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z+g)n6G4#gR=Kz0joq99)9;@@>1bJYB+P1Dj|EG#)b0Mh1909*8n_4FA=avZQui(aD z+8#ceSD`u8w`ux^7^9mL{yqr^@(BI%FCb9{(&)?EYk7oLsARZ?MVnfEfMgbWvdQ~6 zfDG*o(;+GL?EdyT8@l&s|FKWq!{@BZ#xYCbO~CB(IK9Td_~pTh<2*OGHP{EuLpQJE z4U1Ej-q$V?(frj7Jqf}WYD0KR-u!s4sz64VsO^N>EA~6l2qwmuauL%O2Zn%QV~7K0 z113AFo)||UWluSE2VLzM5L_tI2zh9|&1yxpDDDB*fLBseWI1y0-yV#u$dH@lfYX3- zN~I%800slcl)EgNR}yi7t|YUt_ZmayM=#D zJqdG5fBcnOUP^Un*x;uFv?~N6x_p5|7xfl~6=EiN=C>?}o#0|c0DY4^FHt^?6kpPk zzkS&y%}!8O$~@b$$Q;fUMF1<=9$(xNsoJN2vJf(z9469Y$A8^G%{EyG~TiZX32zxl8IRQw^i{%~M2i?wQWasWHu~IJ43^wO6;M**46NOUY7awO5C* zSx>qK7bymf(mMJEi{Amog^U_U^JTN~N$0*=R!cHD9*W2LN!dR7R(&!LIdOkYepdl< z$F)fvp9ddZ`(%VX#gEI9*8M*7{Z!npH_w(O3Hmh0TPM#6v`O~W7U3zKBsbV4{Iy#e zOSyGY1wRB0zxj<2_8Ni)K7eSaLr!qoyqzCVl4vg4;CB-o5HdL{&?R2coWqKj_#dX+ zR#`hS7?NquuPxgp4$;5{e&5&?$9{?RIaqEX&pUwRpu#4JenxHb2VJ2T|3YEll%URi zbp7gSk6OzPOrdI3QMlA;08Di_2^=(>1vX2`1wk!qho}&@N+ZnTgznF~O9@UfC< z!v{&tZ-=blyvjd`ch5gL;)-_I{?}xwASm4Q2gT3_cn3j8=>I1)Iz0$ weil $(x,y)\neq (0,0)$.\\ + Um $\alpha$ zu bestimmen, werden folgende Fälle aufgeführt: + + \begin{kompaktenum}{\bfseries {Fall} 1.}[\rtab][\rtab] + \item + $y=0$. Dann $x\neq 0$ und in diesem Falle gilt $r=|x|$. + Man setze + $\alpha := \begin{cases}[mc]{lcl} + 0 &: &x>0\\ + \pi &: &x<0\\ + \end{cases}$. + Dann $r\cos(\alpha) := + \begin{cases}[mc]{lcl} + r &: &x>0\\ + -r &: &x<0\\ + \end{cases} + = x$ + und $r\sin(\alpha)=0$. + \item + $y>0$. + Man setze $\alpha\in(0,\pi)$ der eindeutige Winkel + mit $\cos(\alpha)=\frac{x}{r}$. + Dann + $r\cos(\alpha)=x$ + und + $r\sin(\alpha)=r\sqrt{1-\cos^{2}(\alpha)} + =\sqrt{r^{2}-(r\cos(\alpha))^{2}} + =\sqrt{(x^{2}+y^{2})-x^{2}} + =\sqrt{y^{2}} + =|y|=y$. + \item + $y<0$. + Man setze $\alpha\in(\pi,2\pi)$ der eindeutige Winkel + mit $\cos(\alpha)=\frac{x}{r}$. + Dann + $r\cos(\alpha)=x$ + und + $r\sin(\alpha)=r\cdot -\sqrt{1-\cos^{2}(\alpha)} + =-\sqrt{r^{2}-(r\cos(\alpha))^{2}} + =-\sqrt{(x^{2}+y^{2})-x^{2}} + =-\sqrt{y^{2}} + =-|y|=y$. + \end{kompaktenum} + + Darum gilt in allen Fällen + $r\cdot\begin{svector}\cos(\alpha)\\\sin(\alpha)\\\end{svector} + =\begin{svector}r\cos(\alpha)\\r\sin(\alpha)\\\end{svector} + =\begin{svector}x\\y\\\end{svector} + =z$. + + Zur {\bfseries Eindeutigkeit}: + Seien $r_{i}\in(0,\infty)$, $\alpha_{i}\in[0,2\pi)$ + mit + $r_{i}\cdot\begin{svector}\cos(\alpha_{i})\\\sin(\alpha_{i})\\\end{svector}=z$ + für $i\in\{1,2\}$. + \textbf{Zu zeigen:} $r_{1}=r_{2}$ und $\alpha_{1}=\alpha_{2}$. + Es gilt + + \begin{mathe}[mc]{rclcl} + r_{1}^{2} &= &r_{1}^{2}(\cos^{2}(\alpha_{1}) + \sin^{2}(\alpha_{1}))\\ + &= &(r_{1}\cos(\alpha_{1}))^{2} + (r_{1}\sin(\alpha_{1}))^{2}\\ + &= &x^{2}+y^{2}\\ + &= &(r_{2}\cos(\alpha_{2}))^{2} + (r_{2}\sin(\alpha))^{2}\\ + &= &r_{2}^{2}(\cos^{2}(\alpha_{2}) + \sin^{2}(\alpha_{2})) + &= &r_{2}^{2},\\ + \end{mathe} + + woraus sich ergibt, dass $r_{1}=r_{2}$, weil $r_{1},r_{2}\geq 0$. + Da $r_{1},r_{2}>0$, folgt + + \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}} + &= &\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}} + &= &\sin(\alpha_{2})\\ + \end{mathe} + + Da $\alpha_{1},\alpha_{2}\in[0,2\pi)$ und wegen Injektivität von $\cos$ + auf $[0,\pi)$ und $[\pi,2\pi)$ und der Symmetrie um $\pi$, + erhalten wir aus + $\cos(\alpha_{1})=\cos(\alpha_{2})$, + dass (i)~$\alpha_{1}=\alpha_{2}$ oder (ii)~$\alpha_{1}=2\pi-\alpha_{2}$ + gelten muss.