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file mode 100644 index 0000000..fb775b1 --- /dev/null +++ b/ws2020/algebra/uebungen/algebra9.tex @@ -0,0 +1,270 @@ +\documentclass[uebung]{../../../lecture} + +\title{Algebra I: Übungsblatt 9} +\author{Lukas Nullmeier, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + Es gilt $f = X^4 - 4X^2 + 9 = (X^2 - 2)^2 + 5$. Damit setze + setze $\alpha \coloneqq \sqrt{2 + i \sqrt{5} } $. Dann gilt + \[ + f( \pm \alpha) = (2 + i \sqrt{5} -2)^2 + 5 = (i \sqrt{5})^2 + 5 = 0 + .\] Damit ist $\alpha ^{4} - 4 \alpha ^2 + 9 = 0$, also + insbesondere $- \alpha ^{4} + 2 \alpha^2 = - 2 \alpha^2 + 9$. Damit folgt + \[ + f\left( \pm \frac{3}{\alpha}\right) = \left( \frac{9}{\alpha^2} -2 \right) ^2 + 5 + = \left( \frac{- \alpha ^{4} + 2 \alpha^2}{\alpha^2} \right) ^2 + 5 + = \left( \alpha^2 - 2 \right)^2 + 5 = f(\alpha) = 0 + .\] Also sind die Nullstellen von $f$ gegeben als $\{ \pm \alpha, \pm 3 / \alpha \} $ und + damit $L = \Q(\alpha)$. + + Z.z.: $f$ irreduzibel. Es ist $f$ g.d. irreduzibel über $\Q$, wenn $f$ irreduzibel über $\Z$. + Ang.: $f$ reduzibel über $\Z$. + Da $f$ keine Nullstellen in $\Z$ hat, folgt + \[ + f = (X^2 + aX + b)(X^2 + cX + d) = X^4 + \underbrace{(a + c)}_{= 0}X^3 + + \underbrace{(b+d + ac)}_{= -4} X^2 + \underbrace{(ad + bc)}_{= 0}X + \underbrace{bd}_{= 9} + \] mit $a, b, c, d \in \Z$. Es folgt direkt $c = -a$ und $a(d-b) = 0$. Falls $a = 0$, folgt + $b + d = -4$ und $bd = 9$, aber $3 + 3 = 6 \neq -4 \neq -6 = -3 -3$. Falls $a \neq 0$ folgt + $d = b$ also $b^2 = 9$ und $2b - a^2 = -4$. Also $a^2 = 2b + 4 = \begin{cases} + 10 & b = 3 \\ + -2 & b = -3 + \end{cases}$, aber weder $10$ noch $-3$ Quadrate in $\Z$. Also folgt $f$ irreduzibel. + + Da $\text{char}(\Q) = 0$ ist $f$ also separabel und $f$ Minimalpolynom von $\alpha$, + also $\#\text{Gal}(L / \Q) = [L : \Q] = 4$ und $\text{Gal}(L / \Q)$ operiert transitiv + auf den Nullstellen von $f$. Mit $\sigma\colon \alpha \mapsto -\alpha $ und + $\tau\colon \alpha \mapsto \frac{3}{\alpha}$ folgt damit + $\text{Gal}(L / \Q) = \langle \sigma, \tau \rangle$. Als Gruppe der Ordnung 4 ist + $\text{Gal}(L / \Q)$ abelsch, also $\sigma \tau = \tau \sigma$ und + durch offensichtlicherweise gilt $\sigma^2 = \tau^2 = (\sigma\tau)^2 = \text{id}$. Nach + Satz von Lagrange hat $\text{Gal}(L / \Q)$ UG der Ordnung $1, 2$ oder $4$. Neben der trivialen Untergruppe + und $\text{Gal}(L / \Q)$ selbst, gibt es also nur UG der Ordnung $2$. Da $2$ Primzahl, sind + diese zyklisch. + Also existieren insgesamt + genau die Untergruppen + \begin{salign*} + H_0 &\coloneqq \langle \text{id} \rangle \\ + H_1 &\coloneqq \langle \sigma \rangle \\ + H_2 &\coloneqq \langle \tau \rangle \\ + H_3 &\coloneqq \langle \sigma \tau \rangle \\ + H_4 &\coloneqq \langle \sigma, \tau \rangle + \intertext{ + Es ist $\sigma(\alpha^2) = \sigma(\alpha)\sigma(\alpha) = (-\alpha)(-\alpha) = \alpha^2$. Weiter + ist $9 = 4 \alpha^2 - \alpha ^{4}$. Damit folgt + $\frac{3}{\alpha} = \frac{4}{3} \alpha - \frac{1}{3} \alpha ^{3}$ und + $\frac{27}{\alpha ^{3}} = \frac{7}{3} \alpha - \frac{4}{3} \alpha ^{3}$. Damit + folgt $\tau(- 7 \alpha + \alpha ^{3}) = -7 \left( \frac{4}{3} \alpha - \frac{1}{3} \alpha ^{3} \right) + + \frac{7}{3} \alpha - \frac{4}{3} \alpha ^{3} = - 7 \alpha + \alpha ^{3}$ und + $\sigma(\tau(\alpha - \alpha ^{3})) = \sigma(\frac{4}{3} \alpha - \frac{1}{3} \alpha ^{3} + - \frac{7}{3} \alpha + \frac{4}{3} \alpha ^{3}) = \sigma(-\alpha + \alpha ^{3}) = \alpha - \alpha ^{3}$. +Diese Elemente sind jeweils unter den anderen Automorphismen nicht invariant. Also folgen nach +dem Hauptsatz} + L^{H_0} &= L \\ + L^{H_1} &= \Q(\alpha^2) \\ + L^{H_2} &= \Q(-7\alpha + \alpha ^{3}) \\ + L^{H_3} &= \Q(\alpha - \alpha ^{3}) \\ + L^{H_4} &= \Q + \end{salign*} als Zwischenkörper von $L / \Q$ +\end{aufgabe} + +\begin{aufgabe} + Im folgenden bezeichne echte Untergruppe eine echte, nicht-triviale Untergruppe. + \begin{enumerate}[(a)] + \item Beh.: Die Aussage ist falsch. + \begin{proof} + Es ist $f = X^4 - 7$ irreduzibel nach Eisenstein mit $p = 7$ und + $f(\sqrt[4]{7}) = 0$, aber auch $f(i \sqrt[4]{7}) = 0$ und + $i \sqrt[4]{7} \not\in \R$, aber $\Q(\sqrt[4]{7}) \subseteq \R$, also zerfällt + $f$ über $\Q(\sqrt[4]{7})$ nicht in Linearfaktoren, also + $\Q(\sqrt[4]{7}) / \Q$ nicht normal und damit nicht galoissch. + \end{proof} + \item Beh.: Die Aussage stimmt. + \begin{proof} + Mit Euklidischem Algorithmus ist durch Rechnung zu sehen, dass + $(f, f') = 1$. Also ist $f$ separabel und damit $[ L \colon \Q ] \mid 4! = 24$. + \end{proof} + \item Beh.: Die Aussage ist falsch. + \begin{proof} + Es ist $f = X^3 - 2$ irreduzibel nach Eisenstein mit $p = 2$ und + $\text{char}(\Q) = 0$, also $f$ separabel. Da + $\alpha_j \coloneqq \sqrt[3]{2} e^{i\frac{2 \pi j}{3}}$ die $3$ paarw. verschiedenen + Nullstellen von $f$. Also + folgt $L = \Q(\alpha_0, \alpha_1, \alpha_2)$ + und $\alpha_0 = \sqrt[3]{2} \in \R$, aber $L \not\subset \R$ folgt + $\Q \subsetneqq \Q(\alpha_0) \subsetneqq L$. Also hat + $\text{Gal}(L / \Q)$ eine echte Untergruppe. Da $[L \colon \Q] \mid 3! = 6$ folgt + $\#\text{Gal}(L / \Q) \in \{1, 2, 3, 6\} $. Da $\{0\} $, $\Z / 2 \Z$ und $\Z / 3 \Z$ + keine echten Untergruppen haben, folgt $\# \text{Gal}(L / \Q) = 6$ und damit + $\text{Gal}(L / \Q) = S_3$, also nicht abelsch, insbesondere nicht zyklisch. + \end{proof} + \item Beh.: $S_3$ hat $4$ echte Untergruppen. + \begin{proof} + Es ist $\# S_3 = 6$, also haben nach Satz von Lagrange + echte Untergruppen Ordnung $2$ oder $3$. In Zykelschreibweise ist + \[ + S_3 = \{ (), (1 2), (2 3), (1 3), (1 2 3), (1 3 2)\} + .