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b/ws2020/algebra/uebungen/algebra4.tex @@ -0,0 +1,204 @@ +\documentclass[uebung]{../../../lecture} + +\title{Algebra I: Übungsblatt 4} +\author{Lukas Nullmeier, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es ist $f = X^{5} - 3$ Minimalpolynom von $\sqrt[5]{3}$, denn + $f(\sqrt[5]{3}) = 3 - 3 = 0$ und $f$ ist primitiv und damit irreduzibel nach Eisenstein + mit $p = 3$. + \item Es ist $f = X^{4} - X^2 + 1$ Minimalpolynom von $\sqrt{2} + \sqrt{3} $. Denn + es ist $f(\sqrt{2} + \sqrt{3}) = 0$. + + Weiter + ist $2 + 2 \sqrt{2} \sqrt{3} + 3 = (\sqrt{2} + \sqrt{3})^2 \in \Q(\sqrt{2} + \sqrt{3})$, also + da $2, 3 \in Q$ auch $\sqrt{2} \sqrt{3} \in \Q(\sqrt{2} + \sqrt{3} )$. Damit folgt + $(\sqrt{2} + \sqrt{3})\sqrt{2}\sqrt{3} = 2 \sqrt{3} + 3\sqrt{2} \in \Q(\sqrt{2} + \sqrt{3})$ + und damit $2 \sqrt{3} + 3\sqrt{2} - 2(\sqrt{2} + \sqrt{3}) = \sqrt{2} \in \Q(\sqrt{2} + \sqrt{3})$ + und damit auch $\sqrt{3} \in \Q(\sqrt{2} + \sqrt{3})$. Damit folgt, da + $\sqrt{2} +\sqrt{3} \in \Q(\sqrt{2} , \sqrt{3} )$, dass + $\Q(\sqrt{2} + \sqrt{3}) = \Q(\sqrt{2} , \sqrt{3} )$. + + Und es gilt weiter + mit $g = X^2 - 2 \in \Q(\sqrt{3})[X] $, $g(\sqrt{2}) = 0$ und $\sqrt{2} \not\in \Q(\sqrt{3})$, + dass $1 < [\Q(\sqrt{2}, \sqrt{3}) \colon \Q(\sqrt{3}) ] \le 2$ also + damit $[\Q(\sqrt{2} , \sqrt{3} ) \colon \Q(\sqrt{3})] = 2$. Analog + mit $h = X^2 - 3$ folgt $[\Q(\sqrt{3}) \colon \Q] = 2$. Insgesamt + folgt mit Gradsatz $[\Q(\sqrt{2} + \sqrt{3}) \colon \Q] = 2 \cdot 2 = 4$. + + Damit ist $f$ bereits normiertes Polynom kleinsten Grades mit $f(\sqrt{2} + \sqrt{3} ) = 0$, also + $f$ Minimalpolynom. + \item Es ist $f = X^4 - \frac{5}{4} X^2 + \frac{5}{16}$ Minimalpolynom + von $\sin\left( \frac{2\pi}{5} \right) $, denn + $\sin\left( \frac{2\pi}{5} \right) = \frac{1}{4}\sqrt{2(5+\sqrt{5})}$ und damit + $f\left(\sin\left( \frac{2\pi}{5} \right)\right) = 0$. Außerdem + ist $16f = 16X^4 - 20 X^2 + 5$ primitiv, denn $5 \nmid 16$ und wegen $5 \mid 20$ + irreduzibel nach Eisenstein mit $p =5$. Also auch $f$ irreduzibel und damit + Minimalpolynom. + \item Es ist $f= X^4 - X^2 + 1$ Minimalpolynom von $e^{\frac{i \pi}{6}} - \sqrt{3} $, denn + es ist mit Euler Identität mit Euler Identität mit Euler Identität mit Euler Identität + \begin{salign*} + e^{\frac{\pi i}{6}} - \sqrt{3} &= \cos\left( \frac{\pi}{6} \right) + i \sin\left( \frac{\pi}{6} \right) - \sqrt{3} \\ + &= \frac{\sqrt{3} }{2} - \frac{i}{2} - \sqrt{3} \\ + &= - \frac{\sqrt{3} }{2} + \frac{i}{2} \\ + &= - \cos\left( \frac{\pi}{6} \right) + i \sin\left( \frac{\pi}{6} \right) \\ + &= - \cos\left( - \frac{\pi}{6} \right) - i \sin\left( - \frac{\pi}{6} \right) \\ + &= - e^{-\frac{\pi i }{6}} + .