\\ + Falls (ii) gilt, + so gilt $\sin(\alpha_{1})=\sin(2\pi-\alpha_{2})=-\sin(\alpha_{2})$. + Da aber $\sin(\alpha_{1})=\sin(\alpha_{2})$, + folgt daraus $\sin(\alpha_{2})=0$, + und damit (iii)~$\alpha_{2}=0$ oder (iv)~$\pi$. + Falls (iii) gilt, so gilt wegen (ii) $\alpha_{1}=2\pi-0=2\pi$, + was ein Widerspruch ist, weil $\alpha_{1}\in[0,2\pi)$. + Darum muss (iv) gelten. + Wegen (ii) gilt also $\alpha_{1}=2\pi-\pi=\pi=\alpha_{2}$.\\ + Zusammegefasst gilt entweder (i) $\alpha_{1}=\alpha_{2}$ + oder (ii), aus dem sich (iv) ergibt, was wiederum $\alpha_{1}=\alpha_{2}$ zur Folge hat. + D.\,h., in allen Fällen gilt $\alpha_{1}=\alpha_{2}$. + + Darum gelten $r_{1}=r_{2}$ und $\alpha_{1}=\alpha_{2}$. + Also ist die Darstellung eindeutig. + \end{proof} + + %% AUFGABE 6-2b + \item + \begin{claim*} + Seien $z_{1},z_{2}\in\kmplx\ohne\{0\}$ mit Darstellungen + $z_{i}=r_{i}\cdot\begin{svector}\cos(\alpha_{i})\\\sin(\alpha_{i})\\\end{svector}$ + für $i\in\{1,2\}$. + Dann gilt die Rechenregel + $z_{1}z_{2}=r_{1}r_{2}\cdot\begin{svector}\cos(\alpha_{1}+\alpha_{2})\\\sin(\alpha_{1}+\alpha_{2})\\\end{svector}$. + \end{claim*} + + \begin{proof} + Multiplikation in $\kmplx$ liefert + + \begin{mathe}[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} + + Die letzte Vereinfachung folgt aus der trigonometrischen Additionsregel. + \end{proof} + %% AUFGABE 6-2c + \item + \begin{claim*}[de Moivre] + Sei $z\in\kmplx\ohne\{0\}$ mit Darstellungen + $z=r\cdot\begin{svector}\cos(\alpha)\\\sin(\alpha)\\\end{svector}$. + Dann gilt die Potenzregel + $z^{n}=r^{n}\cdot\begin{svector}\cos(n\alpha)\\\sin(n\alpha)\\\end{svector}$ + für alle $n\in\ntrlpos$. + \end{claim*} + + \begin{proof} + Wir beweisen dies per Induktion über $n$. + + \begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\bfseries Induktionsanfang:}}] + Die Gleichung gilt offensichtlich für $n=1$. + + \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] + Sei $n>1$. Angenommen, $z^{n-1}=r^{n-1}\cdot\begin{svector}\cos((n-1)\alpha)\\\sin((n-1)\alpha)\\\end{svector}$. + + \item[\uwave{{\bfseries Induktionsschritt:}}] + \textbf{Zu zeigen:} $z^{n}=r^{n}\cdot\begin{svector}\cos(n\alpha)\\\sin(n\alpha)\\\end{svector}$.\\ + Per rekursive Definition vom Potenzieren + gilt zunächst $z^{n}=z^{n-1}\cdot z$ (Multiplikation innerhalb der Algebra $\kmplx$). + Aufgabe 6-2(b) zur Folge gilt somit + + \begin{mathe}[mc]{rcl} + z^{n}=z^{n-1}\cdot z + &\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}{=} + &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} + + Darum gilt die Gleichung für $n$. + \end{kompaktenum} + + 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. +\end{enumerate} + +%% AUFGABE 6-3 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 