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Multiplikation kommutativ ist. + und dieser Ring hat ein Einselement und heißt kommutativ, + weil hier Multiplikation kommutativ ist. \end{proof} + \clearpage \begin{proof}[von \Cref{claim:main:ueb:6:ex:1b}, Ansatz II] Ein scharfes Auge erkennt, dass wir Teilmengen von $X$ mit binären Tupeln identifizieren kann. - Wir wollen $R$ mit einer bekannten algebraischen Struktur in Verbindung - setzen, also betrachten wir konkret die Abbildungen + Vielmehr wollen wir diese Menge von binären Tupeln mit einer bekannten algebraischen Struktur in Verbindung setzen, + also betrachten wir konkret die Abbildungen \begin{mathe}[mc]{rclcl} \Phi &: &\prod_{x\in X}\intgr/2\intgr &\to &\Pot(X)\\ &: &\alpha &\mapsto &\supp(\alpha):=\{x\in X\mid \alpha_{x}=1\}\\ + \\ \Psi &: &\Pot(X) &\to &\prod_{x\in X}\intgr/2\intgr\\ &: &A &\mapsto &(\einser_{A}(x))_{x\in X}\\ \end{mathe} - um Elemente aus dem einen Raum auf Elemente aus dem anderen zu übertragen.\\ + um Elemente aus dem einen Raum auf Elemente aus dem anderen zu übertragen. Nun ist $\intgr/2\intgr$ bekanntermaßen ein kommutativer Ring (eigentlich ein Körper). Darum ist das Produkt $\prod_{x\in X}\intgr/2\intgr$, versehen mit punktweise Addition und punktweise Multiplikation, ebenfalls ein kommutativer Ring. + Darum reicht es aus \textbf{zu zeigen}, dass $\Phi$ eine Bijektion ist, die die Operationen erhält (auch \emph{Isomorphismus} genannt). @@ -4046,17 +4049,17 @@ für alle $A,B\in R$. \item[\uwave{{\bfseries Bijektion:}}] Wir beobachten, dass - \begin{mathe}[mc]{rclclclcl} + \begin{mathe}[mc]{rclcl} \Phi(\Psi(A)) - &= &\{x\in X\mid \Psi(A)_{x}=1\} - &= &\{x\in X\mid \einser_{A}(x)=1\} + &= &\{x\in X\mid \Psi(A)_{x}=1\}\\ + &= &\{x\in X\mid \einser_{A}(x)=1\}\\ &= &\{x\in X\mid x\in A\} &= &A\\ \end{mathe} für alle $A\in\Pot(X)$ und - \begin{mathe}[mc]{rcl} + \begin{mathe}[mc]{rclcl} \Psi(\Phi(\alpha)) &= &(\einser_{\Phi(\alpha)}(x))_{x\in X}\\ &= &(\einser_{\{x'\in X\mid \alpha_{x'}=1\}}(x))_{x\in X}\\ @@ -4071,9 +4074,9 @@ für alle $A,B\in R$. 1 &: &\alpha_{x}=1\\ 0 &: &\alpha_{x}=0\\ \end{cases} - \right)_{x\in X}\\ - &= &(\alpha_{x})_{x\in X}\\ - &= &\alpha\\ + \right)_{x\in X} + &= &(\alpha_{x})_{x\in X} + = \alpha\\ \end{mathe} für alle $\alpha\in\prod_{x\in X}\intgr/2\intgr$. @@ -4085,17 +4088,17 @@ für alle $A,B\in R$. \textbf{Zu zeigen:} $\Phi(\alpha+\beta)=\Phi(\alpha)+\Phi(\beta)$ und - $\Phi(\alpha\cdot\beta)=\Phi(\alpha)\cdot\Phi(\beta)$.