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+\documentclass[uebung]{../../../lecture} + +\title{Algebra 1: Übungsblatt 2} +\author{Lukas Nullmeier, Christian Merten} + +\begin{document} + +\punkte + +\begin{aufgabe}[] + \begin{enumerate}[(a)] + \item Sei $d \in \Z$. Beh.: $\Z[\sqrt{d}]$ ist der kleinste Unterring von $\mathbb{C}$, der $\sqrt{d} $ enthält. + \begin{proof} + Durch Nachrechnen ist sofort offensichtlich, dass $\Z[\sqrt{d}]$ ein Unterring ist, der $\sqrt{d} $ enthält. Sei nun + $R \subseteq \mathbb{C}$ ein Unterring mit $\sqrt{d} \in R$. Dann ist + $\Z \ni 1 \in R$, da $R$ Unterring, d.h. durch Addition der $1$ folgt + $\Z \subseteq R$. Sei nun $x \in \Z[\sqrt{d}]$. Dann ex. $a, b \in \Z$ mit + $x = a + b\sqrt{d}$. Da $a, b, \sqrt{d} \in R$ folgt, da $R$ Unterring, dass auch + $x \in R$, also $\Z[\sqrt{d}] \subseteq R$. + \end{proof} + \item Sei $d = -5$. Beh.: $N$ ist multiplikativ. + \begin{proof} + Seien $x, y \in \Z[\sqrt{-5}]$. Dann ex. $a, b, e, f \in \Z$, s.d. + $x = a + b \sqrt{-5} $ und $y = e + f \sqrt{-5} $. Damit folgt direkt + \begin{salign*} + N(xy) &= N(ae - 5bf + (af + be)\sqrt{-5} ) \\ + &= (ae-5bf)^2 + 5(af + be)^2 \\ + &= a^2e^2 + 25 b^2f^2 + 5 a^2f^2 + 5 b^2e^2 \\ + &= (a^2 + 5b^2) (e^2 + 5f^2) \\ + &= N(x)N(y) + .\end{salign*} + \end{proof} + Beh.: Für $u \in \Z[\sqrt{-5}]$ gilt $u \in \Z[\sqrt{-5}]^{x} \iff N(u) =1$. + \begin{proof} + Sei $u \in \Z[\sqrt{-5}]$. ,,$\implies$'': Sei $u \in \Z[\sqrt{-5}]^{\times }$. Dann + ex. $v \in \Z[\sqrt{-5}]$, s.d. $uv = 1$. Damit folgt + wegen $N$ multiplikativ + \[ + 1 = N(1) = N(uv) = \underbrace{N(u)}_{\in \Z_{\ge 0}} \underbrace{N(v)}_{\in \Z_{\ge 0}} + .\] Es folgt also $N(u) = 1$. + + ,,$\impliedby$'': Sei $N(u) = 1$. Es ex. $a, b \in \Z$, s.d. $u = a + b \sqrt{-5} $. Wegen + \[ + 1 = N(u) = N(a + b\sqrt{-5}) = a^2 + 5 b^2 + \] folgt $b = 0$ und damit $a^2 = 1$, also $u = a$ mit $u^2 = 1$, also $u \in \Z[\sqrt{-5}]^{\times }$. + \end{proof} + Beh.: $\Z[\sqrt{-5}]^{\times } = \{\pm 1\} $. + \begin{proof} + Sei $u \in \Z[\sqrt{-5}]$ mit $u = a + b\sqrt{-5} $ für $a, b \in \Z$. Dann ist + \begin{align*} + N(u) = 1 &\iff a^2 + 5b^2 = 1 \\ + &\iff a^2 = 1 \land b = 0 \\ + &\iff a \in \{\pm 1\} \land b = 0 \\ + &\iff u \in \{\pm 1\} + .\end{align*} + Damit folgt die Behauptung mit der Vorüberlegung. + \end{proof} + \item Beh.: $2 \in \Z[\sqrt{-5}]$ ist irreduzibel. + \begin{proof} + Ang.: $\exists x, y \in \Z[\sqrt{-5}]$, s.d. $2 = xy$ mit + $x, y \not\in \Z[\sqrt{-5}]^{\times }$. Dann folgt + \[ + 4 = N(2) = N(xy) = N(x) N(y) + .\] Da $4 \neq 0 \implies N(x), N(y) \in \N$ und wegen $x, y \not\in \Z[\sqrt{-5}]^{\times }$, + ist $N(x), N(y) \neq 1$, also $N(x) = N(y) = 2$. Das heißt + es ex. $a, b \in \Z$, s.d $N(a + b\sqrt{-5}) = 2$. Dann + folgt $a^2 + 5b^2 = 2$, also $b = 0$ (sonst $a^2 + 5b^2 \ge 5 > 2$). Damit + ist $a^2 = 2 \implies \sqrt{2} \in \Z$ $\contr$. + \end{proof} + Beh.: $\Z[\sqrt{-5}]$ nicht faktoriell. + \begin{proof} + Es gilt + \[ + 2 \cdot 3 = 6 = (1+\sqrt{-5})(1-\sqrt{-5}) + .