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+++ b/ws2020/algebra/uebungen/algebra10.tex @@ -0,0 +1,296 @@ +\documentclass[uebung]{../../../lecture} + +\title{Algebra I: Übungsblatt 10} +\author{Lukas Nullmeier, Christian Merten} +\newcommand{\tageq}{\stepcounter{equation}\tag{\theequation}} + +\begin{document} + +\punkte + +\begin{aufgabe} + Beh.: $f$ irreduzibel + \begin{proof} + Es ist $f(0) = 2$, $f(1) = 1$, $f(2) = 2$ als hat $f$ keine Nullstellen in $\mathbb{F}_3$. + Ang.: $f$ ist reduzibel, dann ex. $g, h \in \mathbb{F}_3[X]$ mit $\text{deg}(g) = \text{deg}(h) = 2$, + da sonst $f$ eine Nullstelle in $\mathbb{F}_3$ hätte. Dann folgt mit $a, b, c, d \in \mathbb{F}_3$: + \begin{salign*} + X^{4} + 2X^2 + 2 &= (X^2 + aX + b)(X^2 + cX + d) \\ + &= X^{4} + \underbrace{(a+c)}_{= 0}X^{3} + (b+d+ac) X^2 + + \underbrace{(bc+ad)}_{=0}X + bd \\ + .\end{salign*} + Es folgt direkt $c = -a$ und damit $a(d-b) = 0$, also $a = 0$ oder $d = b$. + Falls $a = 0$: Dann ist $b + d = 2$ und $bd = 2$ mit $b, d \in \mathbb{F}_3$ $\contr$. + Also ist $b = d$. Dann ist $b^2 = 2$, aber $2$ kein Quadrat in $\mathbb{F}_3$ $\contr$. + + Es folgt also $f$ irreduzibel. + \end{proof} + + Da $f$ irreduzibel und $\mathbb{F}_3[X]$ HIR + ist $\mathbb{F}_{81} = \mathbb{F}_3[X] / (f)$ also ein Körper mit $3^4 = 81$ Elementen. Dann + folgt sofort, dass $f(\overline{X}) = \overline{X}^{4} + 2 \overline{X}^2 + 2 = 0$. + Setze $L \coloneqq \mathbb{F}_3(\overline{X}) \subseteq \mathbb{F}_{81}$. + Da $\mathbb{F}_3$ endlich ist + $L / \mathbb{F}_3$ normal und separabel, also galoissch. + Sei $\sigma$ der Frobenius-Automorphismus für $p = 3$. Dann gilt + \begin{salign*} + \sigma(\overline{X}) &= \overline{X}^{3} \\ + \sigma^2(\overline{X}) &= 2\overline{X} \\ + \sigma ^{3}(\overline{X}) &= 2 \overline{X}^{3} \\ + \sigma ^{4}(\overline{X}) &= \overline{X} + .\end{salign*} + Also ist $\text{ord}(\sigma) = 4$. Da $\text{Gal}( L / \mathbb{F}_3) = \langle \sigma \rangle$ folgt + $\# \text{Gal}( L / \mathbb{F}_3) = 4$ und $[ L : \mathbb{F}_3 ] = 4$, da + aber $[\mathbb{F}_{3^{4}} : \mathbb{F}_3 ] = 4$ folgt $L = \mathbb{F}_{81}$. Außerdem + überführt $\sigma$ Nullstellen von $f$ in Nullstellen von $f$. Da $f$ genau $4$ Nullstellen + in $\overline{\mathbb{F}_3}$ hat, sind die Nullstellen von $f$ bereits + $\{\overline{X}, \overline{X}^{3}, 2 \overline{X}, 2 \overline{X}^{3}\} $ und + $\mathbb{F}_{81}$ ist Zerfällungskörper von $f$ über $\mathbb{F}_3$. + + Wegen $\text{Gal}(\mathbb{F}_{81} / \mathbb{F}_3) = \langle \sigma \rangle$, ist diese zyklisch + der Ordnung $4$, also $\text{Gal}(\mathbb{F}_{81} / \mathbb{F}_3 ) \stackrel{\sim }{=} \Z / 4 \Z$. + Diese hat genau $1$ echte nicht-triviale Untergruppe, da $\text{ord}(1) = \text{ord}(3) = 4$ + und $\text{ord}(2) = 2$. Damit folgen als Untergruppen von $\text{Gal}(\mathbb{F}_{81} / \mathbb{F}_3)$: + \begin{salign*} + H_1 &= \langle \text{id} \rangle \\ + H_2 &= \langle \sigma^2 \rangle \\ + H_3 &= \langle \sigma \rangle + .