\\ + $\Phi(\alpha\cdot\beta)=\Phi(\alpha)\cdot\Phi(\beta)$. Es gilt \begin{mathe}[mc]{rcl} \Phi(\alpha+\beta) &= &\{x\in X\mid (\alpha+\beta)_{x}=1\}\\ - &= &\{x\in X\mid \alpha_{x}+\beta_{x}=1\} - \quad\text{da Operationen auf $\prod_{x\in X}\intgr/2\intgr$ punktweise definiert sind}\\ + &= &\{x\in X\mid \alpha_{x}+\beta_{x}=1\},\\ + &&\quad\text{da Operationen auf $\prod_{x\in X}\intgr/2\intgr$ punktweise definiert sind}\\ &= &\{x\in X\mid \alpha_{x}=1\,\text{od.}\,\beta_{x}=1,\,\text{aber nicht beides}\}\\ - &= &(\{x\in X\mid \alpha_{x}=1\}\cup\{x\in X\mid \beta_{x}=1\}) - \ohne(\{x\in X\mid \alpha_{x}=1\}\cap\{x\in X\mid \beta_{x}=1\})\\ + &= &(\{x\in X\mid \alpha_{x}=1\}\cup\{x\in X\mid \beta_{x}=1\})\\ + &&\,\ohne\,(\{x\in X\mid \alpha_{x}=1\}\cap\{x\in X\mid \beta_{x}=1\})\\ &= &(\Phi(\alpha)\cup\Phi(\beta)) \ohne(\Phi(\alpha)\cap\Phi(\beta)) \quad\text{per Konstruktion von $\Phi$}\\ @@ -4108,8 +4111,8 @@ für alle $A,B\in R$. \begin{mathe}[mc]{rcl} \Phi(\alpha\cdot\beta) &= &\{x\in X\mid (\alpha\cdot\beta)_{x}=1\}\\ - &= &\{x\in X\mid \alpha_{x}\cdot\beta_{x}=1\} - \quad\text{da Operationen auf $\prod_{x\in X}\intgr/2\intgr$ punktweise definiert sind}\\ + &= &\{x\in X\mid \alpha_{x}\cdot\beta_{x}=1\},\\ + &&\quad\text{da Operationen auf $\prod_{x\in X}\intgr/2\intgr$ punktweise definiert sind}\\ &= &\{x\in X\mid \alpha_{x}=1\,\text{und.}\,\beta_{x}=1\}\\ &= &\{x\in X\mid \alpha_{x}=1\}\cap\{x\in X\mid \beta_{x}=1\}\\ &= &\Phi(\alpha)\cap\Phi(\beta) @@ -4122,18 +4125,21 @@ für alle $A,B\in R$. \end{kompaktitem} Zusammegefasst haben wir gezeigt, - dass $(R,+,\cdot)$ zu dem kommutativen Ring, $(\prod_{x\in X}\intgr/2\intgr,+,\cdot)$ isomorph ist + dass $(\Pot(X),+,\cdot)$ zu dem kommutativen Ring, $(\prod_{x\in X}\intgr/2\intgr,+,\cdot)$ isomorph ist (und zwar mittels $\Phi$), und damit dass $(R,+,\cdot)$ selbst ein kommutativer Ring ist. - Man sieht auch, dass - das Nullelement durch $\Phi((0)_{x\in X})=\{x\in X\mid 0=1\}=\leer$ - und dass - das Einselement durch $\Phi((1)_{x\in X})=\{x\in X\mid 1=1\}=X$ - gegeben sind. \end{proof} + + \textbf{Bemerkung.} + Aus diesem Beweis geht hervor, dass + das Nullelement durch $\Phi((0)_{x\in X})=\leer$ + und + das Einselement durch $\Phi((1)_{x\in X})=X$ + gegeben sind. \end{enumerate} %% AUFGABE 6-2 +\clearpage \let\altsectionname\sectionname \def\sectionname{Aufgabe} \section[Aufgabe 2]{} @@ -4232,16 +4238,16 @@ Wir identifizieren $\kmplx$ mit $\reell^{2}$ mittel der Abbildungen \begin{mathe}[mc]{ccccccccccl} \cos(\alpha_{1}) - &= &\frac{r_{1}\cos(\alpha_{1})}{r_{1}} - &= &\frac{x}{r_{1}} - &= &\frac{x}{r_{2}} - &= &\frac{r_{2}\cos(\alpha_{2})}{r_{2}} + &= &\dfrac{r_{1}\cos(\alpha_{1})}{r_{1}} + &= &\dfrac{x}{r_{1}} + &= &\dfrac{x}{r_{2}} + &= &\dfrac{r_{2}\cos(\alpha_{2})}{r_{2}} &= &\cos(\alpha_{2})\\ \sin(\alpha_{1}) - &= &\frac{r_{1}\sin(\alpha_{1})}{r_{1}} - &= &\frac{y}{r_{1}} - &= &\frac{y}{r_{2}} - &= &\frac{r_{2}\sin(\alpha_{2})}{r_{2}} + &= &\dfrac{r_{1}\sin(\alpha_{1})}{r_{1}} + &= &\dfrac{y}{r_{1}} + &= &\dfrac{y}{r_{2}} + &= &\dfrac{r_{2}\sin(\alpha_{2})}{r_{2}} &= &\sin(\alpha_{2})\\ \end{mathe} @@ -4281,13 +4287,13 @@ Wir identifizieren $\kmplx$ mit $\reell^{2}$ mittel der Abbildungen \begin{proof} Multiplikation in $\kmplx$ liefert - \begin{mathe}[mc]{rcl} + \begin{longmathe}[mc]{RCL} z_{1}z_{2} &= &\begin{svector}\ReTeil(z_{1})\ReTeil(z_{2})-\ImTeil(z_{1})\ImTeil(z_{2})\\\ReTeil(z_{1})\ImTeil(z_{2})+\ReTeil(z_{1})\ImTeil(z_{2})\\\end{svector}\\ &= &\begin{svector}r_{1}\cos(\alpha_{1})\cdot r_{2}\cos(\alpha_{2})-r_{1}\sin(\alpha_{1})\cdot r_{2}\sin(\alpha_{2})\\r_{1}\cos(\alpha_{1})\cdot r_{2}\sin(\alpha_{2})+r_{1}\cos(\alpha_{1})\cdot r_{2}\sin(\alpha_{2})\\\end{svector}\\ &= &r_{1}r_{2}\begin{svector}\cos(\alpha_{1})\cos(\alpha_{2})-\sin(\alpha_{1})\sin(\alpha_{2})\\\cos(\alpha_{1})\sin(\alpha_{2})+\cos(\alpha_{1})\sin(\alpha_{2})\\\end{svector}\\ &= &r_{1}r_{2}\begin{svector}\cos(\alpha_{1}+\alpha_{2})\\\sin(\alpha_{1}+\alpha_{2})\\\end{svector}.\\ - \end{mathe} + \end{longmathe} Die letzte Vereinfachung folgt aus der trigonometrischen Additionsregel. \end{proof} @@ -4322,7 +4328,7 @@ Wir identifizieren $\kmplx$ mit $\reell^{2}$ mittel der Abbildungen &\textoverset{IV}{=} &r^{n-1}\cdot\begin{svector}\cos((n-1)\alpha)\\\sin((n-1)\alpha)\\\end{svector} \cdot r\cdot\begin{svector}\cos(\alpha)\\\sin(\alpha)\\\end{svector}\\ - &\textoverset{2b}{=} + &\textoverset{(2b)}{=} &r^{n-1}r\cdot\begin{svector}\cos((n-1)\alpha+\alpha)\\\sin((n-1)\alpha+\alpha)\\\end{svector}\\ &= &r^{n}\cdot\begin{svector}\cos(n\alpha)\\\sin(n\alpha)\\\end{svector}.\\ \end{mathe} @@ -4333,10 +4339,31 @@ Wir identifizieren $\kmplx$ mit $\reell^{2}$ mittel der Abbildungen Also gilt die Gleichung für alle $n\in\ntrlzero$. \end{proof} - \textbf{Bemerkung.