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Also + folgt $\alpha = \frac{\sqrt{5} - 1}{2}$. Also ist $\alpha \not\in \Q$ und + da $\zeta_5 \not\in \R$, folgt $\zeta_5 \not\in \Q(\alpha) \subseteq \R$. Also + ist $\Q(\alpha)$ der eine echte Zwischenkörper. + \item Es ist $\text{Re}(\alpha) = 2 \text{Re}(\zeta_5)$, also + $\text{Re}(\zeta_5) = \frac{\sqrt{5} - 1}{4}$. Dann ex. ein $a \in \R$, s.d. + \[ + \zeta_5 = \frac{\sqrt{5} -1}{4} + a i + .\] Da $|\zeta_5| = 1$, folgt + \[ + a^2 + \frac{(\sqrt{5} -1)^2}{4^2} = 1 \implies a = \pm \sqrt{1 - \frac{(\sqrt{5} -1)^2}{4^2}} + .\] Die Punkte $\zeta_5^{n}$ bilden nach Definition der Einheitswurzeln + ein regelmäßiges Fünfeck der Seitenlänge $|\zeta_5 - \zeta_5^{5}| = |\zeta_5 - 1|$. Es gilt + \[ + |\zeta_5 - 1 | = \sqrt{\left( \frac{\sqrt{5} - 1}{4}- 1\right)^2 + 1 + - \frac{(\sqrt{5} -1)^2}{4^2}} + = \sqrt{2 + \frac{- \sqrt{5} + 1}{2}} = \sqrt{\frac{5 - \sqrt{5} }{2}} + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item OBdA sei $\tau = (12)$. Da $\sigma$ $5$-Zykel, + ex. ein $k \in \{1, \ldots, 4\} $, s.d. $\sigma ^{k}(1) = 2$. Setze + nun $\sigma \coloneqq \sigma ^{k}$ und durch Umbenennung sei oE $\sigma = (1 2 3 4 5)$. Mit + Blatt 11 4a folgt nun + \begin{salign*} + \sigma \tau \sigma ^{-1} &= (\sigma(1) \sigma(2)) = (2 3) + \intertext{Analog folgt} + \sigma^2 \tau \sigma ^{-2} &= (3 4) \\ + \sigma ^{3} \tau \sigma ^{-3} &= (4 5) + .\end{salign*} + Also $(12), (23), (34), (45) \in H$. Seien nun $a, b \in \{1, \ldots, 5\} $ mit + $a \neq b$. OE sei $a < b$. Dann setze + \[ + \rho = (b \quad b-1) \cdots (a + 2 \quad a + 1) \in H + .\] Dann ist $\rho(a+1) = b$ und $\rho(a) = a$. Damit folgt + \[ + H \ni \rho(a \quad a+1) \rho ^{-1} = (\rho(a), \rho(a+1)) = (a b) + .\] Also jede Transposition in $H$ und da jede Permutation als Produkt von + Tranpositionen dargestellt werden kann, folgt $H = \mathfrak{S}_5$. + \item Da $f \in \Q$ irreduzibel und $\Q$ vollkommen, folgt $f$ separabel. Sei + also $L$ der ZK von $f$ über $\Q$. Dann ist $L / \Q$ galoissch. Setze + $G \coloneqq \text{Gal}(L / \Q) \subseteq \mathfrak{S}_5$. + Dann betrachte den $\Q$-Autom. + \begin{salign*} + \tilde{\pi}\colon L &\to \mathbb{C} \\ + z &\mapsto \overline{z} + .\end{salign*} + Da $\mathbb{C}$ ein algebraischer Abschluss von $\Q \subseteq L$ und $L / \Q$ normal, + beschränkt sich $\tilde{\pi}$ zu einem $\Q$-Automorphismus $\pi\colon L \to L$. Es + ist also $\pi \in G$ und da $f$ genau zwei nicht-reelle Nullstellen hat, + $G$ transitiv auf den Nullstellen operiert und $\pi|_\R = \text{id}$, lässt + $\pi$ die $3$ reellen Nullstellen fest und vertauscht die komplexen Nullstellen. $\pi$ ist + also eine Transposition. + + Sei weiter $\alpha \in L$ mit $f(\alpha) = 0$. Dann ist $f$ Mipo von $\alpha$ über $\Q$, also + $[K(\alpha) : \Q] = 5$, damit folgt mit Gradsatz, dass $5 \mid [L : \Q]$, also + ex. nach Sylowsätzen ein $\sigma \in G$ mit $\text{ord}(\sigma) = 5$. Also + ist $\sigma$ ein $5$-Zykel in $\mathfrak{S}_5$ und damit nach (a) $G = \mathfrak{S}_5$. + \item Es ist $f = 2X^{5} - 10x + 5$ irreduzibel über $\Q$, da + $f$ primitiv und damit irreduzibel nach Eisenstein mit $p =5$. + + Es ist $f$ als Funktion $f\colon \R \to \R$ stetig und differenzierbar mit + $f' = 10 x^{4} - 10$. $f'$ hat genau $2$ reelle Nullstellen bei $\pm 1$. Nach + Satz von Rolle, hat damit $f$ maximal $3$ reelle Nullstellen. Es gilt weiter + \[ + f(-2) < 0 \quad f(0) > 0 \quad f(1) < 0 \quad f(2) > 0 + .