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literal 0 HcmV?d00001 diff --git a/ws2020/algebra/uebungen/algebra5.tex b/ws2020/algebra/uebungen/algebra5.tex new file mode 100644 index 0000000..e326db5 --- /dev/null +++ b/ws2020/algebra/uebungen/algebra5.tex @@ -0,0 +1,178 @@ +\documentclass[uebung]{../../../lecture} + +\title{Algebra I: Übungsblatt 5} +\author{Lukas Nullmeier, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $L$ ist ein Zerfällungskörper von $X^{4} - 2 \in \Q[X]$. + \begin{proof} + Es ist zunächst + \[ + X^{4} - 2 = (X^2 - \sqrt{2} )(X^2 + \sqrt{2}) = (X - \sqrt[4]{2})(X + \sqrt[4]{2})(X + i \sqrt[4]{2})(X - i \sqrt[4]{2}) + .\] Weiter ist $i \in \Q(\sqrt[4]{2}, i\sqrt[4]{2})$, denn + \[ + \Q(\sqrt[4]{2}, i\sqrt[4]{2}) \ni \frac{1}{2} \sqrt[4]{2} (i \sqrt[4]{2} )(\sqrt[4]{2})^2 + = i + .\] Also $\Q(\sqrt[4]{2}, i\sqrt[4]{2}) = \Q(\sqrt[4]{2}, i) = L$, also + $L$ Zerfällungskörper von $X^{4} - 2$. + \end{proof} + \item Es ist $\Q \subseteq L$ endliche Körpererweiterung und $\text{char}(\Q) = 0$, also + folgt mit dem letzten Zettel + \[ + 8 = [ L \colon \Q] = [ L \colon \Q]_s = \# \text{Hom}_\Q(L, \overline{\Q}) + \] + mit $\overline{\Q}$ ein algebraischer Abschluss von $\Q$. Da jeder + $\Q$-Automorphismus von $L \subseteq \overline{Q}$ + insbesondere $\Q$-Homomorphismus $L \to \overline{Q}$, folgt + \[ + \# \text{Aut}_\Q(L, L) \le 8 + .\] Sei $x \in \Q(L, L)$. Dann gilt für $a_0, \ldots, a_7 \in \Q$: + \[ + x = a_0 + i a_1 + \sqrt[4]{2} a_2 + \sqrt[4]{2}^{3} a_3 + + \sqrt{2} a_4 + i \sqrt[4]{2} a_5 + i \sqrt{2} a_6 + i \sqrt[4]{2} a_7 + .\] Diese Darstellung ist eindeutig, da die Vorfaktoren der $a_i$ eine + $\Q$ Basis von $L$ darstellen. + Sei nun $\tau_i \colon L \to L$ für $i \ge 1$ mit + $\tau_i$ ändert Vorzeichen des $i$-ten Koeffizienten. Dabei bezeichne + $\tau_0 = \text{id}_L$. Da $\sigma \colon \Q \to \Q, r \mapsto -r$ ein Körperhomomorphismus + folgt durch Nachrechnen, dass die $\tau_i$ ebenfalls Körperhomomorphismen sind. Dabei + ist $\tau_i \mid_\Q = \text{id}$, da für $x \in \Q$ folgt $x = a_0$ und + damit $\tau_i(x) = \tau_i(a_0) = a_0 = x$. + + Die $\tau_i$ sind bereits $8$ $\Q$-Automorphismen von $L$, d.h. mit der Vorüberlegung alle + $\Q$-Automorphismen von $L$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(i)] + \item $L = \Q(\sqrt{2} e^{i \pi /4}, -\sqrt{2}e^{i \pi /4}, \sqrt{2} e^{- i \pi / 4}, - \sqrt{2} e^{- i \pi / 4} $, denn + \[ + X^{4} + 4 = (X^2 - 2i)(X^2 + 2i) + = (X - \sqrt{2} e^{\pi i /4}) + (X + \sqrt{2} e^{\pi i /4})(X - \sqrt{2}e^{- \pi i /4}) + (X + \sqrt{2} e^{- \pi i/4}) + .\] + \item $L = \Q\left(\left( \exp\left( \frac{k \pi}{4} \right) \right)_{k=1}^{8}\right)$, denn + das sind gerade die $8$-ten Einheitswurzeln, also folgt + \[ + X^{8} - 1 = \prod_{k=1}^{8} (X - e^{k \pi i /4}) + .