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z8BIMxU40|`EdOhjb}e?n;eg&{|3>2rEg%2mZj8K@tu(0ssKfSp9w$L6PwCTH%80D_ e?-~e0c!eS&kHw|gkLS146_we6KqDJt_J06!U{`Vg literal 0 HcmV?d00001 diff --git a/ws2020/algebra/uebungen/algebra6.tex b/ws2020/algebra/uebungen/algebra6.tex new file mode 100644 index 0000000..f8aec87 --- /dev/null +++ b/ws2020/algebra/uebungen/algebra6.tex @@ -0,0 +1,317 @@ +\documentclass[uebung]{../../../lecture} + +\title{Algebra I: Übungsblatt 6} +\author{Lukas Nullmeier, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe} + Beh.: $K$ vollkommen $\iff$ $(f \in K[X]$ irreduzibel $\implies$ $f$ separabel). + \begin{proof} + \begin{itemize} + \item ,,$\implies$'': Sei $K$ vollkommen und $f \in K[X]$ irreduzibel + und $\alpha \in \overline{K}$ mit $\overline{K}$ ein algebraischer Abschluss + von $K$. Dann betrachte $L \coloneqq K(\alpha)$. $L$ ist separabel + über $K$, da $K$ vollkommen, insbesondere $\alpha$ separabel über $K$, also + $f$, Minimalpolynom von $\alpha$, separabel über $K$. + \item ,,$\impliedby$'': Sei $L / K$ algebraische Körpererweiterung und $\alpha \in L$. Dann + sei $f \in K[X]$ Minimalpolynom von $\alpha$ über $K$. Dann ist $f$ irreduzibel, + also separabel und damit $\alpha$ separabel und damit $L / K$ separabel. + Da $L$ beliebig, folgt $K$ vollkommen. + \end{itemize} + + Beh.: $\sigma$ surjektiv $\iff$ $K$ vollkommen. + \begin{proof} + \begin{itemize} + \item ,,$\implies$'': Sei $\sigma$ surjektiv und $f \in K[X]$ irreduzibel. Mit + der Vorüberlegung + genügt es zu zeigen, dass $f$ separabel ist. Da $\text{char}(K) = p > 0$ + sei $r \in \N_0$ maximal, s.d. $f(X) = g(X^{p^{r}})$ für ein + $g \in K[X]$. Dann hat nach VL jede Nullstelle von $f$ Vielfachheit + $p^{r}$. Ang.: $r > 0$. Dann ex. insbesondere + ein $f(X) = h(X^{p})$ für $h \in K[X]$. Dann ex. $a_i \in K$ + und da $\sigma$ surjektiv, $\alpha_i \in K$, s.d. $a_i = \alpha_i^{p}$ $\forall i$. Damit + folgt + \[ + f = \sum_{i=1}^{n} a_i X^{ip} = \sum_{i=1}^{n} \alpha_i^{p} (X^{i})^{p} + = \left(\sum_{i=1}^{n} \alpha_i X^{i} \right)^{p} + .\] Also ist $f$ reduzibel $\contr$. Damit folgt $r = 0$, also + hat jede Nullstelle von $f$ genau Vielfachheit $1$, insbesondere + $f$ separabel. + \item ,,$\impliedby$'': Sei $K$ vollkommen und $a \in K$. Dann betrachte + $f \coloneqq X^{p} - a$. Sei $\alpha \in \overline{K}$ mit + $\overline{K}$ ein algebraischer Abschluss von $K$ Nullstelle von $f$. Dann + folgt $\alpha ^{p} = a$. Damit folgt da $\text{char}(K) = p > 0$: + \[ + (X - \alpha)^{p} = X^{p} - \alpha ^{p} = X^{p} - a = f + .