3]{} + \label{ueb:6:ex:3} +\let\sectionname\altsectionname + +Es sei $K$ ein Körper und $F:=K\times K$ versehen mit den Operationen ${+,\cdot:F\times F\to F}$, +definiert vermöge + + \begin{mathe}[mc]{rcl} + (a,b)+(a',b') &= &(a+a',b+b')\\ + (a,b)\cdot (a',b') &= &(aa'-bb',ab'+a'b)\\ + \end{mathe} + +für alle $a,b,a',b'\in K$. +Wir beobachten zuerst folgendes Ergebnis. + +\begin{claim} + \makelabel{claim: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} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + Da die Teilstruktur, $(F,+)$, durch die Produktstruktur $(K,+)\times (K,+)$ gegeben ist, + dessen Faktoren all kommutative Gruppen sind, ist $(F,+)$ eine kommutative Gruppe. + Das heißt, die {\bfseries Additionsaxiome} unter den Körperaxiomen sind allesamt erfüllt. + (Insbesondere ist das Nullelement durch $0_{F}=(0,0)$ gegeben.) + + Bei den {\bfseries Multiplikationsaxiomen} sehen wir dass Kommutativität offensichtlich gilt, + weil die o.\,s. Definitions von Multiplikation in den Argumenten offensichtlich symmetrisch ist, + und weil die Operationen in $K$ kommutativ sind. + Es gilt auch $(a,b)\cdot(1,0)=(a\cdot 1-b\cdot 0,a\cdot 0+1\cdot b)=(a,b)$, + sodass $1_{F}:=(1,0)$ das Einselement von $F$ ist. + Assoziativität von Multiplikation ist auch erfüllt, weil + + \begin{mathe}[mc]{rcl} + (a,b)\cdot((a',b')\cdot(a'',b'')) + &= &(a,b)\cdot(a'a''-b'b'', a'b''+a''b')\\ + &= &(a(a'a''-b'b'')-b(a'b''+a''b'), a(a'b''+a''b')+(a'a''-b'b'')b)\\ + &= &(aa'a'' - ab'b'' - ba'b'' - ba''b', aa'b'' + aa''b' + a'a''b - b'b''b)\\ + &= &\boxed{(aa'a'' - ab'b'' - a'bb'' - a''bb', aa'b'' + aa''b' + a'a''b - bb'b'')}\\ + \end{mathe} + + und + + \begin{mathe}[mc]{rcl} + ((a,b)\cdot(a',b'))\cdot(a'',b'') + &= &(aa'-bb',ab'+a'b)\cdot(a'',b'')\\ + &= &((aa'-bb')a'' - (ab'+a'b)b'', (aa'-bb')b'' + a''(ab'+a'b))\\ + &= &(aa'a'' - bb'a'' - ab'b'' - a'bb'', aa'b'' - bb'b'' + a''ab' + a''a'b)\\ + &= &\boxed{(aa'a'' - ab'b'' - a'bb'' - a''bb', aa'b'' + aa''b' + a'a''b - bb'b'')}\\ + &= &(a,b)\cdot((a',b')\cdot(a'',b'')) + \end{mathe} + + für alle $(a,b),(a',b'),(a'',b'')\in F$. + Darum ist $(F,\cdot)$ assoziativ, kommutativ, und hat ein Neutralelement. + + Wegen Kommutativität von Multiplikation in $F$, ist {\bfseries Distributitivität} + zu Linksdistributivität äquivalent, und da + + \begin{mathe}[mc]{rcl} + (a,b)\cdot((a',b')+(a'',b'')) + &= &(a,b)\cdot(a'+a'',b'+b'')\\ + &= &(a(a'+a'')-b(b'+b''), a(b'+b'')+(a'+a'')b)\\ + &= &((aa'-bb')+(aa''-bb''), (ab'+a'b)+(ab''+a''b))\\ + &= &(aa'-bb',ab'+a'b)+(aa''-bb'', ab''+a''b)\\ + &= &(a,b)\cdot(a',b')+(a,b)/cdot (a'',b'')\\ + \end{mathe} + + für alle $(a,b),(a',b'),(a'',b'')\in F$, ist dies erfüllt. + + Anhand der o.\,s. Erkenntnisse darüber, welchen Axiomen $(F,+,\cdot)$ bereits genügt, + erhalten wir, dass $(F,+,\cdot)$ genau dann ein Körper, wenn in $(F,\cdot)$ jedes Element ein Inverses hat. + \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 + + \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} + +Da $(a,b)\neq 0_{F}=(0,0)$ gibt es folgende Fälle + + \begin{kompaktenum}{\bfseries {Fall} 1.