} Wir können eigentlich zeigen, dass dies für all $n\in\intgr$ gilt. - Für $n=0$, beachte, dass $z^{0}=1=1+\imageinh 0$, $r^{0}=1$, $\cos(0)=1$ und $\sin(0)=0$. - Für negative Zahlen reicht es aus, $z^{-1}=r^{-1}\cdot\begin{svector}\cos(-\alpha)\\\sin(-\alpha)\\\end{svector}$ zu zeigen, - und dann $z^{n}=(z^{-1})^{|n|}$ für $n<0$ zu verwenden. + \begin{rem*} + Wir können eigentlich zeigen, + dass dies für alle $n\in\intgr$ gilt. + Für $n=0$, gilt + $% + z^{0} + =1+\imageinh 0 + =\begin{svector}1\\0\\\end{svector} + =r^{0}\cdot\begin{svector}\cos(0\alpha)\\\sin(0\alpha)\\\end{svector}% + $. + Für $n=-1$ liefert uns die Rechenregel für Multiplikation innerhalb $\kmplx$, + dass + $r^{-1}\cdot\begin{svector}\cos(-\alpha)\\\sin(-\alpha)\\\end{svector}$ + eine hinreichende Konstruktion für ein Inverses von $z$ ist, + und darum ist dies wegen Eindeutigkeit des Inverses gleich $z^{-1}$. + Für $n<0$ allgemein wenden wir schließlich + $% + z^{n} + =(z^{-1})^{|n|} + =(r^{-1}\cdot\begin{svector}\cos(-\alpha)\\\sin(-\alpha)\\\end{svector})^{|n|} + =(r^{-1})^{|n|}\cdot\begin{svector}\cos(|n|\cdot-\alpha)\\\sin(|n|\cdot-\alpha)\\\end{svector} + =r^{n}\cdot\begin{svector}\cos(n\alpha)\\\sin(n\alpha)\\\end{svector}% + $ + an. + \end{rem*} \end{enumerate} %% AUFGABE 6-3 @@ -4356,13 +4383,78 @@ definiert vermöge \end{mathe} für alle $a,b,a',b'\in K$. -Wir beobachten zuerst folgendes Ergebnis. -\begin{claim} - \makelabel{claim:1:ueb:6:ex:3} +Wir werden folgendes klassifizierendes Ergebnis verwenden, +um die Aufgaben zu behandeln +(und dieses Resultat dann anschließend beweisen). + +\begin{satz} + \makelabel{satz:1:ueb:6:ex:3} + $(F,+,\cdot)$ ist genau dann ein Körper, + wenn $a^{2}+b^{2}\neq 0$ innerhalb $K$ + für alle $(a,b)\in F\ohne\{(0,0)\}$. +\end{satz} + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 6-3a + \item + \begin{schattierteboxdunn} + \begin{claim*} + Sei $K=\mathbf{F}_{2}=\intgr/2\intgr$. + Dann ist $(F,+,\cdot)$ kein Körper. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + Da $(a,b):=(1,1)\in F\ohne\{(0,0)\}$ + und + $a^{2}+b^{2}=1+1=0$ + innerhalb $K=\intgr/2\intgr$, + ist \Cref{satz:1:ueb:6:ex:3} zufolge $F$ kein Körper. + Es scheitert genau das Axiom der Existenz multiplikativer Inverser. + (Nichtsdestotrotz bildet $F$ einen kommutativen Ring mit Einselement.) + \end{proof} + + %% AUFGABE 6-3b + \item + \begin{schattierteboxdunn} + \begin{claim*} + Sei $K=\mathbf{F}_{3}=\intgr/3\intgr$. + Dann ist $(F,+,\cdot)$ ein Körper. + \end{claim*} + \end{schattierteboxdunn} + + \begin{proof} + Laut \Cref{satz:1:ueb:6:ex:3} reicht es aus + für alle $(a,b)\in F\ohne\{(0,0)\}$ + \textbf{zu zeigen}, dass $a^{2}+b^{2}\neq 0$ innerhalb $K=\intgr/3\intgr$. + Sei also $(a,b)\in F\ohne\{(0,0)\}$ beliebig. + Da $a\neq 0$ oder $b\neq 0$ gibt es folgende Fälle: + + \begin{kompaktenum}{\bfseries {Fall} 1.