\] + Also ist $\alpha$ Nullstelle von $f$ mit $p$-facher Vielfachheit. Also + $f$ nicht separabel und mit der Vorüberlegung also $f$ reduzibel über $K$. Sei + $g \in K[X]$ ein irreduzibler Faktor von $f$ mit $\text{deg}(g) = r$. Dann + ist $\alpha$ Nullstelle von $g$ mit $r$-facher Vielfachheit und da $g$ irreduzibel, + ist $g$ separabel nach der Vorüberlegung. Also folgt $r = 1$ und $g = X - \alpha \in K[X]$. + Insbesondere folgt $\alpha \in K$ und wegen $\alpha ^{p} = a$ folgt + $\sigma$ surjektiv. + \end{itemize} + \end{proof} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Ist $K$ vollkommen, so auch $L$. + \begin{proof} + Sei $M / L$ eine algebraische Körpererweiterung. Da $M / L$ und $L / K$ algebraisch, + ist auch $M / K$ algebraisch und da $K$ vollkommen damit $M / K$ separabel. + Also insbesondere $M / L$ separabel. Da $M / L$ beliebig, folgt $L$ vollkommen. + \end{proof} + \item Beh.: $L / K$ separabel $\implies$ $K$ vollkommen. + \begin{proof} + Sei $M / K$ algebraische Körpererweiterung und sei $\alpha \in M$ beliebig. Betrachte + das Kompositum $E \coloneqq K(\alpha)L$. Es ist $E / L$ endliche algebraische Erweiterung, + da $K(\alpha) / K$ und $L / K$ endlich und algebraisch. Da $L$ vollkommen, ist + also $E / L$ separabel. Da außerdem $L / K$ separabel, + ist auch $E / K$ separabel. + Insbesondere ist also $\alpha \in K(\alpha) \subseteq E$ separabel + über $K$. Da $\alpha$ beliebig, folgt $M / K$ separabel. + \end{proof} + \item Beh.: $L / K$ separabel. + \begin{proof} + Setze $n \coloneqq [ L \colon K ] < \infty$. Dann sei $\alpha \in L$ und + $f \in K[X]$ Minimalpolynom zu $\alpha$. + Da $\text{char}(K) = p > 0$ sei + $r \in \N_0$ maximal, s.d. $f(X) = g(X^{p^{r}})$ + für ein $g \in K[X]$. Da $f$ irreduzibel ist $g$ nach VL separabel und + $f(\alpha) = 0 \implies g(\alpha ^{p^{r}}) = 0$. Also $\alpha ^{p^{r}}$ Nullstelle + eines separablen Polynoms über $K$, also separabel über $K$. Es folgt + also $\alpha ^{p^{r}} \in K_s$. Ang.: $r > n$. Da + \[ + n = [L : K] = [L : K(\alpha)] [K(\alpha) : K] = [L : K(\alpha)] \text{deg}(f) + ,\] folgt $\text{deg}(f) \le n$. Also insbesondere $r > \text{deg}(f) $. Dann + \[ + \text{deg}(f) = \underbrace{\text{deg}(g)}_{\ge 1} p^{r} > p^{r} > p^{n} > \text{deg}(f) + \quad \contr + .\] Also folgt $r \le n$. Damit folgt da $K_s$ Körper + \[ + \alpha ^{p^{n}} = \alpha ^{p^{n-r} p^{r}} = (\alpha ^{p^{r}})^{p^{n-r}} \in K_s + .\] + + Sei nun $a \in L$. + Da $L$ vollkommen, ist nach Aufgabe 1 $\sigma$ bijektiv, damit + ist auch $\sigma ^{n}\colon L \to L, x \mapsto \sigma(x)^{n} = x^{p^{n}}$ bijektiv + als Komposition bijektiver Abbildungen. Also ex. ein $\alpha \in L$, s.d. + $\sigma ^{n}(\alpha) = a$. Also folgt mit der Vorüberlegung $a = \alpha ^{p^{n}} \in K_s$, + also $a$ separabel über $K$ und damit $L / K$ separabel. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Jede quadratische Erweiterung $L$ ist von der Form $L = K(\sqrt{a})$ für + ein $a \in K^{\times } \setminus (K^{\times })^2$. + \begin{proof} + Sei $L / K$ quadratische Erweiterung. Dann ist $[L : K] = 2$ insbesondere + algebraisch und endlich. Also $L = K(\alpha_1, \ldots, \alpha_n)$ für + $\alpha_1, \ldots, \alpha_n \in L$. Da der Grad von echten Erweiterungen echt größer + als $1$ ist, folgt mit Gradformel $L = K(\alpha)$ für ein $\alpha \in L$ mit + Minimalpolynom + \[ + f = X^2 + \tilde{b}X + c + \] für $b, c \in K$. Da $\text{char}(K) \neq 2 \implies \exists b \in K$, s.d. + $2b = \tilde{b}$. Damit ist + \[ + f = X^2 + 2bX + c + .\] Da $f(\alpha) = 0$ folgt + \begin{salign*} + \alpha^2 + 2b \alpha + c = \alpha^2 + 2b\alpha + b^2 - b^2 + c = (\alpha + b)^2 + c - b^2 = 0 + .\end{salign*} + Also insbesondere + \[ + (\alpha + b)^2 = b^2 - c + \] und damit $\alpha \in K(\sqrt{b^2 -c})$. Also + $K(\alpha) \subseteq K(\sqrt{b^2 - c})$. Außerdem gilt für + $g \coloneqq X^2 - b^2 +c$: $g(\sqrt{b^2 - c}) = 0$, also + $[ K(\sqrt{b^2 - c}) : K] \le 2$. Also folgt mit Gradformel + \[ + 2 \ge [K(\sqrt{b^2 -c} : K] = [K(\sqrt{b^2 - c} ) : K(\alpha) ] + [K(\alpha) : K] = 2 [K(\sqrt{b^2 -c}) : K(\alpha) ] + .\] Damit folgt $[K(\sqrt{b^2 - c}) : K(\alpha)] = 1$ also $K(\alpha) = K(\sqrt{b^2 - c})$. + + Es ist $b^2 - c \neq 0$, denn sonst $\alpha = -b \in K$, also $L = K$. + Ang.: $b^2 - c \in (K^{\times })^2$, dann ist $\sqrt{b^2 -c} \in K^{\times }$, also + insbesondere $K(\alpha) = K$ also $[ L : K] = 1 \neq 2$ $\contr$. + + Also folgt $L = K(\sqrt{b^2 - c})$ und $b^2 - c \in K^{\times} \setminus (K^{\times })^2$. + \end{proof} + \item Beh.: Zwei Erweiterungen $K(\sqrt{a})$ und $K(\sqrt{b})$ mit $a, b \in K^{\times }$ sind + genau dann $K$-isomorph, wenn $\frac{a}{b} \in (K^{\times})^2$. + \begin{proof} + \begin{itemize} + \item ,,$\implies$'': Sei $\varphi\colon K(\sqrt{a}) \to K(\sqrt{b})$ + $K$-Isomorphismus. Dann $\exists x \in K(\sqrt{a})$, s.d. + $\varphi(x) = \sqrt{b} $. Da $\varphi$ $K$-Iso + folgt außerdem $\varphi(b) = b$. Also folgt damit + \[ + \varphi(x^2) = \varphi(x)^2 = b = \varphi(b) + .\] Da $\varphi$ injektiv, folgt $x^2 = b$. Also insbesondere + $\sqrt{b} \in K(\sqrt{a})$. + + Falls $K(\sqrt{a}) = K$, dann ist $\sqrt{a}, \sqrt{b} \in K^{\times }$, da + $a, b \in K^{\times }$, + insbesondere $\frac{a}{b} \in (K^{\times})^2$. + + Sei nun $K(\sqrt{a}) \neq K$. Da $\sqrt{b} \in K(\sqrt{a})$ + ex. $u, v \in K$, s.d. + $\sqrt{b} = u + v \sqrt{a} $. + + Ang.: $v = 0$, dann ist $\sqrt{b} \in K$, also $K(\sqrt{b}) = K$ + und da $\varphi$ $K$-Isomorphismus auch $K(\sqrt{a}) = K$ $\contr$. + + Ang.: $u \neq 0$: Dann gilt wegen $\text{char}(K) \neq 2$: + \[ + b = (u + v \sqrt{a})^2 = u^2 + 2uv\sqrt{a} + a + \implies \sqrt{a} = \frac{b - u^2 - a}{2uv} \in K + .\] Also $\sqrt{a} \in K$, also insbesondere $K(\sqrt{a}) = K$ $\contr$. + + Also folgt $\sqrt{b} = v \sqrt{a}$, also + $\sqrt{\frac{a}{b}} = \frac{1}{v} \in K$, also da $a, b \in K^{\times }$ + auch $\frac{a}{b} \in (K^{\times })^2$. + \item ,,$\impliedby$'': Es sei $\sqrt{\frac{a}{b}} \in K$. Dann ist + \[ + K(\sqrt{a}) = K\left(\sqrt{\frac{a}{b} b}\right) + = K\left(\sqrt{\frac{a}{b}} \sqrt{b}\right) + \stackrel{\sim }{=} K(\sqrt{b}) + .