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zzFvuj)2nmsw1(2omtR9f+&v-}F%o}@b6ln_Qu<^WGW^B_Ir4SqI0Xv_>`X!a%;==? zz~3HES$91Q7qf{?oJ?s0E@z6lObLNbh{nd=Ocb}^y(#b2=}!uR!Bh!AE4b;GU*6I9 qrr`nCy`PS#*b&^h|KBz4;c4dX>FfTwsM$HW|DjaT(MhX7F#Z>+bQFvL literal 0 HcmV?d00001 diff --git a/ws2020/algebra/uebungen/algebra11.tex b/ws2020/algebra/uebungen/algebra11.tex new file mode 100644 index 0000000..331c2af --- /dev/null +++ b/ws2020/algebra/uebungen/algebra11.tex @@ -0,0 +1,277 @@ +\documentclass[uebung]{../../../lecture} + +\title{Algebra I: Übungsblatt 11} +\author{Lukas Nullmeier, Christian Merten} +\usepackage[]{gauss} + +\begin{document} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es ist $L$ Zerfällungskörper von $f$ über $K$, also $L / K$ normal. + Außerdem ist $f$ irreduzibel und $f' = n x^{n-1}$ und $\text{char }K \nmid n$ also + $f' \neq 0$. Also $f$ separabel. Also sind die Nullstellen von $f$ separabel und + da $L$ von den Nullstellen von $f$ über $K$ erzeugt, folgt $L / K$ separabel, + insgesamt also galoissch. + + Sei nun $b \in L$ eine Nullstelle von $f$ und $\mu_n = \{\zeta_1, \ldots, \zeta_n\} $. + Diese sind paarw. verschieden. + Setze nun $\alpha_k \coloneqq \zeta_k b$. Dann + ist für $k \in \{1, \ldots, n\} $: + \[ + f(\alpha_k) = (\zeta_k b)^{n} - a = \zeta_k^{n} b^{n} - a = b^{n} - a = 0 + .\] Das heißt $\alpha_k$ sind Nullstellen von $f$ und + paarweise verschieden (Ang. es gäbe $i \neq j$ mit $\alpha_i = \alpha_j \implies \zeta_i = + \zeta_j$ $\contr$), also sind die $n$ Nullstellen von $f$ genau die $(\alpha_k)_{k=1}^{n}$. + Da $\mu_n \subseteq K$ folgt $\alpha_k \in K(b)$ $\forall k \in \{1, \ldots, n\} $, also + $L = K(b)$. + \item Sei $\beta \in L$ eine Nullstelle von $f$. Dann sind analog zu (a) + $\alpha_k = \zeta_k \beta$ die Nullstellen von $f$. + + \begin{itemize} + \item Z.z.: $\psi$ wohldefiniert. + Seien $b, b'$ Nullstellen von $f$. Dann ex. $i, j \in \{ 1, \ldots, n\} $, s.d. + $b = \zeta_i \beta$ und $b' = \zeta_j \beta$. Dann folgt + \[ + \frac{\sigma(b)}{b} = \frac{\sigma(\zeta_i \beta)}{\zeta_i \beta} + = \frac{\zeta_i \sigma(\beta)}{\zeta_i \beta} + = \frac{\sigma(\beta)}{\beta} + = \frac{\zeta_j \sigma(\beta)}{\zeta_j \beta} + = \frac{\sigma(\zeta_j \beta)}{ \zeta_j \beta} + = \frac{\sigma(b')}{b'} + .\] + \item Z.z.: $\psi$ Gruppenhomomorphismus + Seien $\sigma, \tau \in \text{Gal}(L / K)$. Dann ex. ein $i \in \{ 1, \ldots, n\}$, + s.d. $\tau(\beta) = \alpha_i$. Dann gilt + \[ + \psi(\sigma \circ \tau) = \frac{\sigma \circ \tau(\beta)}{\beta} + = \frac{\sigma(\zeta_i \beta)}{\beta} + = \frac{\beta \zeta_i \sigma(\beta)}{\beta^2} + = \frac{\tau(\beta)}{\beta} \frac{\sigma(\beta)}{\beta} + .