\end{salign*} + Es ist $\sigma^2(\overline{X}^2) = \sigma^2(\overline{X})^2 = (2 \overline{X})^2 = \overline{X}^2$. + Also ist $\overline{X}^2 \in \mathbb{F}_{81}^{H_2}$, + insbesondere $\mathbb{F}_3(\overline{X}^2) \subseteq \mathbb{F}_{81}^{H_2}$. + Es ist $\text{Gal}( \mathbb{F}_{81} / \mathbb{F}_{3}(\overline{X}^2)) = H_2$, insbesondere + $[ \mathbb{F}_{81} : \mathbb{F}_{81}^{H_2} ] = 2$ und da $[ \mathbb{F}_{81} : \mathbb{F}_3 ] = 4$ folgt + auch $[ \mathbb{F}_{81}^{H_2} : \mathbb{F}_3 ] = 2$. Da aber $\overline{X}^2 \not\in \mathbb{F}_3$ gilt + ebenfalls $[ \mathbb{F}_3(\overline{X}^2) : \mathbb{F}_3 ] \ge 2$. Mit Gradformel folgt + also $\mathbb{F}_3(\overline{X}^2) = \mathbb{F}_{81}^{H_2}$. + Damit folgen nach dem Hauptsatz die Zwischenkörper: + \begin{salign*} + \mathbb{F}_{81}^{H_1} &= \mathbb{F}_{81} \\ + \mathbb{F}_{81}^{H_2} &= \mathbb{F}_3(\overline{X}^2) \\ + \mathbb{F}_{81}^{H_3} &= \mathbb{F}_3 + .\end{salign*} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es ist $\text{deg}(\Phi_n(-X)) = \text{deg}(\Phi_n(X))$. Außerdem + ist $(2, n) = 1$, da $n$ ungerade. Damit folgt $\varphi(2n) = \varphi(2) \varphi(n) = \varphi(n)$. + Also folgt insgesamt + \[ + \text{deg}(\Phi_{2n}) = \varphi(2n) = \varphi(n) = \text{deg}(\Phi_n(X)) + = \text{deg}(\Phi_{n}(-X)) + .\] + Außerdem sei $\zeta$ Nullstelle von $\Phi_{2n}$. Dann ist $\zeta \in \mu_{2n}$ und primitiv. + Also ex. ein $k \in \N$ mit $(2n, k) = 1$ und + $\zeta = \exp(\frac{2\pi i}{2n} k)$. Betrachte nun + \[ + - \zeta = \exp\left(\pi i + \frac{\pi i}{n} k \right) + = \exp\left( \frac{\pi i (n + k)}{n} \right) + .\] Da $(2n, k) = 1$ und $2n$ gerade, folgt $k$ ungerade. Da $n$ ebenfalls ungerade, + folgt $n + k \in 2 \N$. Also ex. ein $l \in \N$, s.d. $n + k = 2l$. Damit folgt + \[ + - \zeta = \exp\left( \frac{2 \pi i l}{n} \right) + .\] Es ist $(2n, k) = 1$, d.h. es existieren $a, b \in \Z$, s.d. + \[ + 1 = a 2n + bk = n (2a + b - b) + bk = b(n+k) + (2a-b)n = \underbrace{2b}_{\in \Z} + l + \underbrace{(2a-b)}_{ \in \Z} n + .\] Es folgt $(l, n) = 1$, also ist $-\zeta$ primitive $n$-te Einheitswurzel. + Also $- \zeta$ Nullstelle von $\Phi_n(X)$ und damit $\zeta$ Nullstelle von $\Phi_n(-X)$. + + Da nun $\text{deg}(\Phi_{2n}) = \text{deg}(\Phi_{n}(-X)) $ und jede Nullstelle von + $\Phi_{2n}$ auch Nullstelle von $\Phi_n(-X)$ folgt + $\Phi_n(-X) \stackrel{\sim }{=} \Phi_{2n}$. Da $\Phi_n$ normiert für $n \in \N$ + folgt $a \in \{\pm 1\} $ und $a$ ist gerade der Leitkoeffizient von $\Phi_n(-X)$. + Es genügt jetzt zu zeigen, dass $\phi(n)$ gerade ist für $n \ge 3$ und $n$ ungerade. Denn + dann ist der Leitkoeffizient $e(\Phi_n(-X)) = (-1)^{2l} = 1$ für ein $l \in \N$ + und damit $\Phi_n(-X) = \Phi_{2n}$. + + Da $\Z$ faktoriell und $n \ge 3$, ungerade ex. $p_i$ und $r_i$ mit $p_i$ + paarweise verschieden und $r_i \in \N$, s.d. + \[ + n = \prod_{i=1}^{s} p_i^{r_i} + .