\] + \end{itemize} + \end{proof} + \item Beh.: Für $p$ ungerade Primzahl und $a \in \mathbb{F}_p^{\times }$ gilt + \[ + a^{\frac{p-1}{2}} = \begin{cases} + 1 & a \in (\mathbb{F}_p^{\times })^2 \\ + -1 & a \not\in (\mathbb{F}_p^{\times })^2 + \end{cases} + .\] + \begin{proof} + Es ist zunächst $\left( a^{\frac{p-1}{2}} \right)^2 = a^{p-1} = 1$, da + $\text{ord}(\mathbb{F}_p^{\times}) = p-1$ und kl. Satz v. Fermat. Damit + folgt + \[ + \left( a^{\frac{p-1}{2}} \right)^2 -1 = 0 + \implies \left( a^{\frac{p-1}{2}} - 1 \right) \left( a^{\frac{p-1}{2}} + 1 \right) = 0 + .\] Da $\mathbb{F}_p$ nullteilerfrei, folgt $a^{\frac{p-1}{2}} = 1$ oder + $a^{\frac{p-1}{2}} = -1$. Es genügt also zu zeigen, dass + $a^{\frac{p-1}{2}} = 1 \iff a \in (\mathbb{F}_p^{\times })^2$. + + \begin{itemize} + \item ,,$\implies$'': Sei $a \in (\mathbb{F}_p^{\times })^{2}$. Dann + ex. ein $x \in \mathbb{F}_p^{\times }$ mit $x^2 = a$. Damit folgt wieder + mit Satz v. Fermat + \[ + a^{\frac{p-1}{2}} = (x^2)^{\frac{p-1}{2}} = x^{p-1} = 1 + .\] + \item ,,$\impliedby$'': Sei $a^{\frac{p-1}{2}} = 1$. Es ist $\mathbb{F}_p^{\times }$ + zyklisch, also ex. ein $x \in \mathbb{F}_p^{\times }$ und ein $r \in \N$, s.d. + $x^{r} = a$. Dann folgt + \[ + 1 = a^{\frac{p-1}{2}} = (x^{r})^{\frac{p-1}{2}} = x^{r \frac{p-1}{2}} + .\] Da $\text{ord}(x) = p-1$ folgt $(p-1) \mid r \frac{p-1}{2}$. Also + folgt $\frac{r}{2} \in \N$ also $r$ gerade. Damit $\exists k \in \N$, s.d. $r = 2k$. + Damit folgt + \[ + a = x^{r} = x^{2k} = (\underbrace{x^{k}}_{\in \mathbb{F}_p^{\times }})^2 + .\] Also $a \in (\mathbb{F}_p^{\times })^2$. + \end{itemize} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $f \in \mathbb{F}_p[X]$ irreduzibel: $f \mid (X^{p^{n}} - X) \iff \text{deg}(f) \mid n$. + \begin{proof} + Sei $f \in K[X]$ irreduzibel. + \begin{itemize} + \item ,,$\implies$'': Es sei $f \mid (X^{p^{n}} - X)$. Es zerfällt + $X^{p^{n}} - X$ über $\mathbb{F}_{p^{n}}$ in Linearfaktoren und da + $f \mid (X^{p^{n}} - X)$ zerfällt auch $f$ über $\mathbb{F}_p^{n}$ in Linearfaktoren. + + Sei $\alpha \in \mathbb{F}_{p^{n}}$ eine Nullstelle von $f$. Dann ist $\alpha$ auch + Nullstelle von $X^{p^{n}} - X$ und da $\mathbb{F}_{p^{n}}$ gerade + von diesen Nullstellen erzeugt, folgt + $\mathbb{F}_p(\alpha) \subseteq \mathbb{F}_{p^{n}}$. Außerdem + ist $f$ irreduzibel, also gerade das Minimalpolynom von $\alpha$ über $\mathbb{F}_p$. + Damit folgt mit Gradformel + \[ + n = [ \mathbb{F}_{p^{n}} : \mathbb{F}_{p} ] + = [ \mathbb{F}_{p^{n}} : \mathbb{F}_p(\alpha) ] + [ \mathbb{F}_p(\alpha) : \mathbb{F}_p ] + = [ \mathbb{F}_{p^{n}} : \mathbb{F}_p(\alpha) ] \text{deg}(f) + .