\] Also ex. nach Zwischenwertsatz mindestens $3$ reelle Nullstellen jeweils eine zwischen + $f(-2)$ und $f(0)$, $f(0)$ und $f(1)$ und $f(1)$ und $f(2)$. Also hat $f$ genau $3$ reelle + Nullstellen. + + Nach (b) ist also die Galoisgruppe der Gleichung $f(X) = 0$ die ganze + $\mathfrak{S}_5$, insbesondere ist $f(X) = 0$ nicht durch Radikale auflösbar, da + $\mathfrak{S}_5$ nicht auflösbar ist. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $G$ eine endliche Gruppe und bezeichne $s_p$ die Anzahl der $p$-Sylowgruppen von $G$. + \begin{enumerate}[(a)] + \item Zunächst sei $\#G = 6 = 2 \cdot 3$. Dann + ist $s_3 \in \{1, 2, 3, 6\} $, aber $2, 3, 6 \not\equiv 1 \text{ mod 3}$, also + $s_3 = 1$. Sei $H \subseteq G$ die einzige $3$-Sylowgruppe von $G$. Damit + folgt $H \triangleleft G$. und $\#H = 3$, also $H$ $3$-Gruppe, insb. auflösbar. Außerdem + $\# G / H = 6 / 3 = 2$, also $G / H $ $2-$Gruppe, insb. auflösbar. Insgesamt + folgt also nach 5.48 $G$ auflösbar. + + Sei nun $\#G = 42 = 2 \cdot 3 \cdot 7$. Dann ist $s_7 = 1$ nach Sylowsätzen. Sei + $H \subseteq G$ die einzige $7$-Sylow. von $G$. Dann ist $H \triangleleft G$ und + $\# H = 7$, also $H$ $7$-Gruppe, insb. auflösbar. Außerdem + ist $\# G / H = 42 / 7 = 6$, also $G / H$ nach Vorüberlegung auflösbar. + Insgesamt folgt mit 5.48 $G$ auflösbar. + \item Sei zunächst $\#G = 10 = 2\cdot 5$. Dann ist wieder analog $s_2 = 1$ und + da $10 / 2 = 5$ und $5$ prim, folgt analog zu (a), dass $G$ auflösbar. + + Sei nun $\# G = 30 = 2 \cdot 3 \cdot 5$. Dann ist $s_5 \in \{1, 6\} $ + und $s_3 \in \{1, 10\} $. + Falls $s_5 = 1$ und $30 / 5 = 6$ folgt analog zu (a), dass $G$ auflösbar. Sei + also nun $s_5 = 6$. Seien $H_1, H_2 \subseteq G$, $H_1 \neq H_2$ mit $\# H_1 = H_2 = 5$. + Da $5$ Primzahl, haben $H_1$ und $H_2$ keine echten Untergruppen. Also folgt + $H_1 \cap H_2 = \{1\} $. Da $s_5 = 6$ ex. also mindestens $6 \cdot (5 - 1) = 24$ Elemente + der Ordnung $5$ in $G$. Es kann maximal $6$ Elemente der Ordnung $3$ geben. Da + $3$ prim, also die $3$-Sylows gerade die von diesen Elementen erzeugten Untergruppen sind, + folgt $s_3 < 10 \implies s_3 = 1$. Argumentiere nun wieder analog zu (a), da + $30 / 3 = 10$ und Gruppen der Ordnung $10$ nach Vorüberlegung auflösbar. + \item Seien $G, H$ auflösbar und + \begin{salign*} + 1 &= G_n \subset G_{n-1} \subset \ldots \subset G_0 = G \\ + 1 &= H_m \subset H_{m-1} \subset \ldots \subset H_0 = H + \end{salign*} Normalreihen mit abelschen Faktoren. Sei o.E. $n \ge m$. Dann + setze $U_i \coloneqq G_i \times H_i$ mit $H_{m+1} = \ldots = H_n = 1$. Es gilt + trivialerweise $H_{m+k} \triangleleft H_{m+k-1}$, also $(H_i)_{i = 0}^{n}$ immer noch + Normalreihe mit abelschen Faktoren von $H$. + + Sei nun $a \in U_{i-1}$ mit $a = (g, h)$ und $g \in G_{i-1}, h \in H_{i-1}$. Dann + ist, da $G_i \triangleleft G$ und $H_i \triangleleft H$: + \begin{salign*} + a U_i a^{-1} = a (G_i \times H_i)a^{-1} = (g G_i g^{-1}) \times (h H_i h^{-1}) + = G_i \times H_i = U_i + .\end{salign*} + Außerdem sei $\varphi\colon G_{i-1} \times H_{i-1} \to G_{i-1} / G_i \times H_{i-1} / H_i$, + $(g,h) \mapsto (\overline{g}, \overline{h})$. Dann ist $\varphi$ Grp.hom. und + $\text{ker } \varphi = G_i \times H_i$. Also folgt mit Hom.satz: + \[ + (G_{i-1} \times H_{i-1}) / (G_{i} \times H_i) + \stackrel{\sim }{=} G_{i-1} / G_i \times H_{i-1} / H_i + .\] + Also ist + \[ + 1 = U_n \subset U_{n-1} \subset \ldots \subset U_0 = G \times H + \] eine Normalreihe mit abelschen Faktoren. + \end{enumerate} +\end{aufgabe} + +\end{document}