}[\rtab][\rtab] + %% FALL 1 + \item $a=0$, $b\neq 0$. Dann $b=\pm 1\mod 3$. + Also $a^{2}+b^{2}=0+1=1\nequiv 0\mod 3$. + %% FALL 2 + \item $a\neq 0$, $b=0$. Dann $a=\pm 1\mod 3$. + Also $a^{2}+b^{2}=1+0=1\nequiv 0\mod 3$. + %% FALL 3 + \item $a\neq 0$, $b\neq 0$. Dann $a,b=\pm 1\mod 3$. + Also $a^{2}+b^{2}=1+1=2\nequiv 0\mod 3$. + \end{kompaktenum} + + Also gilt in jedem Falle $a^{2}+b^{2}\neq 0$. + Darum bildet $F$ einen Körper. + \end{proof} +\end{enumerate} + +Um \Cref{satz:1:ueb:6:ex:3} zu beweisen, brauchen wir zunächst folgendes Zwischenresultat. + +\begin{lemm} + \makelabel{lemm:1:ueb:6:ex:3} $(F,+,\cdot)$ ist genau dann ein Körper, wenn in der Teilstruktur $(F,\cdot)$ multiplikative Inverse existieren für jedes Element. -\end{claim} +\end{lemm} \begin{einzug}[\rtab][\rtab] \begin{proof} @@ -4419,167 +4511,71 @@ Wir beobachten zuerst folgendes Ergebnis. \end{proof} \end{einzug} -Aus diesem Ergebnis geht hervor, dass wir uns der Existenz multiplikativer Inverser widmen müssen. -Sei also $(a,b)\in F\ohne\{0_{F}\}$ beliebig und sei $(a',b')\in F$. -Dann +Jetzt können wir uns dem Beweis von \Cref{satz:1:ueb:6:ex:3} widmen - \begin{mathe}[mc]{rcl} - \eqtag[eq:0:ueb:6:ex3] - (a,b)\cdot(a',b') = 1_{F} - &\Longleftrightarrow - &(a,b)\cdot(a',b') = (1,0)\\ - &\Longleftrightarrow - &(aa'-bb',ab'+a'b) = (1,0)\\ - &\Longleftrightarrow - &aa'-bb'=1\,\text{und}\,ab'+a'b=0\\ - &\Longleftrightarrow - &aa'-bb'=1\,\text{und}\,ab'=-a'b.\\ - \end{mathe} +\begin{proof}[von \Cref{satz:1:ueb:6:ex:3}][\Cref{satz:1:ueb:6:ex:3}] + Laut \Cref{lemm:1:ueb:6:ex:3} gilt $(F,+,\cdot)$ ein Körper gdw. jedes Element in $F$ hat ein multiplikatives Inverses. + Darum reicht es aus \textbf{zu zeigen}, dass -Da $(a,b)\neq 0_{F}=(0,0)$ gibt es folgende Fälle + \begin{mathe}[mc]{rcl} + \forall{(a,b)\in F\ohne\{0_{F}\}:~}\exists{(a',b')\in F:~}(a,b)\cdot (a',b')=1_{F} + &\Longleftrightarrow + &\forall{(a,b)\in F\ohne\{0_{F}\}:~}a^{2}+b^{2}\neq 0 + \end{mathe} - \begin{kompaktenum}{\bfseries {Fall} 1.}[\rtab][\rtab] - %% FALL 1 - \item - $a=0$, $b\in K\ohne\{0\}$. - Dann ist $b$ invertierbar (innerhalb $K$) und + gilt.\footnote{ + Man beachte: Wegen multiplikativer Kommutativität folgt aus + $(a,b)\cdot (a',b')=1_{F}$, + dass auch + $(a',b')\cdot (a,b)=1_{F}$ + gilt. + } - \begin{mathe}[mc]{rcl} - (a,b)\cdot(a',b') = 1_{F} - &\eqcrefoverset{eq:0:ueb:6:ex3}{\Longleftrightarrow} - &0a'-bb'=1\,\text{und}\,0b'=-a'b\\ - &\Longleftrightarrow - &b'=-b^{-1}\,\text{und}\,a'=-b^{-1}0=0\\ - &\Longleftrightarrow - &(a',b')=(0,-b^{-1}).