\] Damit folgt $\text{deg}(f) \mid n$. + \item ,,$\impliedby$'': Es sei $\text{deg}(f) \mid n$. Dann betrachte + $K \coloneqq \mathbb{F}_p(X) / (f)$. Da $\mathbb{F}_p$ Körper, ist $\mathbb{F}_p[X]$ HIR und + da $f$ irreduzibel ist also $K$ Körper. Falls + $f = X$, dann ist $X \mid (X^{p^{n}} - X)$. Sei also $f \neq X$. + Es ist $\text{dim}_{\mathbb{F}_p}K = \text{deg}(f) \eqqcolon d$ + nach Zettel 3, da $f \neq 0$, da + $f$ irreduzibel. Da $\# \mathbb{F}_p = p$ folgt $\# K = p^{d}$. Da + $f \neq X$ folgt $f \nmid X$, denn $f$ irreduzibel und damit $d \ge 1$. Also + ist $X \in K^{\times}$ und damit mit Satz von Fermat: + $X^{p^{d}} = X$ in $K$. Damit folgt + $X^{p^{d}} - X \in (f)$. + + Da nun $d \mid n$ ex. ein $q \in \N$, s.d. $n = qd$. Damit folgt + $\mathbb{F}_{p^{d}} \subseteq \mathbb{F}_{p^{dq}} = \mathbb{F}_{p^{n}}$, also sind + Nullstellen von $X^{p^{d}} - X$ auch Nullstellen von $X^{p^{n}} - X$ (da + $\mathbb{F}_{p^{d}}$ bzw. $\mathbb{F}_{p^{n}}$ Zerfällungskörper über diese Polynome), also + $(X^{p^{d}} - X) \mid (X^{p^{n}} - X)$. + + Insgesamt + folgt also $f \mid (X^{p^{n}} - X)$. + \end{itemize} + \end{proof} + \item Beh.: + \[ + X^{p^{n}} - X = \prod_{f} f(X) + \] wobei $f$ die irreduziblen normierten Polynome in $\mathbb{F}_p[X]$ mit $\text{deg}(f) \mid n$ + durchlaufen. + \begin{proof} + Die Teiler von $X^{p^{n}} - X$ sind wegen (a) genau die irreduziblen + Polynome $f \in K[X]$ mit $\text{deg}(f) \mid n$. Da $X^{p^{n}} - X$ normiert können diese + normiert gewählt werden. Da außerdem $X^{p^{n}} - X$ separabel über $\mathbb{F}_p$, sind + die normierten Teiler von $X^{p^{n}} - X$ paarweise verschieden. Das zeigt die Behauptung. + \end{proof} + \item Beh.: $\sum_{d|n} d a_d(p) = p^{n}$ + \begin{proof} + Es ist für $d \in \N$ und $f$ durchlaufe die normierten irreduziblen Polynome gilt + \[ + \sum_{\text{deg}(f)= d} \text{deg}(f) = d a_d(p) + .\] Damit folgt wenn $f \in \mathbb{F}_p[X]$ wieder die normierten irreduziblen Polynome + mit $\text{deg}(f) \mid n$ durchläuft mit Umordnung der vorletzten Summe und + der Vorüberlegung: + \[ + p^{n} = \text{deg}(X^{p^{n}} - X) \stackrel{\text{(b)}}{=} \sum_{f} \text{deg}(f) + = \sum_{d|n} d a_d(p) + .\] + \end{proof} + \item Beh.: $a_6(2) = 9$. + \begin{proof} + Es genügt die positiven Teiler zu betrachten, da $\text{deg}(f) \ge 1$ für + $f \in \mathbb{F}_p[X]$ irreduzibel. Damit folgt mit Anwendung von (c): + + \begin{enumerate}[(i)] + \item $1$ hat Teiler $1$, also $2^{1} = 1 \cdot a_1(2) \implies a_1(2) = 2$. + \item $2$ hat Teiler $1$ und $2$, also $2^{2} = a_1(2) + 2a_2(2) \implies a_2(2) = 1$. + \item $3$ hat Teiler $1$ und $3$, also $2^{3} = a_1(2) + 3 a_3(2) \implies a_3(2) = 2$. + \end{enumerate} + $6$ hat Teiler $1, 2, 3$ und $6$. Damit folgt + \[ + 2^{6} = a_1(2) + 2a_2(2) + 3a_3(2) + 6a_6(2) \implies a_6(2) = 9 + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}