\] Es ist $\text{ord}((12)) = \text{ord}((2 3)) = \text{ord}((1 3)) = 2$ + und $\text{ord}((123)) = \text{ord}((1 3 2)) = 3$. + + Da Gruppen der Ordnung $2$ und $3$ zyklisch sind + und $\langle (123) \rangle = \langle (1 32)\rangle$ gibt es genau + eine Untergruppe der Ordnung $3$ und $3$ Untergruppen der Ordnung $2$. + \end{proof} + + Beh.: Die Aussage stimmt. + \begin{proof} + Falls $f$ reduzibel existiert ein $g \in \Q[X]$ mit $f = g (X - \alpha)$ für $\alpha \in \Q$ + mit $\text{deg}(g) = 2$. Da $\alpha \in \Q$ ist $L$ auch Zerfällungskörper von $g$ über $\Q$ + und $[ L \colon \Q ] \le 2! = 2$. Also hat $L / \Q$ keine echten Zwischenkörper. + + Sei nun $f$ irreduzibel. Da $\text{char}(\Q) = 0$ ist $f$ separabel und damit + $\# \text{Gal}(L / \Q) \in \{1, 2, 3, 6\} $. Für + $\# \text{Gal}(L / \Q) \in \{1, 2, 3\} $ hat diese keine echten Untergruppen, da + $2$ und $3$ prim und $\{0\} $ die triviale Gruppe. Falls + $\# \text{Gal}( L / \Q) = 6$ folgt $\text{Gal}(L / \Q) \stackrel{\sim }{=} S_3$. Dann + folgt die Behauptung aus der Vorbemerkung. + \end{proof} + \item Beh.: Die Aussage stimmt. + \begin{proof} + Sei $ K \subseteq L \subseteq \overline{K}$ ein + algebraischer Abschluss von $K$ und $\alpha \in \overline{K}$ mit + $\alpha ^{4} = a$. Es ist $f$ irreduzibel und $f' = 4 X^3 \neq 0$, da $\text{char}(K) \neq 2$, + also $f$ separabel. Sei $\zeta \in \mu_4$ mit $\langle \zeta \rangle = \mu_4$. Dann + sind $\{ \alpha \zeta, \alpha \zeta^2, \alpha \zeta^{3}, \alpha \zeta^{4}\}$ die + vier paarweise verschiedenen Nullstellen von $f$. Es ist $L = K(\alpha, \zeta)$. + Dann ex. $\sigma, \tau \in \text{Gal}( L / K)$ mit + \begin{salign*} + &\sigma\colon \alpha \mapsto \alpha \quad \zeta \mapsto \zeta^2 \\ + &\tau\colon \alpha\mapsto \alpha \zeta \quad \zeta \mapsto \zeta + ,\end{salign*} da $\sigma(\alpha \zeta^{k}) = \alpha \zeta^{2k}$ und + $\tau(\alpha \zeta^{k}) = \alpha \zeta^{2k}$ Nullstellen von $f$ in Nullstellen + von $f$ überführen. Aber es gilt + \[ + \sigma(\tau(\alpha)) = \sigma(\alpha \zeta) = \alpha \zeta^2 \neq \alpha \zeta + = \tau(\alpha) = \tau(\sigma(\alpha)) + .