\] + \item Z.z.: $\psi$ injektiv. + Sei $\sigma \in \text{Gal}(L / K)$ mit $\psi(\sigma) = 1$. Dann ex. + ein $i \in \{1, \ldots, n\} $, s.d. $\sigma(\beta) = \alpha_i$. Damit folgt + \[ + 1 = \psi(\sigma) = \frac{\sigma(\beta)}{\beta} + = \frac{\zeta_i \beta}{\beta} = \zeta_i + .\] Also $\sigma(\beta) = \beta$. Da $L / K$ von $\beta$ erzeugt wird, legt + $\sigma(\beta)$ $\sigma$ eindeutig fest, d.h. $\sigma = \text{id}$. + \item Z.z.: $\text{Bild}(\psi) = \mu_n$. Sei $\sigma \in \text{Gal}(L / K)$. Dann + ex. ein $i \in \{1, \ldots, n\} $, s.d. $\sigma(\beta) = \zeta_i \beta$. Also + \[ + \psi(\sigma) = \frac{\sigma(\beta)}{\beta} = \frac{\zeta_i \beta}{\beta} = \zeta_i \in \mu_n + .\] Also folgt $\text{Bild}(\psi) \subseteq \mu_n$. Da + $L = K(\beta)$ und $f$ Mipo von $\beta$, folgt $[L : K ] = n$, also + $\# \text{Gal}(L / K) = n$. Da $\psi$ injektiv, ist + also $\#\text{Bild}(\psi) = n$ und da $\# \mu_n = n$ folgt + $\text{Bild}(\psi) = \mu_n$. + \end{itemize} + Damit ist also $\text{Gal}(L / K) \stackrel{\sim }{=} \mu_n$. Da $\mu_n$ zyklisch, folgt + $\text{Gal}(L / K)$ zyklisch. + \item Betrachte $K = \Q$, $f = X^{4} - 2$. Dann ist $f$ irred. nach Eisenstein und + nach Blatt 5 ist $L = \Q(\sqrt[4]{2}, i)$ und $\text{Gal}(L / \Q) \stackrel{\sim }{=} D_4$, + aber $D_4$ ist nicht abelsch, insbesondere nicht zyklisch. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Zunächst beachte: + \[ + \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in \text{GL}_2(\mathbb{F}_p) + \iff \begin{gmatrix}[v] a & b \\ 0 & d \end{gmatrix} = ad \neq 0 + \quad \stackrel{\mathbb{F}_p \text{ nullt.frei.}}{\iff} \quad a \neq 0 \land d \neq 0 + .\] Damit ist + \[ + G = \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in M_{2,2}(\mathbb{F}_p) \; + \Big| \;a, b, d \in \mathbb{F}_p, a \neq 0 \neq d \right\} + .\] Damit folgen + \begin{salign*} + G\begin{pmatrix} 0 \\ 0 \end{pmatrix} &= \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \end{pmatrix} \; \Big| \; a, b, d \in \mathbb{F}_p, a\neq 0 \neq d\right\} \\ + &= \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix} \right\} \\ + G\begin{pmatrix} 1 \\ 0 \end{pmatrix} &= \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix} \; \Big| \; a, b, d \in \mathbb{F}_p, a\neq 0 \neq d\right\} \\ + &= \left\{ \begin{pmatrix} a \\ 0 \end{pmatrix} \; \Big| \; a \in \mathbb{F}_p^{\times } \right\} \\ + G\begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \; \Big| \; a, b, d \in \mathbb{F}_p, a\neq 0 \neq d\right\} \\ + &= \left\{ \begin{pmatrix} b \\ d \end{pmatrix} \; \Big| \; b, d \in \mathbb{F}_p, d \neq 0\right\} + .