}[\rtab][\rtab] + %% FALL 1 + \item + $a=0$, $b\in K\ohne\{0\}$. + Dann ist $b$ invertierbar (innerhalb $K$) und + + \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} + + 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') = 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 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 + + \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} + + 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} + \setcounternach{part}{2} \part{Selbstkontrollenaufgaben} \def\chaptername{SKA Blatt} -%% ******************************************************************************** -%% FILE: body/ska/ska1.tex -%% ******************************************************************************** - -%% ******************************************************************************** -%% FILE: body/ska/ska2.tex -%% ******************************************************************************** - -%% ******************************************************************************** -%% FILE: body/ska/ska3.tex -%% ******************************************************************************** - %% ******************************************************************************** %% FILE: body/ska/ska4.tex %% ******************************************************************************** diff --git a/notes/berechnungen_wk7.md b/notes/berechnungen_wk7.md new file mode 100644 index 0000000..e69de29 diff --git a/protocol/README.md b/protocol/README.md index 1351c3c..035e463 100644 --- a/protocol/README.md +++ b/protocol/README.md @@ -12,6 +12,7 @@ Jede Woche werden Anmerkungen in Markdown-Dateien hier festgehalten: - Woche 4: [/protocol/woche4/README.md](./woche4). - Woche 5: [/protocol/woche5/README.md](./woche5). - Woche 6: [/protocol/woche6/README.md](./woche6). +- Woche 7: [/protocol/woche7/README.md](./woche7). ## Übungsgruppen ### diff --git a/protocol/woche6/README.md b/protocol/woche6/README.md index a67b2e8..cbf41a6 100644 --- a/protocol/woche6/README.md +++ b/protocol/woche6/README.md @@ -2,12 +2,12 @@ ## Ablauf ## -- ( ) allgemeine Ankündigungen +- (√) allgemeine Ankündigungen - Fortschritt mit dem VL-Stoff? - Bewältigung von Aufgaben? - TeX / Markdown - Collaborationstool: -- ( ) SKA 6 +- (√) SKA 6 - Breakout-Rooms (10min) - Gruppe 1: 1 - Gruppe 2: 2 + *3 @@ -16,4 +16,4 @@ - Gruppe 5: 7[a, b, *c, *d] - Präsentationen (5 x 2–3 ≈ 15min) - Gruppendiskussion (~ 15min) -- ( ) VL-Stoff + allg. Fragen (Rest) +- (√) VL-Stoff + allg. Fragen (Rest) diff --git a/protocol/woche7/README.md b/protocol/woche7/README.md index e69de29..b0db07e 100644 --- a/protocol/woche7/README.md +++ b/protocol/woche7/README.md @@ -0,0 +1,9 @@ +## Woche 7 (KW 50, 7.–13.12.) ## + +## Ablauf ## + +- ( ) allgemeine Ankündigungen + - Berechnungen —> Beweise? +- ( ) ÜB5 ? +- ( ) SKA 7 +- ( ) VL-Stoff + allg. Fragen (Rest) diff --git a/protocol/woche8/README.md b/protocol/woche8/README.md new file mode 100644 index 0000000..e69de29