\] Also ist $\text{Gal}(L / K)$ nicht abelsch, insbesondere nicht zyklisch. + \end{proof} + \item Beh.: Die Aussage stimmt. + \begin{proof} + Falls $[L \colon K] = \infty$: Dann hat $[ L \colon \Q]$ maximal $\infty$ ZK. + Falls $[L \colon K] = n < \infty$: Dann korrespondieren die Zwischenkörper + von $L / K$ zu Untergruppen von $\text{Gal}(L / K)$. Da $\# \text{Gal}(L / K) = n$, + hat diese nur maximal $2^{n}$ paarweise verschiedene Teilmengen, + insbesondere maximal $2^{n}$ paarweise verschiedene Untergruppen. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item + + Es ist $\mu_n = \{ \zeta_n^{k}\}_{k = 0}^{n-1}$ mit $\zeta_{n}^{k} = e^{\frac{2\pi i k}{n}}$. + Weiter bezeichne $\sigma$ die komplexe Konjugation. Dann gilt + \[ + \sigma(\zeta_{n}^{k}) = e^{- \frac{2\pi i}{n} k} = e ^{\frac{2\pi i}{n}(n-k)} + = \zeta_n^{n-k} + .\] Beachte, dass $\zeta_{n}^{n} = \zeta_{n}^{0} = 1$. Damit ergibt sich + die Permutation + \begin{salign*} + \pi\colon \{0, \ldots, n-1\} &\to \{0, \ldots, n-1\} \\ + k &\mapsto \begin{cases} + n - k & k < n \\ + 0 & k = n + \end{cases} + .\end{salign*} + Auf $\Z / n \Z$ ergibt das die Permutation + \begin{salign*} + \tau\colon \Z / n \Z &\to \Z / n \Z \\ + k &\mapsto \overline{n-k} + .\end{salign*} + \item Es ist $\zeta_n = e^{\frac{2 \pi i}{n}} \in \R \iff \frac{2\pi i}{n} \in \{ \pi i, 2 \pi i\} + \iff n = 1 \lor n = 2$. Also $\zeta_n \in \mathbb{C} \setminus \R$ für $n \ge 3$. Also + $[L \colon L \cap \R ] > 1$, also $L = L \cap \mathbb{C} = (L \cap \R)(i)$ und + $X^2 + 1$ irreduzibel über $L \cap \R$, da $\pm i \not\in L \cap \R$. Also folgt + $[ L \colon L \cap \R ] = [ (L \cap \R)(i) \colon L \cap \R ] = 2$. + + Es ist + \[ + \zeta_n + \zeta_n^{-1} = \cos\left( 2 \frac{\pi}{n} \right) + i \sin\left( 2 \frac{\pi}{n} \right) + \cos\left( - \frac{2\pi}{n} \right) + i \sin\left( - 2 \frac{\pi}{n} \right) = + 2 \cos \left( 2 \frac{\pi}{n} \right) \in \R + .\] Also folgt $\Q(\zeta_n + \zeta_n^{-1}) \in L \cap \R$. Betrachte + nun $f = X^2 - (\zeta_n + \zeta_n^{-1})X + 1 \in \Q(\zeta_n + \zeta_{n}^{-1})$. + Es ist $f(\zeta_n) = 0$, also folgt + $[ L : \Q(\zeta_n + \zeta_{n}^{-1} ] \le 2$. Da $[ L \colon L \cap \R ] = 2$ folgt + \[ + 2 \ge [ L \colon \Q( \zeta_{n} + \zeta_n^{-1} ] = \underbrace{[ L \colon L \cap \R ]}_{=2} + [ L \cap \R \colon \Q(\zeta_n + \zeta_n^{-1} ] + ,\] und damit $L \cap \R = \Q(\zeta_n + \zeta_{n}^{-1})$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $a \in K$. Dann definiere + \begin{salign*} + \sigma_a \colon L &\to L \\ + f &\mapsto f(Y + a) + .\end{salign*} + Dann ist $\sigma_a|_K = \text{id}$ und $\sigma_a(Y) = Y + a$. + Es ist $\sigma_{-a} \circ \sigma = \text{id} = \sigma \circ \sigma_{-a}$, also + $\sigma_a$ bijektiv. Nachrechnen ergibt, dass $\sigma_a$ $K$-Homomorphismus ist. Also + $\sigma_a \in \text{Aut}_K(L)$. Das zeigt die Existenz. + + Seien nun $\sigma, \tau \in \text{Aut}_K(L)$ mit $\sigma(Y) = Y + a = \tau(Y)$. Dann + sei $f \in K(Y)$. Dann gilt wegen $\sigma|_K = \text{id} = \tau|_{K}$: + \[ + \sigma(f) = f(Y + a) = \tau(f) + .