+1) + \varphi(h) \stackrel{(b)}{=} \varphi(h) = 0 + .\] Wegen $\varphi(h) = 0$, $\text{deg}(h) < 2$ und (b) folgt also $h = 0$. + Damit ist $f = g \cdot (t^2 + 1) \in (t^2+1)$. + + ,,$\supset$'': Sei $f \in (t^2+1)$. Dann ex. $g \in \R[t]$ mit $f = g\cdot(t^2+1)$. Damit + folgt $\varphi(f) = \varphi(g) \varphi(t^2+1) \stackrel{(b)}{=} 0 \implies f \in \text{ker } \varphi$. + \end{proof} + \item Beh.: $\R[t] / (t^2+1) \stackrel{\sim}{=} \mathbb{C}$ und $(t^2+1) \subset \R[t]$ ist + maximales Ideal. + + \begin{proof} + Folgt mit $\text{ker } \varphi = (t^2+1)$ und $\varphi$ surjektiv, also + $\text{im } \varphi = \mathbb{C}$ aus dem Homomorphiesatz. + + Damit ist $\R[t] / (t^2+1)$ isomorph zum Körper $\mathbb{C}$, damit selbst ein + Körper. Damit folgt mit Bem. 1.23, dass $(t^2+1)$ ein maximales Ideal ist. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[] + \begin{enumerate}[(a)] + \item Beh.: $\sqrt{I} $ ist ein Ideal in R mit $I \subset \sqrt{I} $. + \begin{proof} + $r \in I \implies r^{1} \in I \implies r \in \sqrt{I} \implies I \subset \sqrt{I} $. + + \begin{enumerate}[label=(I\arabic*)] + \item $I$ Ideal $\implies 0 \in I \implies 0^{1} \in I \implies 0 \in \sqrt{I}$. + \item Seien $a,b \in \sqrt{I}$: Dann ex. $n, m \in \N$ mit $a^{n} \in I$ und + $b^{m} \in I$. Definiere $k := n \cdot m$. Dann ist wegen $I$ Ideal + $a^{k} = (a^{n})^{m} \in I$ und $b^{k} = (b^{m})^{n} \in I$. Dann folgt + mit binomischer Formel und da jeder Summand entweder einen + Faktor $a^{n}$ oder $b^{m}$ enthält: + \[ + (a + b)^{k} = \sum_{l=0}^{k} \binom{k}{l} a^{k-l}b^{l} + = a^{k} + \ldots + \binom{\ldots}{\ldots} a^{n}b^{m} + \ldots + b^{k} \in I + .\] Mit $\binom{k}{l}\cdot $ ist hier die $\binom{k}{l}$-fache Summe des Einselements + in $R$ gemeint. Damit ist $a+b \in \sqrt{I}$. + \item Sei $a \in \sqrt{I} $: Dann ex. ein $n \in \N$ mit $a^{n} \in I$. Sei + $r \in R$ beliebig. Dann ist wegen $I$ Ideal + \[ + r^{n} a^{n} \in I \implies (r\cdot a)^{n} \in I \implies r\cdot a \in \sqrt{I} + .\] + \end{enumerate} + $\implies \sqrt{I} $ ist ein Ideal + \end{proof} + \item Beh.: Ist $I$ ein Primideal, so gilt $\sqrt{I} = I$. + \begin{proof} + Sei $r \in \sqrt{I} $ und $I$ Primideal. Dann ex. $n \in \N$ mit $r^{n} \in I$. + Mit $r^{n} = \underbrace{r \cdot \ldots \cdot r}_{n\text{-mal}}$ folgt + durch sukzessives Anwenden der Primidealeigenschaft auf $I$: $r \in I$. + $\implies \sqrt{I} \subset I$ und damit wegen (a) $\sqrt{I} = I$. + \end{proof} + \item Wähle $R$ beliebig und $I = R$, dann ist $I$ nach Definition kein Primideal, aber + wegen $R = I \subset \sqrt{I} \subset R \implies \sqrt{I} = R = I$. + \end{enumerate} + +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\Phi\colon \{\text{Ideale in } R / I\} \to \{\text{Ideale } \tilde{I} + \text{ in } R \text{ mit } I \subset \tilde{I}\}$, $J \mapsto \pi^{-1}(J)$ + ist wohldefiniert und inklusionserhaltend. + + \begin{proof} + Sei $J \in \{\text{Ideale in } R / I\} $ beliebig. Wegen $J$ Ideal in $R / I$ und + $\pi$ Ringhomomorphismus ist $\Phi(J) = \pi^{-1}(J) := \tilde{I}$ Ideal in $R$. + + Außerdem ist wegen $J$ Ideal, $I = 0 + I = 0 \in J$. Damit folgt + $I = \text{ker } \pi \subset \pi^{-1}(J) = \tilde{I}$. + + Seien nun $J, J'$ Ideale in $R / I$ mit $J \subset J'$. Sei $r \in \pi^{-1}(J)$. + Dann ist $\pi(r) \in J \subset J' \implies r \in \pi^{-1}(J')$. Damit folgt + $\Phi(J) \subset \Phi(J')$. + \end{proof} + Beh.: $\Psi\colon \{\text{Ideale } \tilde{I} \text{ in } R \text{ mit } I \subset \tilde{I}\} \to \{\text{Ideale in } R / I\}$, $\tilde{I} \mapsto \pi(\tilde{I})$ und inklusionserhaltend. + \begin{proof} + Sei $\tilde{I}$ ein Ideal in $R$ mit $I \subset \tilde{I}$. Wegen $\pi$ surjektiver + Ringhomomorphismus ist $\pi(\tilde{I})$ ein Ideal in $R / I$. + + Seien nun $\tilde{I}, \tilde{I}'$ Ideale in R mit $I \subset \tilde{I} \subset \tilde{I}'$. + Sei $A \in \Psi(\tilde{I})$. Dann ex. ein $r \in \tilde{I} \subset \tilde{I}'$ mit $\pi(r) = A$. + Dann ist $\pi(r) = A \in \pi(\tilde{I}') = \Psi(\tilde{I}')$. Damit folgt + $\Psi(\tilde{I}) \subset \Psi(\tilde{I}')$. + \end{proof} + \item Beh.: $\Psi \circ \Phi = \text{id}$ + \begin{proof} + Sei $J$ ein Ideal in $R / I$. Definiere $\tilde{I} := \Phi(J) = \pi^{-1}(J)$. + Nach (a) ist $\tilde{I}$ ein Ideal in $R$ mit $I \subset \tilde{I}$. + + \begin{itemize} + \item Zz.: $J \subset \Psi(\Phi(J))$: Sei $A \in J$. Dann ex. $r \in R$ + mit $\pi(r) = A$. Es gilt $r \in \pi^{-1}(J) = \tilde{I}$. Damit folgt + $A = \pi(r) \in \pi(\tilde{I}) = \Psi(\Phi(J))$. + + \item Zz.: $\Psi(\Phi(J)) \subset J$. Sei $A \in \Psi(\Phi(J))$. Wegen + $A \in \pi(\Phi(J))$ ex. ein $r \in \Phi(J) = \pi^{-1}(J)$ + mit $\pi(r) = A$. Damit folgt $\exists B \in J$ mit + $B = \pi(r) = A \implies A \in J$. + \end{itemize} + \end{proof} + Beh.: $\Phi \circ \Psi = \text{id}$ + \begin{proof} + Analog. + \end{proof} + \item Beh.: Ideale in $\Z / 15 \Z$ sind $\{ (\overline{1}), (\overline{3}), (\overline{5})\} $. + \begin{proof} + Bestimme alle Teiler von 15: $1, 3, 5$. Diese bilden alle Ideale $\tilde{I}$ + in $\Z$ mit $(15) \subset \tilde{I}$. Mit (b) und $(1) = \Z$ folgt damit direkt für + alle Ideale in $\Z / 15 \Z$: + $\{\Psi(\Z), \Psi((3)), \Psi((5))\} = \{(\overline{1}), (\overline{3}), (\overline{5})\} $. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}