\] + \item $L = \Q\left(\sqrt{\sqrt{3} -1}, - \sqrt{\sqrt{3} -1}, \sqrt{-\sqrt{3} -1}, - \sqrt{-\sqrt{3} - 1} \right)$ + , denn + \begin{salign*} + X^{4} + 2 X^2 -2 &= (X^2 + 1 - \sqrt{3}) (X^2 +1 + \sqrt{3}) \\ + &= \left(X - \sqrt{\sqrt{3} -1} \right) + \left( X + \sqrt{-1 + \sqrt{3} } \right) + \left( X - \sqrt{-1 - \sqrt{3} } \right) + \left( X + \sqrt{-1 - \sqrt{3} } \right) + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $L = K(\alpha)$. + \begin{proof} + Es ist $[L \colon K] = n$. Da $\sigma_i$ $K$-Automorphismus, + ist $\sigma \coloneqq \sigma_i |_K = \text{id}$. + + Sei nun $f$ Minimalpolynom von $\alpha$ über $K$. + Dann + ist $f^{\sigma} = f^{\text{id}} = f$. Betrachte + $\sigma_i'\colon K(\alpha) \to L$ mit $\sigma_i' = \sigma_i|_{K(\alpha)}$. Dann + setzt $\sigma_i'$ $\sigma$ fort, es folgt also mit 3.40 + $\sigma_i'(\alpha)$ Nullstelle von $f^{\sigma} = f$. Da + die $\sigma_i'(\alpha)$ paarweise verschieden, folgt + $\text{deg}(f) \ge n$. Da $\text{deg}(f) = [K(\alpha) \colon K] \le [ L \colon K] = n$ folgt + $\text{deg}(f) = n$. Damit folgt mit Gradsatz + \[ + n = [L \colon K] = [ L \colon K(\alpha)] [ K(\alpha) \colon K] + = [L \colon K(\alpha) ] n \implies [ L \colon K(\alpha)] = 1 + .\] Also $L = K(\alpha)$. + \end{proof} + \item Beh.: Jeder $K$ Homomorphismus $\sigma\colon L \to L$ ist Automorphismus. + \begin{proof} + Sei $\sigma\colon L \to L$ ein $K$-Hom. + Sei $\alpha \in L$ und $f \in K[X]$ Minimalpolynom von $\alpha$ über $L$. Seien + weiter $\gamma_1, \ldots, \gamma_n$ Nullstellen von $f$ über $L$ mit + $\Gamma \coloneqq \{ \gamma_1, \ldots, \gamma_r\} $. Dann gilt, da + $\sigma|_K = \text{id}$ und $\sigma$ Ringhom, folgt + \[ + f(\sigma(\gamma)) = \sigma (f(\gamma)) = \sigma(0) = 0 \qquad \forall \gamma \in \Gamma + .\] Damit gilt $\sigma(\Gamma) \subseteq \Gamma$. Da $\sigma$ als Körperhom. + injektiv und $\Gamma$ endlich (da $\# \Gamma \le \text{deg}(f) < \infty$) + folgt $\sigma(\Gamma) = \Gamma$, also $\exists \gamma \in \Gamma\colon \sigma(\gamma) = \alpha$. + + Also ist $\sigma$ auch surjektiv und damit bijektiv. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[] + \begin{enumerate}[(a)] + \item Sei $K \subseteq R \subseteq L$ Unterring. Beh.: $R$ Körper. + \begin{proof} + Sei $\alpha \in R \setminus \{0\} $. Dann ist $\alpha \in L$ algebraisch über $K$, also + sei $f \in K[X]$ Minimalpolynom zu $\alpha$ mit + \[ + f = X^{n} + c_{n-1} X^{n-1} + \ldots + c_0 + \] für $c_i \in K$. Es ist $c_0 \neq 0$, da $f$ irreduzibel. In $L$ existiert + $\alpha ^{-1}$. Damit folgt + \begin{salign*} + 0 = \alpha ^{-1}f(\alpha) = \alpha ^{-1} (\alpha^{n} + c_{n-1}\alpha^{n-1} + \ldots + c_0) + = \alpha^{n-1}+ c_{n-1}\alpha^{n-2} + \ldots + c_0 \alpha^{-1} + .\end{salign*} + Da $c_0 \neq 0$ folgt + \begin{salign*} + \alpha^{-1} = - c_0^{-1} (\alpha^{n-1}+c_{n-1}\alpha^{n-2} + \ldots + c_1) + .