\end{salign*} + \begin{enumerate}[(a)] + \item Es ist $\begin{pmatrix} 0 \\ 0 \end{pmatrix} \not\in G\begin{pmatrix} 1 \\ 0 \end{pmatrix} + \cup G \begin{pmatrix} 0 \\ 1 \end{pmatrix} $ + und $\begin{pmatrix} 1 \\ 0 \end{pmatrix} \not\in G\begin{pmatrix} 0 \\ 1 \end{pmatrix} $. Da + $V$ in die disjunkte Vereinigung der Bahnen zerfällt, sind also + $G \begin{pmatrix} 0 \\ 0 \end{pmatrix}, G \begin{pmatrix} 1 \\ 0 \end{pmatrix}, + G \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ paarw. verschieden. + + Sei nun $x \in V$ beliebig. Dann ex. $a, b \in \mathbb{F}_p$ mit $x = \begin{pmatrix} a \\ b \end{pmatrix} $. + Falls $a = b = 0$, dann folgt $x \in G \begin{pmatrix} 0 \\ 0 \end{pmatrix} $. + + Falls $b \neq 0$. Dann ist + \[ + \underbrace{\begin{pmatrix} 1 & a \\ 0 & b \end{pmatrix}}_{\in G} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} + = \begin{pmatrix} a \\ b \end{pmatrix} = x + \implies x \in G \begin{pmatrix} 0 \\ 1 \end{pmatrix} + .\] Falls $b = 0$, dann ist $a \neq 0$ und es gilt + \[ + \underbrace{\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}}_{\in G} + \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ 0 \end{pmatrix} = x + \implies x \in G \begin{pmatrix} 1 \\ 0 \end{pmatrix} + .\] Das zeigt die Behauptung. + \item Seien nun $x_1 \coloneqq \begin{pmatrix} 0 \\ 0 \end{pmatrix} $, $x_2 \coloneqq \begin{pmatrix} 1 \\ 0 \end{pmatrix} $ und $x_3 \coloneqq \begin{pmatrix} 0 \\ 1 \end{pmatrix} $. + Es ist offensichtlich $G_{x_1} = G$. + + Es gilt + \[ + \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} + = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \iff \begin{pmatrix} a \\ 0 \end{pmatrix} + = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \iff a = 1 + .\] Also folgt + \[ + G_{x_2} = \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in M_{2,2}(\mathbb{F}_p) + \; \Big| \; a,b, d \in \mathbb{F}_p, d \neq 0, a = 1 \right\} + .\] Weiter gilt + \[ + \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} + = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \iff \begin{pmatrix} b \\ d \end{pmatrix} + = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \iff b = 0 \land d = 1 + .\] Also folgt + \[ + G_{x_3} = \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in M_{2,2}(\mathbb{F}_p) + \; \Big| \; a,b, d \in \mathbb{F}_p, a \neq 0, b = 0, d=1 \right\} + .\] + \item Für die Bahnen siehe Vorbemerkung. + Es ist ebenfalls nach Vorbemerkung $\# G = (p-1)^2 p$. Mit (b) folgt nun: + \begin{itemize} + \item $\#(G x_1 )= 1$ und $\#G_{x_1} = \#G$, also $\# G_{x_1} \#(G x_1) = \#G$. + \item $\#(G x_2) = p-1$ und $\#G_{x_2} = p(p-1)$, also + $\# G_{x_2} \#(G x_2) = (p-1)^2 p = \#G$. + \item $\#(G x_3) = p(p-1)$ und $\# G_{x_3} = p-1$, also + $\# G_{x_3} \#(G x_3) = (p-1)^2 p = \# G$. + \end{itemize} + \item Es ist nach Lagrange + \[ + (G : G_x) = \frac{\# G}{\# G_x} = \frac{(p-1)^2 p}{\# G_x} + .\] Also folgt durch Einsetzen der Ergebnisse aus (c): + $(G : G_{x_1}) = 1$, $(G : G_{x_2}) = p-1$ und $(G : G_{x_3}) = (p-1)p$. Damit folgt + \[ + 1 + (p-1)p + p-1 = 1 + p^2 -p +p -1 = p^2 = \#V + .\] + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $G$ endlich mit $\#G = 2020$. Dann + ist $2020 = 2^{2} \cdot 5 \cdot 101$. Da $101 \mid \#G$, ex. + ein $a \in G$ mit $\text{ord}(a) = 101$, also $H \coloneqq \langle a \rangle$ + ist $101$-Gruppe und $101 \nmid 2^2 \cdot 5 = (G : H)$, also + $H$ $101$-Sylowgruppe. Sei $s$ die Anzahl der $101$-Sylowgruppen. Nach Sylowsätzen gilt + nun $s \mid \#G$ und $s \equiv 1 \text{ mod 101}$. Da aber $2^2 \cdot 5 < 101$ sind + alle Teiler $t \neq 1$ von $2020$ mit $101 \nmid 101$, bereits $t \not\equiv 1 \text{ mod } 101$. + Also folgt $s = 1$. + + Da alle $101$-Sylowgruppen zueinander konjugiert sind, gilt für $g \in G$: $gHg^{-1} = H$. + Also ist $H$ Normalteiler in $G$. Außerdem ist $\# H = 101$, also $H \stackrel{\sim }{=} \Z / 101 \Z$, + insbesondere abelsch. + \item Sei $G$ endlich mit $\#G = 2021$. + Es ist $2021 = 43 \cdot 47$. Da $43 < 47$ und $43 \nmid 46 = 47 - 1$ folgt nach VL, dass + $G$ zyklisch ist und damit $G \stackrel{\sim }{=} \Z / 2021 \Z$. + \item Sei $G$ endlich mit $\#G = 36 = 2^2 \cdot 3^2$. Sei $s$ die Anzahl der + $3$-Sylowgruppen. Es ist $s \mid \#G$ und $s \equiv 1 \text{ mod }3$, also + folgt $s \in \{1, 4\} $, da $2 \equiv 2 \text{ mod }3$. + Falls $s = 1$: Wende Argument aus (a) an. + + Falls $s = 4$: Dann sei $X$ die Menge der $3$-Sylowgruppen auf denen $G$ mittels Konjugation + wirkt. Es ist dann $\#X = 4$ und + $\forall H \in X$ ist $\#H = 9$, also $\{1\} \neq H \neq G$. + Sei $\varphi\colon G \to \mathfrak{S}(X) + \stackrel{\sim }{=} \mathfrak{S}_4$, der zur Konjugationswirkung assoziierte + Gruppenhomomorphismus. Da + \[ + \# \mathfrak{S}_4 = 4! = 24 < 36 = \#G + \] + folgt $\text{ker } \varphi \neq \{1\}$. + Falls $\varphi(G) = \{\text{id}\} $: Dann + ist für $H \in X$ und $g \in G$: + \[ + g H g^{-1} = H^{g} = \varphi(g)H = H + .\] + Also $H$ nicht-trivialer + Normalteiler in $G$. + Sei nun $\#\varphi(G) > 1$: Da $G / \text{ker }\varphi \stackrel{\sim }{=} \varphi(G)$, folgt + $\# \varphi(G) \# \text{ker } \varphi = \#G$. Da $\#\varphi(G) > 1$, + folgt also $\# \text{ker }\varphi < 36$, also insgesamt $G \neq \text{ker } \varphi \neq \{1\} $. + Damit ist $\text{ker } \varphi$ nicht-trivialer Normalteiler in $G$. + + %Es ist $\#X = s = 4 = 1 + 3 = 1 + 1 + 2 = 1 + 1 + 1 + 1 = 2 + 2$. Nach Bahnengleichung + %ex. also entweder eine Bahn der Kardinalität $4$, + %mindestens eine Bahn der Kardinalität $1$ oder zwei Bahnen der Kardinalität $2$. + + %Falls es nur eine Bahn der Kardinalität $4$ gibt, dann gibt es insbesondere + + %Falls es mindestens eine Bahn der Kardinalität $1$ gibt, dann ex. ein $H \in X$, s.d. + %$GH = \{H\} $. Dann ist aber $H \in \text{Fix}_{G}(X)$ und damit ist nach VL + %$H \neq \{1\}$ Normalteiler in $G$. + + %Falls es eine Bahn der Kardinalität $2$ gibt, dann sein $H \in X$ ein + %Vertreter dieser Bahn. Dann gilt $2 = \#(GH) = (G : G_{H})$. Also + %ist $G_H \neq \{1\}$ Normalteiler in $G$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $(x_1 \ldots x_r)$ ein $r$-Zykel in $\mathfrak{S}_n$ und $\sigma \in \mathfrak{S}_n$ beliebig. + Sei $x \in \{1, \ldots, n\} $. + + Falls es ex. ein $i \in \{1, \ldots, r\} $ mit $\sigma ^{-1}(x) = x_i $, dann + ist $\sigma(x_i) = x$ und $(x_1 \ldots x_r) x_i = x_{i+1}$ mit $x_{r+1} \coloneqq x_1$. Dann + gilt + \[ + \sigma(x_1 \ldots x_r) \sigma ^{-1}(x) = \sigma (x_1 \ldots x_r) x_i + = \sigma(x_{i+1}) + = (\sigma(x_1) \ldots \sigma(x_r))\sigma(x_i) + = (\sigma(x_1) \ldots \sigma(x_r))x + .\] + + Falls $\sigma ^{-1}(x) \not\in \{x_1, \ldots, x_r\} $, dann + ist $x \not\in \{ \sigma(x_1), \ldots, \sigma(x_r)\} $ und + \[ + \sigma(x_1 \ldots x_r) \sigma ^{-1}(x) = \sigma \sigma ^{-1}(x) = x + = \left( \sigma(x_1) \ldots \sigma(x_r) \right) x + .\] + \item Für ein Produkt aus zwei $2$-er Zykeln gilt für $\sigma \in \mathfrak{S}_n$: + \[ + \sigma (x_1 x_2) (x_3 x_4) \sigma ^{-1} + = \sigma (x_1 x_2) \sigma ^{-1} \sigma (x_3 x_4) \sigma ^{-1} + \stackrel{\text{(a)}}{=} (\sigma(x_1) \sigma(x_2)) (\sigma(x_3) \sigma(x_4)) + .\] Also ist ein Produkt aus zwei $2$-er Zykeln nach Konjugation wieder + ein Produkt aus zwei $2$-er Zykeln. Außerdem ist, falls + $x_1, \ldots, x_4$ paarweise verschieden, auch $\sigma(x_1), \ldots, \sigma(x_4)$ paarweise + verschieden, da $\sigma$ injektiv. + Außerdem gilt $\sigma \text{id} \sigma ^{-1} = \text{id}$. + + Da + $\mathfrak{V}_4$ gerade aus $\text{id}$ und allen Produkten aus + zwei $2$-er Zykeln mit paarweise verschiedenen Einträgen besteht, + folgt $\sigma \mathfrak{V}_4 \sigma ^{-1} = \mathfrak{V}_4$, also + $\mathfrak{V}_4$ Normalteiler in $\mathfrak{S}_4$. + \item Es ist $1 \triangleleft \mathfrak{V}_4$ trivial und + $\# \mathfrak{V}_4 / 1 = \# \mathfrak{V}_4 = 4 = 2^2$, also $\mathfrak{V}_4 / 1 $ abelsch. + + Da $\mathfrak{V}_4 \triangleleft \mathfrak{S}_4$ und $\text{sgn}(\sigma) = 1$ + $\forall \sigma \in \mathfrak{V}_4$ folgt $\mathfrak{V}_4 \triangleleft \mathfrak{A}_4$. Da + $\# \mathfrak{A}_4 = \frac{4!}{2} = 12$ folgt $\# \mathfrak{A}_4 / \mathfrak{V}_4 = 3$, + also $\mathfrak{A}_4 / \mathfrak{V}_4$ abelsch. + + Da $\mathfrak{A}_4 = \text{ker } \text{sgn}$ ist $\mathfrak{A}_4 \triangleleft \mathfrak{S}_4$ + und $\# \mathfrak{S}_4 / \mathfrak{A}_4 = 2$, also $\mathfrak{S}_4 / \mathfrak{A}_4$ abelsch. + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2020/wtheo/uebungen b/ws2020/wtheo/uebungen index 4c121ea..25ed8ff 160000 --- a/ws2020/wtheo/uebungen +++ b/ws2020/wtheo/uebungen @@ -1 +1 @@ -Subproject commit 4c121ead88cc592b2a998f94b046554c103717d3 +Subproject commit 25ed8ff2ef2e122a6fc1ff1127ee1245d9d17843