\] Aber $2, 3, 1+\sqrt{-5}, 1-\sqrt{-5} $ irreduzibel, also hat $6$ zwei + verschiedene Darstellungen als Produkt von irreduziblen Elementen. + Damit ist $\Z[\sqrt{-5}]$ nicht faktoriell. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $(*):$ Für $R$ Ring, ex. ein $n \in \N_{\ge 2}$, s.d. $x^{n} = x$ für alle $x \in R$. + \begin{enumerate}[(a)] + \item Beh.: Endliche nullteilerfreie Ringe $R$ sind Körper. + \begin{proof} + Sei $R$ nullteilerfreier, endlicher Ring. Zunächst ist $R \neq 0$, da + $R$ nullteilerfrei. Sei nun $x \in R \setminus \{0\} $. Dann betrachte + $f_x\colon R \to R, y \mapsto xy$. Es ist $f_x$ injektiv, denn + für $y_1, y_2 \in R$ folgt + \[ + f_x(y_1) = f_x(y_2) \implies xy_1 = xy_2 + \qquad + \stackrel{R \text{ nullteilerfrei}, x\neq 0}{\implies} + \qquad + x(y_1 - y_2) = 0 \implies y_1 = y_2 + .\] Da $R$ endlich ist $f_x$ auch surjektiv, inbes. ex. ein $y \in R$, s.d + $f_x(y) = 1 \implies xy = 1$, also $x \in R^{x}$. Insgesamt + also $R^{x} = R \setminus \{ 0 \}$. + \end{proof} + \item Beh.: $R$ nullteilerfrei mit $(*)$ $\implies$ $R$ endlicher Körper. + \begin{proof} + Sei $x \in R \setminus \{0\} $. Wegen $n \ge 2$ ist $n-2 \ge 0$. Es gilt + \[ + x^{n} = x \implies x^{n-1} x = x + .\] Da $x \neq 0$ und $R$ nullteilerfrei, folgt $x^{n-1} = 1$. Damit ist + \[ + x^{n-2} x = 1 \implies x \in R^{x} + .\] + \end{proof} + \item Beh.: Jedes Primideal in $R$ mit $(*)$ ist maximal. + \begin{proof} + Sei $I \subsetneqq R$ Primideal. Dann sei $\overline{x} \in R / I$. Dann ist + $\overline{x}^{n} = \overline{x^{n}} = \overline{x}$, wegen $(*)$. + Da $I$ Primideal, ist + $R / I$ nullteilerfrei mit Eigenschaft $(*)$. Mit (b) ist + $R / I$ also Körper, also $I$ Maximalideal. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Zunächst sind Polynome vom Grad 1 immer irreduzibel über $\mathbb{F}_2[X]$, da $\mathbb{F}_2$ + nullteilerfrei und damit auch $\mathbb{F}_2[X]$ nullteilerfrei. Damit folgt für ein + $f \in \mathbb{F}_2[X]$ mit $\text{deg}(f) = 1$ und $f = gh$ für $g, h \in \mathbb{F}_2$[X], dass + \[ + 1 = \text{deg}(f) = \text{deg}(gh) = \text{deg}(g) + \text{deg}(h) + .\] Da $\text{deg}(f) = 1$ ist $g, h \neq 0$, also $\text{deg}(g), \text{deg}(h) \ge 0$ und + damit $\text{deg}(g) = 0$ oder $\text{deg}(h) = 0$. Da $\mathbb{F}_2$ Körper ist damit + $g \in \mathbb{F}_2[X]^{\times }$ oder $h \in \mathbb{F}_2[X]^{\times }$, also $f$ irreduzibel. + + Da $\mathbb{F}_2$ Körper sind zudem alle Polynome vom Grad 0 Einheiten also nicht irreduzibel. + + Weiter ist $\# \mathbb{F}_2 = 2$, also existieren genau $2^{k}$ paarweise verschiedene Polynome + vom Grad $k$, für $k \ge 0$. Da $\mathbb{F}_2$ Körper, ist $\mathbb{F}_2$ HIR, also faktoriell + und nach Satz von Gauß ist auch $\mathbb{F}_2[X]$ faktoriell. Das heißt die irreduziblen + Polynome sind diejenigen die nicht als Produkt von irreduziblen Polynomen enstehen. + Damit sind also + \begin{salign*} + f_1 &\coloneqq X \\ + f_2 &\coloneqq X+1 + \intertext{ + alle Polynome vom Grad 1 und irreduzibel. + Daraus entstehen als Produkte die Polynome} + f_3 &\coloneqq X^2 = X \cdot X \\ + f_4 &\coloneqq X^2 + X = X (X+1) \\ + f_5 &\coloneqq X^2 + 1 = (X+1)^2 \\ + \intertext{Es bleibt als viertes Polynom vom Grad 2 nur} + f_6 &\coloneqq X^2 + X + 1 + \intertext{das damit irreduzibel sein muss. Aus den irreduziblen Polynome vom Grad $\le 2$ + ergeben sich} + f_7 &\coloneqq X^3 \\ + f_8 &\coloneqq X^3 + X^2 = X^2 (X+1) \\ + f_9 &\coloneqq X^3 + X^2 + X = X(X^2 + X +1) \\ + f_{10} &\coloneqq X^3 + X^2 + X + 1 = (X+1)^3 \\ + f_{11} &\coloneqq x^3 + X = X(X+1)^2 \\ + f_{12} &\coloneqq X^3 + 1 = (X^2 + X + 1)(X +1) \\ + \intertext{Damit bleiben als Polynome vom Grad 3 nur noch} + f_{13} &\coloneqq X^3 + X^2 + 1 \\ + f_{14} &\coloneqq X^3 + X + 1 + \end{salign*} + die damit irreduzibel sein müssen. + Damit sind alle irreduziblen Polynome vom Grad $\le 3$ in $\mathbb{F}_2[X]$ gegeben als + \[ + \{f_1, f_2, f_6, f_{13}, f_{14}\} + .\] + Es existieren genau $2^{4} = 16$ Polynome vom Grad $4$ in $\mathbb{F}_2[X]$. Die reduziblen + ergeben sich wieder als Produkte der irreduziblen Polynome vom Grad $\le 3$. Da sich + die Grade bei Produktbildung addieren, treten folgende Kombinationen auf: + \begin{salign*} + \text{(i)} \qquad 4 &= 1 + 1 + 1 +1 \\ + \text{(ii)} \qquad 4 &= 1 + 1 + 2 \\ + \text{(iii)} \qquad 4 &= 1 + 3 \\ + \text{(iv)} \qquad 4 &= 2 + 2 + .\end{salign*} + Die Anzahl der Kombinationen pro Fall ergibt sich durch Produktbildung der Möglichkeiten Polynome + der entsprechenden Grade auszuwählen. Damit ergeben sich nach Ana I für (i) $\binom{2 + 4 - 1}{4} = 5$ + Kombinationen, für (ii): $1 \cdot \binom{2 + 3 - 1}{2} = 3$, für (iii): $2 \cdot 2$ und + für (iv): $1 \cdot 1 = 1$ Kombinationen. In Summe also $5 + 3 + 4 + 1 = 13$ reduzible Polynome, + also $16 - 13 = 3$ irreduzible Polynome vom Grad $4$. +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item $f = \frac{(X-1)^{3}}{1} \in Q(\R[X]) \implies v_{X-1}(f) = 3$ und + $g = \frac{1}{(X-1)^2} \in Q(\R[X]) \implies v_{X-1}(g) = -2$. Folgt direkt aus + der Definition. + \item $X^2 + 1$ ist irreduzibel über $\R$, da $X^2 + 1$ keine Nullstellen in $\R$ hat. + \item $(2,X)$ ist kein Hauptideal in $\Z[X]$. + \begin{proof} + Ang. es ex. ein $f \in \Z[X]$ mit $(f) = (2,X)$, dann ex. $g \in \Z[X]$ mit + $2 = fg$, da $\Z[X]$ nullteilerfrei, folgt mit Gradformel, dass $\text{deg}(f) = 0$, also + $f \in \Z$. Außerdem ex. ein $h \in \Z[X]$ mit $X = f h$. Dann ist + $e(X) = 1 = e(f) e(g)$, also $f \in \Z^{\times }$. Damit ist + $(f) = (2,X) = (1) = \Z$. Insbesondere ist $1 \in (2, X)$, d.h. es ex. $h, g \in \Z[X]$ + mit + \[ + 1 = 2h + Xg + .