From 868ad7c61d9796ab9224fe97fcff611f892e66ae Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Wed, 9 Dec 2020 10:01:23 +0100
Subject: [PATCH] =?UTF-8?q?master=20>=20master:=20=C3=9CB6=20vereinfacht?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
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---
 docs/loesungen.pdf | Bin 528758 -> 530422 bytes
 docs/loesungen.tex | 390 ++++++++++++++++++++++-----------------------
 2 files changed, 193 insertions(+), 197 deletions(-)

diff --git a/docs/loesungen.pdf b/docs/loesungen.pdf
index d4025160b0ffae83382b2454a5427d41844a95c2..b86e5aa7bd7f8cbae37f24a1611e8ace5afbaa8d 100644
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diff --git a/docs/loesungen.tex b/docs/loesungen.tex
index 81a6e33..0dd6791 100644
--- a/docs/loesungen.tex
+++ b/docs/loesungen.tex
@@ -1315,7 +1315,7 @@
 
 \def\leer{\emptyset}
 \def\restr#1{\vert_{#1}}
-\def\ohne{\setminus}
+\def\ohne{\mathbin{\setminus}}
 \def\Pot{\mathop{\mathcal{P}}}
 \def\einser{\mathbf{1}}
 \def\supp{\mathop{\mathrm{supp}}}
@@ -4016,28 +4016,31 @@ für alle $A,B\in R$.
             \end{kompaktitem}
 
             Darum erfüllt $(R,+,\cdot)$ die Axiome eines Rings
-            und dieser Ring hat ein Einselement und ist kommutativ,
-            weil 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}