\\ - \end{mathe} + \herRichtung + Angenommen, $a^{2}+b^{2}\neq 0$ für alle $(a,b)\in F\ohne\{0_{F}\}$. + Sei $(a,b)\in F\ohne\{0_{F}\}$ beliebig. + \textbf{Zu zeigen:} $(a,b)$ sei innerhalb $F$ invertierbar.\\ + Per Annahme gilt nun $r:=a^{2}+b^{2}\neq 0$ und somit ist $r$ innerhalb $K$ invertierbar. + Setze $(a',b'):=(r^{-1}a,-r^{-1}b)$. + Dann - Also existiert in diesem Falle ein multiplikatives Inverses für $(a,b)$. - %% FALL 2 - \item - $b=0$, $a\in K\ohne\{0\}$. - Dann ist $a$ invertierbar (innerhalb $K$) und + \begin{mathe}[mc]{rcl} + (a,b)\cdot(a',b') + &= &(a\cdot r^{-1}a-b(-r^{-1}b),a(-r^{-1}b)+(r^{-1}a)b)\\ + &= &(r^{-1}(a^{2}+b^{2}), 0)\\ + &= &(r^{-1}r, 0)\\ + &= &(1, 0)\\ + &= &1_{F}.\\ + \end{mathe} - \begin{mathe}[mc]{rcl} - (a,b)\cdot(a',b') = 1_{F} - &\eqcrefoverset{eq:0:ueb:6:ex3}{\Longleftrightarrow} - &aa'-0b'=1\,\text{und}\,ab'=-a'0\\ - &\Longleftrightarrow - &a'=a^{-1}\,\text{und}\,b'=a^{-1}0=0\\ - &\Longleftrightarrow - &(a',b')=(a^{-1},0).\\ - \end{mathe} + Also ist jedes $(a,b)\in F\ohne\{0_{F}\}$ innerhalb $F$ invertierbar. - Also existiert in diesem Falle ein multiplikatives Inverses für $(a,b)$. - %% FALL 2 - \item - $a,b\in K\ohne\{0\}$. - Dann sind $a,b$ invertierbar (innerhalb $K$) und + \hinRichtung + Angenommen, jedes Element in $F\ohne\{0_{F}\}$ sei invertierbar.\\ + Wie oben erklärt, ist $(F,+,\cdot)$ somit ein Körper.\\ + Sei $(a,b)\in F\ohne\{0_{F}\}$ beliebig. + \textbf{Zu zeigen:} $a^{2}+b^{2}\neq 0$.\\ + Da $(a,b)\neq 0_{F}=(0,0)$, + gilt $a\neq 0$ oder $b\neq 0$ + und damit gilt auch $(a,-b)\in F\ohne\{0_{F}\}$.\\ + Da $F$ ein Körper ist, + sind $(a,b)$ und $(a,-b)$ + und folglich auch das Produkt + $(a,b)\cdot(a,-b)$ + invertierbar.\\ + Da $F$ ein Körper ist, bedeutet dies wiederum, + dass $(a,b)\cdot(a,-b)\neq 0_{F}$ gilt. + Nun, - \begin{mathe}[mc]{rcl} - (a,b)\cdot(a',b') = 1_{F} - &\eqcrefoverset{eq:0:ueb:6:ex3}{\Longleftrightarrow} - &aa'-bb'=1\,\text{und}\,ab'=-a'b\\ - &\Longleftrightarrow - &aa'-bb'=1\,\text{und}\,b'b^{-1}=-a^{-1}a'\\ - &\Longleftrightarrow - &\exists{t\in K:~} - b'b^{-1}=-t - \,\text{und}\, - a^{-1}a'=t - \,\text{und}\, - aa'-bb'=1\\ - &\Longleftrightarrow - &\exists{t\in K\ohne\{0\}:~} - b'=-tb - \,\text{und}\, - a'=at - \,\text{und}\, - a(at)-b(-tb)=1\\ - &\Longleftrightarrow - &\exists{t\in K\ohne\{0\}:~} - (a',b')=(ta,-tb) - \,\text{und}\, - t(a^{2}+b^{2})=1\\ - &\Longleftrightarrow - &a^{2}+b^{2}\,\text{invertierbar} - \,\text{und}\, - (a',b')=((a^{2}+b^{2})^{-1}a,-(a^{2}+b^{2})^{-1}b)\\ - &\Longleftrightarrow - &a^{2}+b^{2}\neq 0\,\text{und}\,(a',b')=((a^{2}+b^{2})^{-1}a,-(a^{2}+b^{2})^{-1}b).