\] Da $n$ ungerade ist $p_i \neq 2$ und damit $p_i - 1$ gerade für alle $i$. + Da die $p_i$ paarweise verschieden sind diese teilerfremd und es gilt + \[ + \varphi(n) = \prod_{i=1}^{s} \varphi(p_i^{r_i}) = \prod_{i=1}^{s} (p_i - 1) p_i^{r_i-1} + .\] Da die $p_i -1$ gerade und $s \ge 1 $ wegen $n \ge 3$, folgt $2 \mid \varphi(n)$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Da $\overline{K}$ nullteilerfrei folgt + \begin{salign*} + \Delta_f = 0 \iff \alpha_i - \alpha_j = 0 \text{ für } i \neq j + \iff \alpha_i = \alpha_j \text{ für } i \neq j \iff f \text{ nicht separabel} + .\end{salign*} + \item Die Aussage ist offensichtlich falsch für $a = 0$. Sei also $a \neq 0$ + und im folgenden $\text{char}(K) \neq 2$. Sei außerdem $\sqrt{a} \in \overline{K}$ + mit $\sqrt{a}^2 = a$ und $\sqrt{b^2 - 4 a c} \in \overline{K}$ + mit $\sqrt{b^2 - 4 a c}^2 = b^2 - 4 a c$. Dann sind die Nullstellen von + $f$ gegeben als $\alpha_{1,2} = \frac{-b \pm \sqrt{b^2 - 4 a c} }{2a} \in \overline{K}$. + + Denn es gilt + \begin{salign*} + f &= aX^2 + bX + \left( \frac{b}{2 \sqrt{a} } \right)^2 - \left( \frac{b}{2 \sqrt{a} } \right)^2 + + c = \left( \sqrt{a} X + \frac{b}{2 \sqrt{a} } \right)^2 - \frac{b^2}{4a} +c + \intertext{Damit folgt} + f(\alpha_{1,2}) &= \left( \sqrt{a} \frac{-b \pm \sqrt{b^2 - 4ac} }{2a} + \frac{b}{2 \sqrt{a} } \right)^2 - \frac{b^2}{4a} + c + = 0 + .\end{salign*} + Damit folgt + \begin{salign*} + \Delta_f &= (\alpha_1 - \alpha_2)^2 \\ + &= \left[\frac{-b + \sqrt{b^2 - 4ac} }{2a} - \frac{-b - \sqrt{b^2 - 4ac} }{2a} \right]^2\\ + &= \frac{1}{a^2} (b^2 - 4ac) + .\end{salign*} + \item Sei $\sigma \in G$. Dann ist $\sigma(\alpha_i) = \alpha_j$ für geeignete + $i, j \in \{1, \ldots, n\}$. Damit folgt + \begin{salign*} + \sigma(\delta_f) &= \prod_{1\le i < j \le n}^{} (\sigma(\alpha_i) - \sigma(\alpha_j)) \\ + &= \prod_{1\le i < j \le n}^{} \left( \alpha_{\varphi(\sigma)(i)} - \alpha_{\varphi(\sigma)(j)} \right) + \intertext{ + Es ist $\varphi(\sigma)$ als Produkt von $s$ Transpositionen darstellbar für + $s \in \N$. Jede + dieser $s$ Transpositionen ändert das Vorzeichen von $\delta_f$ um $(-1$). + Also folgt} + \sigma(\delta_f) &= (-1)^{s} \prod_{1\le i < j \le n}^{} (\alpha_{i} - \alpha_{j})\\ + &= \text{sgn}(\varphi(\sigma)) \delta_f + .\end{salign*} + \item Es ist für $\sigma \in G$: + \[ + \sigma(\Delta_f) = \sigma(\delta_f^2) = \sigma(\delta_f)^2 + \; \stackrel{\text{(c)}}{=} (\text{sgn}(\varphi(\sigma)) \delta_f)^2 = \Delta_f + .\] Also folgt $\Delta_f \in L^{G} = K$. + \item Sei zunächst $\varphi(G) \subseteq \mathfrak{A}_n$. Dann ist für $\sigma \in G$ wegen (c) + $\sigma(\delta_f) = \sigma(\delta_f)$, also $\delta_f \in L^{G} = K$, also da + $f$ separabel und nach (a) also $\Delta_f \neq 0$ auch + $\Delta_f = \delta_f^2 \in (K^{x})^2$. + + Sei nun $\Delta _f \in (K^{\times })^2$. Dann ist $\delta_f \in K^{\times}$. Sei + nun $\sigma \in G$. Dann folgt + \begin{salign*} + \delta_f = \sigma(\delta_f) = \text{sgn}(\varphi(\sigma)) \delta_f + \implies \delta_f (\text{sgn}(\varphi(\sigma)) - 1) = 0 + .