\end{salign*} + Da $\alpha \in R$, $c_i \in K$ und $R$ Ring mit $K \subseteq R$ folgt + $\alpha^{-1} \in R$. + \end{proof} + \item Es bezeichne $R$ die Menge aus der Aufgabenstellung. Beh.: $EF = R$. + \begin{proof} + \begin{enumerate}[(i)] + \item Z.z.: $R$ Ring. Da $K \subseteq E, F$ ist $1 = 1_K 1_K \in R$ und $0 = 0_K 0_K \in R$. + Weiter seien $x, y \in R$ mit $x = \sum_{i=1}^{n} a_i b_i$ + und $y = \sum_{i=1}^{m} c_i d_i$ mit $a_i, c_i \in E$, $b_i, d_i \in F$. Durch + Ergänzung der $a_i$ und $b_i$ am Ende durch $c_i$ bzw. $d_i$ folgt direkt + \[ + x + y = \sum_{i=1}^{n} a_i b_i + \sum_{i=1}^{m} c_i d_i + = \sum_{i=1}^{n+m} a_i b_i \in R + .\] Für $x y$ entstehen durch Ausmultiplizieren Summanden + der Form $a_i b_i c_j d_j = \underbrace{a_i c_j}_{\in E} \underbrace{b_i d_j}_{\in F}$, + also folgt durch Umbenennung $xy \in R$ aus $x + y \in R$. + Da $E$ und $F$ Körper übertragen sich Assoziativ und Distributivgesetz + direkt auf $R$. Auch additive Inverse existieren, da $E, F$ Körper sind folgt + \[ + x + \underbrace{\sum_{i=1}^{n} -(a_i b_i)}_{=: -x} = \sum_{i=1}^{n} (a_i b_i - a_i b_i) = 0 + .\] Also $-x \in R$. + \item Sei nun $H \subseteq L$ Körper, der $E$ und $F$ enthält, dann + folgt insbesondere $R \subseteq H$. + \item Nach (i) ist $R$ Ring und offensichtlich Teilring von $L$, damit + nach (a) Körper und wegen (ii) kleinster Teilkörper von $L$, der + $E$ und $F$ enthält. Also $EF = R$. + \end{enumerate} + \end{proof} + \item Beh.: Sind $[ E \colon K]$ und $[F \colon K]$ endlich, so auch $[EF \colon K]$ und es + gilt + \[ + [EF \colon K] \le [ E \colon K ] [ F \colon K] + .\] + \begin{proof} + Seien $[E \colon K]$, $[F \colon K]$ endlich. Dann sind $E$ und $F$ e.d. + als $K$ VR, insbesondere existieren $\alpha_i$, $\beta_i$, s.d. + $(\alpha_i)_{i=1}^{n}$ Basis von $E$ und $(\beta_i)_{i=1}^{m}$ Basis von $F$. Wegen (b) + ist damit $(\alpha_i \beta_j)_{i, j =1}^{n, m}$ endliches Erzeugendensystem + von $EF$ der Länge $nm$. Damit folgen beide Behauptungen. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2020/tut-ana/praesenzaufgaben4.xopp b/ws2020/tut-ana/praesenzaufgaben4.xopp index 31fb5cfdc6c45bf919af7ec44a79c4f0b6719a53..6e1f6925dab1cfd8fa61986cf6f8538b1ecf9dcf 100644 GIT binary patch delta 382798 zcmV(-K-|B!&K1Pa7k?j%2mk;800003w7u8T+&Ho=e7~PU(dV+_s0e@{fInvT36}TX z@{Z-bW!b0ySAuG(_{~i3*%2d6#0W}Ou}C0u3>Hq?YC0PdpPewJU{k#`SzgW9-59Bj_Q++xXfCeX|b5`{p_H;e)jAA zY9ZgROrzIupzfWcXV#;+=$}UY%Bim>h5k9`9jbky3itJF&<*;F?eRRDUaE`_-&Zp! z%rC=1jqMASUB1&TV}i)_VtUD@t+s^Pfq%$&+bOpnLTq&Tkh_tU4YFIU;|p4IYCGj+si9_*FL zwo7GpK7Ry!E{CH?<_~EnYM!ji&xh<~>*?hjNPILFdm&4WJa?BaW zcOtLYIM#M19QXFU-6&V^Do)wFdZk(M7+pGje}5LW8t2QUQfWTD)8XQ&U^Q+n(yR7x zmao(rYijA$+_!X2N_;oc+E@c!DHCmL~|F*XohmYIL6CLTVw?m0hNt zdw+F=?q9R7*R8PG%?ED(5#%@NbAPv+&pHd0zN^MAJCIYi$?hj_RoMq-)a>0<6q);a zI&Jc0SX+vnY}KmcM$Wr+)~#G=@qC5*S+QAal~QuEI33dLs!YR8wRK+Jrs=fWs>Zoi z-i*yCvkdfR*&OXy-*4wO^}JK9cKWkUt$#A!J;Lna&QkTxRb<*7*XNB|wvn5u<+k*= 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