\end{salign*} + Damit folgt $f(e^{\pi i / 6} - \sqrt{3}) = f\left( - e^{- \pi i /6} \right) = 0$. + + Da $e^{\frac{i \pi}{6}} -\sqrt{3} = \frac{i}{2} - \frac{\sqrt{3} }{2}$, bleibt es analog + zu (b) zu zeigen, dass $\sqrt{3}, i \in \Q(i - \sqrt{3})$. Es + ist $(i - \sqrt{3})^2 = 2 - 2 i \sqrt{3}$, also $i \sqrt{3} \in \Q(i - \sqrt{3})$. Damit folgt + $\Q(i - \sqrt{3}) \ni i\sqrt{3} (i - \sqrt{3}) -(i - \sqrt{3}) = - 4i$, also + $i \in \Q(i - \sqrt{3}) $ und damit auch $\sqrt{3} \in \Q(i-\sqrt{3})$, also analog + zu (b): $\Q(i - \sqrt{3}) = \Q(i, \sqrt{3})$. Mit + $g = X^2 + 1$, $g(i) = 0$ und $h = X^2 - 3$, $h(\sqrt{3}) = 0$, folgt damit analog zu (b) + $[\Q(i - \sqrt{3}) \colon \Q] = 2 \cdot 2 = 4$. + + Also ist $f$ bereits normiertes Polynom kleinsten Grades mit $f(e^{\pi i /6} - \sqrt{3}) = 0$, + also $f$ Minimalpolynom. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es ist $f = X^4 - 2$ Minimalpolynom von $\sqrt[4]{2}$ über $\Q$, denn + $f(\sqrt[4]{2}) = 0 $ und $f$ normiert und irreduzibel nach Eisenstein mit $p = 2$. + Damit folgt $[ K \colon \Q] = \text{deg}(f) = 4$. + \item Es ist $f = X^2 + 1$ Minimalpolynom von $i$ über $K$, denn + $f(i) = 0$ und $f$ irreduzibel über $\R$, also insbesondere über $K \subseteq \R$. + Damit ist $[L \colon K] = \text{deg}(f) = 2$ und damit mit Gradsatz + $[L \colon \Q] = [L \colon K] \cdot [K \colon \Q] \stackrel{\text{(a)}}{=} 2 \cdot 4 = 8$. + \item Es ist $\sqrt[4]{2} \in L$, also auch $L \ni \sqrt[4]{2}^2 = \sqrt{2}$. + + Weiter gilt $f = X^2 - 2$ normiert und irreduzibel nach Eisenstein mit $p = 2$, also + Minimalpolynom von $\sqrt{2} $ über $\Q$, denn $f(\sqrt{2}) = 0$. + Damit folgt $[\Q(\sqrt{2} \colon \Q] = \text{deg}(f) = 2$. + + Außerdem ist $\Q(\sqrt{2}) \subseteq \R$ und $f = X^2 + 1$ normiert und irreduzibel über $\R$, also + insbesondere über $\Q(\sqrt{2})$. Also ist $f$, wegen $f(i) = 0$, Minimalpolynom + von $i$ über $\Q(\sqrt{2})$. Damit folgt $[\Q(\sqrt{2})(i) \colon \Q(\sqrt{2})] = 2$. + + Insgesamt folgt damit mit Gradsatz: $[\Q(\sqrt{2} , i) \colon \Q] = 2 \cdot 2 = 4$. + \item Es ist $f = X^2 - 2 \sqrt{2} X + 3$ Minimalpolynom von $\sqrt{2} + i$ über $\Q(\sqrt{2})$, + denn $\sqrt{2} + i \not\in \Q(\sqrt{2})$, also insbesondere + $\Q(\sqrt{2} + i) \neq \Q(\sqrt{2})$. Damit ist $f$ bereits normiertes Polynom + kleinsten Grades, also Minimalpolynom von $\sqrt{2} + i$ über $\Q(\sqrt{2})$. + + Es ist $i \in \Q(\sqrt{2} + i)$, denn mit + $(\sqrt{2} + i)^2 = 1 + 2i\sqrt{2}$ folgt $i \sqrt{2} \in \Q(\sqrt{2} + i)$, also + auch + $\Q(\sqrt{2} +i) \ni \sqrt{2} i (\sqrt{2} + i) + (\sqrt{2} +i) = 2i - \sqrt{2} + \sqrt{2} + i = 3i$. + Damit folgt auch $\sqrt{2} \in \Q(\sqrt{2} + i)$. Da $\sqrt{2} + i \in \Q(\sqrt{2} , i)$ + folgt insgesamt $\Q(\sqrt{2} + i) = \Q(\sqrt{2} , i)$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Ist $[L : K] = p$ eine Primzahl, dann ex. ein $\alpha \in L$ mit $L = K(\alpha)$. + \begin{proof} + Es ist $[L \colon K] = p < \infty$. Das heißt + $\exists \alpha_1, \ldots, \alpha_n \in L$ mit $\alpha_1, \ldots, \alpha_n$ algebraisch + mit $L = K(\alpha_1, \ldots, \alpha_n)$. Dann folgt + \[ + K \subseteq K(\alpha_1) \subseteq K(\alpha_1, \alpha_2) \subseteq \ldots + \subseteq K(\alpha_1, \ldots, \alpha_n) = L + .\] Mit Gradsatz folgt damit + \[ + p = [L \colon K] = [ K(\alpha_1, \ldots, \alpha_n) \colon K(\alpha_1, \ldots, \alpha_{n-1}] + \cdot \ldots \cdot [K(\alpha_1) \colon K] + .\] Da $p$ Primzahl sind alle Faktoren $1$ bis auf einer. oE.: $[K(\alpha_1) \colon K] = p$. + Damit sind $K(\alpha_1) = K(\alpha_1, \alpha_2) = \ldots = K(\alpha_1, \ldots, \alpha_n) = L$. + \end{proof} + \item Beh.: Ist $[L : K] = 2^{k}$ und $f \in K[X]$ Polynom vom Grad 3 mit Nullstelle in $L$, dann + hat $f$ auch Nullstelle in $K$. + \begin{proof} + Sei $[L \colon K] = 2^{k}$ und $f \in K[X]$ mit $\alpha \in L$ und $f(\alpha ) = 0$. + Ang.: $\alpha \not\in K$. Dann ist $f$ bereits irreduzibel vom Grad 3 über $K$, da + $f$ keine Nullstelle hat über Körper $K$. + Damit $\exists g \in K[X]$ mit $g \stackrel{\wedge}{=} f$ und + $g$ normiert mit $\text{deg}(g) = \text{deg}(f) = 3$. Dann ist + $g$ Minimalpolynom von $\alpha $ über $K$. Also folgt + $[K(\alpha) \colon K] = 3$ und damit nach Gradsatz + \[ + [ L \colon K] = [L \colon K(\alpha)] \cdot [K(\alpha) \colon K] + = 3n + \] für ein $n \in \N$. Also $2^{k} = 3n$ $\contr$, denn einziger Primfaktor + von $2^{k}$ ist $2$. + \end{proof} + \item Beh.: Ist $[L \colon K]$ ungerade und $L = K(\alpha)$ für ein $\alpha \in L$, so + gilt $L = K(\alpha^2)$. + \begin{proof} + Sei $\alpha \in L$ mit $L = K(\alpha)$ und $K(\alpha) \neq K(\alpha^2)$. Da + $\alpha^2 \in K(\alpha)$ + ist $K \subseteq K(\alpha^2) \subseteq K(\alpha)$. Da aber $K(\alpha) \neq K(\alpha^2)$ + ist $[K(\alpha) \colon K(\alpha^2)] \ge 2$. Da außerdem + mit $f = X^2 - \alpha^2 \in K(\alpha^2)[X]$ gilt, dass $f(\alpha) =0$, folgt + $[K(\alpha) \colon K(\alpha^2)] \le 2$, also insgesamt + $[K(\alpha) \colon K(\alpha^2)] = 2$. Damit folgt mit Gradsatz + \[ + [ L \colon K ] = [ K(\alpha) \colon K(\alpha^2) ] \cdot [K(\alpha^2) \colon K] + = 2 n + \] für ein $n \in \N$, also insbesondere $[ L \colon K]$ gerade. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\overline{K}$ ist unendlich. + \begin{proof} + Sei $\overline{K}$ endlich. Dann sei $\overline{K} = \{a_1, \ldots, a_n\} $ mit + $n \in \N$. + Betrachte $f = (X - a_1) (X - a_2) \ldots (X - a_n) + 1$. Dann ist + $\text{deg}(f) \ge 1$, denn $\overline{K}$ Körper, insbesondere $0, 1 \in K$, also $n \ge 2$. + + Aber + $f$ hat keine Nullstelle in $\overline{K}$, denn + $\forall a \in \overline{K}\colon (a - a_1)(a - a_2) \ldots (a-a_n) = 0$, also + $f(a) = 1$. Also $\overline{K}$ nicht algebraisch abgeschlossen. + \end{proof} + \item Beh.: $K$ abzählbar $\implies$ $\overline{K}$ abzählbar. + \begin{proof} + Sei $K$ abzählbar. Dann existieren abzählbar viele Polynome über $K$, insbesondere + abzählbar viele normierte, irreduzible Polynome, also insbesondere + abzählbar viele algebraische Elemente $\alpha \in \overline{K}$ über $K$. + Also $\overline{K}$ abzählbar. + \end{proof} + \item Beh.: Es existieren Zahlen $z \in \mathbb{C}$, die transzendent über $\overline{Q}(\pi)$ sind. + \begin{proof} + Ang. alle Elemente $z \in \mathbb{C}$ sind algebraisch über $\overline{\Q}(\pi)$. Da + $\mathbb{C}$ algebraisch abgeschlossen folgt damit, dass + $\mathbb{C}$ ein algebraischer Abschluss von $\overline{Q}(\pi)$ ist. Allerdings + ist $\Q$ abzählbar, nach (b) insbesondere $\overline{Q}$ abzählbar und + $\overline{Q}(\pi) \stackrel{\sim }{=} \overline{Q}^2$ abzählbar. Damit ist + erneut nach (b) $\mathbb{C}$ abzählbar $\contr$. + \end{proof} + \item Beh.: $\overline{K}(X)$ ist nicht algebraisch abgeschlossen. + \begin{proof} + Betrachte $f = T^2 - X \in \overline{K}(X)[T]$. Ang. $f$ hat Nullstelle $g \in \overline{K}(X)$. + Also $g = \frac{p}{q}$ für $p, q \in \overline{K}[X]$ mit $p, q \neq 0$ da $f \neq 0$, also + $\text{deg}(p) , \text{deg}(q ) \ge 0$. Aus $f(g) = 0$ folgt + $\left( \frac{p}{q} \right) ^2 = X$, also $p^2 = X q^2$. Damit + \[ + \text{deg}(p^2) = \text{deg}(X) + \text{deg}(q^2) = 1 + \text{deg}(q^2) + .\] + + Nach Gradsatz + ist $\text{deg}(p^2) = 2\text{deg}(p)$ und $\text{deg}(q^2) = 2 \text{deg}(q) $, also folgt + \[ + \underbrace{2 \text{deg}(p)}_{\in 2 \Z} = \underbrace{1 + 2 \text{deg}(q) }_{\not\in 2\Z} + \quad \contr + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2020/wtheo/uebungen b/ws2020/wtheo/uebungen index e8291be..ea0d034 160000 --- a/ws2020/wtheo/uebungen +++ b/ws2020/wtheo/uebungen @@ -1 +1 @@ -Subproject commit e8291be73e8184bc41a874aef00757fa71d0a848 +Subproject commit ea0d034763cd32cf8afc9772522af1cfaf1cb98f