\end{salign*} + Da $\delta_f \in K^{x}$ folgt + $\text{sgn}(\varphi(\sigma)) = 1 \qquad \stackrel{\text{char } K \neq 2}{\neq } \qquad -1$. + Also $\varphi(G) \subseteq \mathfrak{A}_n$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Seien $a, b \in \Z$ mit $(a, q) = 1$ und $(b, q) = 1$. Dann + ist auch $(ab, q) = 1$, da $\Z$ faktoriell. Damit + ist $\overline{a}, \overline{b} \in \mathbb{F}_p^{\times }$. Damit folgt + nach Zettel 6, Aufgabe 3(c): + \begin{salign*} + \left( \frac{ab}{q} \right) = \left( \overline{a b} \right)^{\frac{q-1}{2}} + = \overline{a}^{\frac{q-1}{2}} \overline{b}^{\frac{q-1}{2}} + = \left( \frac{a}{q} \right) \left( \frac{b}{q} \right) + .\end{salign*} + Erneut nach Zettel 6 ist $\left( \frac{-1}{q} \right) = (-1)^{\frac{q-1}{2}} \in \{\pm 1\}$. + Da $q$ ungerade ist weiter $q \equiv 3 \text{ (mod 4)}$ oder $q \equiv 1 \text{ (mod 4)}$, + also g.z.z., dass $(-1)^{\frac{q-1}{2}} = 1 \iff q \equiv 1 \text{ (mod 4)}$. Es gilt + \[ + (-1)^{\frac{q-1}{2}} = 1 \iff 2 \mid \frac{q-1}{2} + \iff 4 | (q-1) + \iff q \equiv 1 \text{ (mod 4)} + .\] + \item Es ist $f = X^{p} - 1$ nach Vorlesung separabel, da $p \neq q$ Primzahlen + und damit $\text{char}(\mathbb{F}_q) = q \nmid p$. Außerdem + ist mit der angegebenen Formel: + \[ + \Delta_f = (-1)^{\frac{p(p-1)}{2}} p^{p} (-1)^{p-1} + .\] Da $q$ ungerade ist $\text{char}(\mathbb{F}_q) \neq 2$. Nach 3(e) ist damit + das Bild von $G$ in $\mathfrak{S}_p$ g.d. in $\mathfrak{A}_p$ enthalten, wenn + \stepcounter{equation} + \begin{align*} + \Delta_f \in \left( \mathbb{F}_q^{\times } \right)^2 + \iff \left( \frac{\Delta_f}{q} \right) = 1 + \tageq \label{e:1} + .\end{align*} + Da $p$ ungerade ist $(-1)^{p} = -1$. Damit ist auch $\left( \frac{p}{q} \right)^{p} = + \left( \frac{p}{q} \right) $ $(*)$ Außerdem ist + deshalb $p-1$ gerade und damit $\left( \frac{-1}{q} \right)^{p-1} = 1$ $(**)$. Damit folgt + \begin{salign*} + \left( \frac{\Delta_f}{q} \right) + &\stackrel{\text{(a)}}{=} \left( \frac{(-1)^{\frac{p(p-1)}{2}}}{q} \right) + \left( \frac{p^{p}}{q} \right) \left( \frac{(-1)^{p-1}}{q} \right) \\ + &\stackrel{\text{(a)}}{=} \left( \frac{-1}{q} \right)^{\frac{p(p-1)}{2}} + \left( \frac{p}{q} \right)^{p} \underbrace{\left( \frac{-1}{q} \right)^{p-1}}_{= 1 \text{(**)}} \\ + &\stackrel{\text{(*), (a)}}{=} + (-1)^{ p \frac{q-1}{2} \frac{p-1}{2}} \left( \frac{p}{q} \right) \\ + &\stackrel{p \text{ ungerade}}{=} + \left( -1 \right)^{\frac{q-1}{2} \frac{p-1}{2}} \left( \frac{p}{q} \right) + .