\] Das zeigt die Eindeutigkeit. + \item Setze $\phi\colon K \to G$, $a \mapsto \sigma_a$. Es ist $\phi$ wohldefiniert nach (a) und + \[ + \phi(a) = \phi(b) \implies \sigma_a(Y) = \sigma_b(Y) \implies Y + a = Y + b \implies a = b + ,\] also $\phi$ injektiv. Nach Konstruktion von $G$ ist $\phi$ surjektiv. Es gilt außerdem + für $a, b \in K$: + \[ + \phi(a+b)(Y) = Y + a + b = \sigma_a(Y) + b = \sigma_a (Y + b) = \sigma_a(\sigma_b(Y)) + = \sigma_a \circ \sigma_b (Y) + .\] Nach (a) ist $\phi(a + b)$ damit eindeutig festgelegt und es folgt + $\phi(a+b) = \phi(a) \phi(b)$, also $\phi$ Gruppeniso. Da + $G = \text{Bild}(\phi)$ folgt $G$ Untergruppe von $\text{Aut}_K(L)$. + + Zeige letzte Behauptung per Kontraposition. Sei also $ K \subsetneqq L^{G}$. Dann ex. + ein $f \in L^{H} \setminus K$ mit $g, h \in K[Y], g \neq 0$ und $f = \frac{g}{h}$.\\ + Falls + $\forall a \in K\colon h(a) = 0$, folgt wegen $g \neq 0$ $\#K \le \text{deg}(h) < \infty$, also + $K$ endlich.\\ + Sonst ex. ein $a \in K$ mit $h(a) \neq 0$. Dann setze $d \coloneqq f(a) \in K$. + Betrachte nun $p \coloneqq f - d \in K(Y)$. Es ist $f \in L^{G}$, denn + $\forall r \in K$ ist $\sigma_r(p) = \sigma_r(f) - d = f - d = p$. Es + ist außerdem $p(a) = f(a) - d = d - d = 0$ und + damit $\forall r \in K\colon p(a + r) = p(a) = 0$. Da $f \not\in K$ ist $p = f - d \not\in K$ + und damit $p \neq 0$. Weiter existieren $\tilde{g}, \tilde{h} \in K[Y]$ mit + $f = \frac{\tilde{g}}{\tilde{h}}$ und damit + $\forall r \in K\colon \tilde{g}(a + r) = \tilde{g}(a) = 0$. Also + folgt, da $p \neq 0$ und damit $\tilde{g} \neq 0$: $\#K \le \text{deg}(\tilde{g}) < \infty$. + Also ist $K$ endlich. + \item Es ist zunächst $K(Z) \subseteq L^{H}$, denn für $f \in K(Z)$ und $a \in \mathbb{F}_p$ + ist + \begin{salign*} + \sigma_a(f(Y^{p} - Y)) &= f(\sigma_a(Y^{p} - Y)) \\ + &= f((Y + a)^{p} - Y - a) \\ + &\stackrel{\text{char}(K) = p > 0}{=} f(Y^{p} + a^{p} - Y - a) \\ + &\stackrel{\text{Fermat}}{=} f(Y^{p} - Y) + .\end{salign*} + Weiter ist $\mathbb{F}_p$ endlich und nach (a) ist damit $\# H = \# \mathbb{F}_p = p$. Damit + ist $L / L^{H}$ galoissch mit $[ L \colon L^{H} ] = \# H = p$. Betrachte nun weiter + \[ + f = X^{p} - X - (Y^{p} - Y) \in K(Z)[X] + .\] Dann ist $\text{deg}(f) = p$ und $f(Y) = 0$. Also folgt $[ L : K(Z) ] \le p$. Damit gilt + wegen $K(Z) \subseteq L^{H}$: + \[ + p \ge [ L : K(Z) ] = [ L : L^{H} ] [ L^{H} : K(Z) ] = p [ L^{H} : K(Z) ] + .\] Also folgt $L^{H} = K(Z)$. + \end{enumerate} +\end{aufgabe} + +\end{document}