\\ - \end{mathe} + \begin{mathe}[mc]{rclcl} + (a,b)\cdot(a,-b) + &= &(aa-b(-b),a(-b)+ab) + &= &(a^{2}+b^{2},0).\\ + \end{mathe} - Folglich existiert dann ein Inverses, wenn $a^{2}+b^{2}\neq 0$. - \end{kompaktenum} - -Beachte, dass im Fall 1, $a^{2}+b^{2}=b^{2}\neq 0$ und im Fall 2 $a^{2}+b^{2}=a^{2}\neq 0$. -Darum können wir diese Fälle zusammenfassen als -$(a,b)\in F\ohne\{0_{F}\}$ hat genau dann ein multiplikatives Inverse, -wenn $a^{2}+b^{2}\neq 0$. -Angesichts \Cref{claim:1:ueb:6:ex:3} haben wir darum bewiesen: - -\begin{satz} - \makelabel{satz:1:ueb:6:ex:3} - $(F,+,\cdot)$ ist genau dann ein Körper, - wenn $a^{2}+b^{2}\neq 0$ innerhalb $K$ - für alle $(a,b)\in F\ohne\{(0,0)\}$. -\end{satz} - -Wir können dieses allgemeine klassifizierende Resultat verwenden, -um die Aufgaben zu behandeln. - -\begin{enumerate}{\bfseries (a)} - %% AUFGABE 6-3a - \item - \begin{schattierteboxdunn} - \begin{claim*} - Sei $K=\mathbf{F}_{2}=\intgr/2\intgr$. - Dann ist $(F,+,\cdot)$ kein Körper. - \end{claim*} - \end{schattierteboxdunn} - - \begin{proof} - Da $(a,b):=(1,1)\in F\ohne\{(0,0)\}$ - und - $a^{2}+b^{2}=1+1=0$ - innerhalb $K=\intgr/2\intgr$, - ist \Cref{satz:1:ueb:6:ex:3} zufolge $F$ kein Körper. - Es scheitert genau das Axiom der Existenz multiplikativer Inverser. - (Nichtsdestotrotz bildet $F$ einen kommutativen Ring mit Einselement.) - \end{proof} - - %% AUFGABE 6-3b - \item - \begin{schattierteboxdunn} - \begin{claim*} - Sei $K=\mathbf{F}_{3}=\intgr/3\intgr$. - Dann ist $(F,+,\cdot)$ ein Körper. - \end{claim*} - \end{schattierteboxdunn} - - \begin{proof} - Laut \Cref{satz:1:ueb:6:ex:3} reicht es aus - für alle $(a,b)\in F\ohne\{(0,0)\}$ - \textbf{zu zeigen}, dass $a^{2}+b^{2}\neq 0$ innerhalb $K=\intgr/3\intgr$. - Sei also $(a,b)\in F\ohne\{(0,0)\}$ beliebig. - Da $a\neq 0$ oder $b\neq 0$ gibt es folgende Fälle: - - \begin{kompaktenum}{\bfseries {Fall} 1.}[\rtab][\rtab] - %% FALL 1 - \item $a=0$, $b\neq 0$. Dann $b=\pm 1\mod 3$. - Also $a^{2}+b^{2}=0+1=1\nequiv 0\mod 3$. - %% FALL 2 - \item $a\neq 0$, $b=0$. Dann $a=\pm 1\mod 3$. - Also $a^{2}+b^{2}=1+0=1\nequiv 0\mod 3$. - %% FALL 3 - \item $a\neq 0$, $b\neq 0$. Dann $a,b=\pm 1\mod 3$. - Also $a^{2}+b^{2}=1+1=2\nequiv 0\mod 3$. - \end{kompaktenum} - - Also gilt in jedem Falle $a^{2}+b^{2}\neq 0$. - Darum bildet $F$ einen Körper. - \end{proof} -\end{enumerate} + Darum gilt $(a^{2}+b^{2},0)=(a,b)\cdot(a,-b)\neq 0_{F}=(0,0)$, + woraus sich $a^{2}+b^{2}\neq 0$ ergibt. +\end{proof} \setcounternach{part}{2} \part{Selbstkontrollenaufgaben}