\end{salign*} + Mit (\ref{e:1}) zeigt das die gewünschte Äquivalenz. + \item Für $a \in \Z / p \Z$ gilt + $\pi(a)^{r} = a q^{r}$ für $r \in \N_0$. Es gilt + $\pi(a)^{k} = a q^{k} = a$. Jedes Element in $a \in \Z / p \Z$ + wird also von $\pi$ zyklisch vertauscht. $\pi$ erzeugt also disjunkte + Zykeln der Länge $k$. Zwei Elemente $a, b \in (\Z / p \Z)^{\times}$ liegen genau dann + im selben Zykel, wenn $b^{-1}a \in \langle q \rangle$. Also g.d.w + $b \sim_{\langle q \rangle} a$. Also besteht $\pi$ aus genau + $((\Z / p \Z)^{\times } : \langle q \rangle) \quad \stackrel{\text{Lagrange}}{=} \quad \frac{p-1}{k}$ + disjunkten Zykeln. Das Signum eines Zykels der Länge $k$ ist genau $(-1)^{k}$. Da + weiterhin $\text{sgn}$ Gruppenhomomorphismus folgt: + \[ + \text{sgn}(\pi) = ((-1)^{k-1})^{\frac{p-1}{k}} + = (-1)^{k-1 \frac{p-1}{k}} + .\] + \item Sei $\gamma \in \mathbb{F}_p^{\times } $ ein Erzeuger. + + Beh.: $\gamma^{n} \in (\mathbb{F}_p^{\times })^2 \iff 2 \mid n$. + \begin{itemize} + \item $2 \mid n \implies n = 2l$ für ein $ l \in \N \implies \gamma^{n} = \gamma^{2l} + = (\gamma^{l})^2$. + \item $\gamma^{n} \in (\mathbb{F}_p^{x})^2 \implies \exists l \in \N\colon (\gamma^{l})^2 + = \gamma^{n} \implies \gamma^{2l \text{ mod } p} = \gamma^{n \text{ mod } p} + \implies 2l \equiv n \text{ mod p} \implies 2 \mid n$. + \end{itemize} + Da $\gamma$ Erzeuger, ex. ein $n \in \N$, s.d. + $q = \gamma^{n}$. Es ist $\gamma^{nk} = q^{k} = 1 = \gamma^{p-1} \implies n = \frac{p-1}{k}$. + Damit folgt + \[ + 1 = \left( \frac{q}{p} \right) \iff q = \gamma^{2l} \text{ für ein } l \in \N + \iff 2 \mid \frac{p-1}{k} \tageq \label{e:3} + .\] + Es gilt außerdem $p-1 = (\left( \Z / p \Z \right)^{\times } : \langle q \rangle ) k + = \underbrace{\frac{p-1}{k}}_{\in \N} k$. Falls $\frac{p-1}{k}$ ungerade, + dann ist also $k$ gerade, da $p-1$ gerade und damit $k-1$ ungerade. + Damit folgt $2 \mid (k-1) \frac{p-1}{k} \iff 2 \mid \frac{p-1}{k}$. + + Das Bild von $G$ ist g.d. in $\mathfrak{A}_n$ enthalten, wenn $\text{sgn}(\pi) = 1$. Weiter + gilt mit (c) + \[ + \text{sgn}(\pi) = 1 \iff 1 = (-1)^{(k-1) \frac{p-1}{k}} + \iff 2 \mid (k-1) \frac{p-1}{k} \iff 2 \mid \frac{p-1}{k} + .\] + Das zeigt mit (\ref{e:3}) die behauptete Äquivalenz. + + Nun gilt also mit (b), dass + \[ + 1 = \left( \frac{q}{p} \right) \iff 1 = (-1)^{\frac{p-1}{2}\frac{q-1}{2}} \left( \frac{p}{q} \right) = 1 + .\] Damit folgt: + \begin{salign*} + \left( \frac{q}{p} \right) &= (-1)^{\frac{p-1}{2} \frac{q-1}{2}} \left( \frac{p}{q} \right) + \intertext{Mit $\left( \frac{p}{q} \right)^2 = 1$ folgt} + \left( \frac{p}{q} \right) \left( \frac{q}{p} \right) + &= (-1)^{\frac{p-1}{2}\frac{q-1}{2}} \\ + &= \left( (-1)^{\frac{p-1}{2}} \right)^{\frac{q-1}{2}} \\ + &\stackrel{\text{(a)}}{=} \begin{cases} + (-1)^{\frac{p-1}{2}} & q \equiv 3 \text{ (mod } 4) \\ + 1 & q \equiv 1 \text{ (mod } 4 ) + \end{cases} \\ + &\stackrel{\text{(a)}}{=} \begin{cases} + -1 & q, p \equiv 3 \text{ (mod } 4) \\ + 1 & \text{sonst} + \end{cases} + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\end{document}