\] Es ist offensichtlich $h, g \neq 0$ also $\text{deg}(2h + Xg) \ge 1 > 0 = \text{deg}(1)$. + $\contr$. + \end{proof} + \item $(X)$ in $\Z[X,Y]$. Es ist offensichtlich $(X) \subsetneqq (X,Y) \subsetneqq \Z[X, Y]$, + denn $Y \not\in (X)$ und $1 \not\in \Z[X, Y]$. Da $X$ irreduzibel über $\Z[X, Y]$ und + $\Z[X,Y]$ faktoriell ist $X$ auch Primelement also $(X)$ Primideal, aber kein Maximalideal. + \item $R = \Z[X,Y]$ und $a = X$, $b = Y$. Es ist $\text{ggT}(X,Y) = 1$, da + $X, Y$ prim aber $1 \not\in (X) + (Y)$, also $(X) + (Y) \neq (1)$. + \item $K = Q(\Z / 2 \Z[X])$ ist Körper mit Charakteristik $2$, denn + \[ + Q(\Z / 2 \Z[X]) \ni 1 = \frac{\overline{1}}{\overline{1}} + \frac{\overline{1}}{\overline{1}} + = \frac{\overline{1} + \overline{1}}{\overline{1}} = \frac{\overline{0}}{\overline{1}} + = 0 \in Q(\Z / 2 \Z[X]) + ,\] aber $Q(\Z / 2 \Z[X])$ unendlich, da $\Z / 2 \Z[X]$ Polynomring und hat damit unendlich viele + Elemente, also auch $Q( \Z / 2 \Z [X])$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item + \begin{figure}[h] + \begin{tikzpicture} + \begin{axis}[ytick={1, 2, -1, -2}, yticklabels={$\sqrt{2}$, $2\sqrt{2}$, $-\sqrt{2}$, + $-2\sqrt{2} $}, + grid=both, axis lines=middle, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + enlargelimits={abs=0.2}, + xlabel=$\text{Re}(z)$, + ylabel=$\text{Im}(z)$ + ] + \addplot[only marks, mark=x] + coordinates{ % plot 1 data set + (0, 0) + (0, 1) + (0, 2) + (0, -1) + (0, -2) + (1, 0) + (1, 1) + (1, 2) + (1, -0) + (1, -1) + (1, -2) + (2, 0) + (2, 1) + (2, 2) + (2, 0) + (2, -1) + (2, -2) + (-1, 0) + (-1, 1) + (-1, 2) + (-1, -0) + (-1, -1) + (-1, -2) + (-2, 0) + (-2, 1) + (-2, 2) + (-2, 0) + (-2, -1) + (-2, -2) + % more points... + }; + \end{axis} + \end{tikzpicture} + \centering + \caption{Ausschnitt von $\Z[\sqrt{-2}] \subseteq \mathbb{C}$ in der komplexen Ebene} + \end{figure} + Beh.: $\Z[\sqrt{-2}]$ euklidisch. Sei dazu + $\text{rd}\colon \R \to \Z$ die Rundungsfunktion. Bei zwei Möglichkeiten wähle die größere. + Dann ist $\forall x \in \R\colon |x - \text{rd}(x)| \le \frac{1}{2}$. + + Schritt 1: Sei zunächst + $z \in \mathbb{C}$ mit $z = c + id$ mit $c, d \in \R$. Dann wähle + $a = \text{rd}(c)$ und $b = \text{rd}(d)$. Dann ist + $a + b \sqrt{-2} \in \Z[\sqrt{-2}]$ und + \begin{salign*} + |z - (a + \sqrt{-2}b)| = |(c - a) + i \sqrt{2} (d -b)| + = \sqrt{(c-a)^2 + 2(d-b)^2} + = \sqrt{\frac{1}{4} + \frac{2}{4}} = \frac{\sqrt{3} }{2} < 1 + .\end{salign*} + + Schritt 2: $\Z[\sqrt{-2}]$ ist nullteilerfrei. + + Schritt 3: Seien nun $z, w \in \Z[\sqrt{-2}]$ mit $w \neq 0$. Dann ist $\frac{z}{w} \in C$ und es + ex. nach Schritt 1 + ein $q \in \Z[\sqrt{-2}]$ mit $\left| \frac{z}{w} - q \right|\le \frac{\sqrt{3} }{2}$. + Dann setze $r \coloneqq z - qw$. Dann gilt, da der komplexe Betrag multiplikativ ist: + \[ + N(r) = N(z - qw) = |z - qw|^2 = |\frac{z}{w} - q|^2 |w|^2 + \le \frac{3}{4} N(w) < N(w) + .\] Damit ist $\Z[\sqrt{-2}]$ euklidisch mit der Normfunktion $N$. + \item Beh.: $\Z[\sqrt{-2}]^{\times } = \{\pm 1\} $ + \begin{proof} + Es ist schnell nachgerechnet, dass $N$ multiplikativ ist. Wende dann das + exakt selbe Argument wie in 1(b) an. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}