From 62a37b6af48da89d9fe9699873de0b3a2c2aaf22 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 12 May 2020 10:30:01 +0200 Subject: [PATCH] fix typos in la --- sose2020/la/uebungen/la2.pdf | Bin 115928 -> 116186 bytes sose2020/la/uebungen/la2.tex | 16 ++++++++++------ 2 files changed, 10 insertions(+), 6 deletions(-) diff --git a/sose2020/la/uebungen/la2.pdf b/sose2020/la/uebungen/la2.pdf index 94bf126e0542b5a5f706e1c8ae5d6264c3cbf317..5c73d2cfc01db8a15b8ecaba7c821784192ae755 100644 GIT binary patch delta 23236 zcmV(^K-It4i3i$?2e9-H12!}`lOaDSf9)OHk{mbn-Cr?0FCvfA##&-?G*9?Gw7X zUeVR>o_z8A$*+FPJw7f5Y|e z>Shyu`0C=4oAp=E!cQ*%#MS-r`u6Iuxw{L$;}@4qFmBghzJ0Sl?*A5k`{m}i-yNQY zAAbG%W`BFP-yFk_-|TMgcE^h!pMU-2H_xB^;|ar!(iN*WMlAu$S1;c@`Qb;p+TwqG zy`uOlvwG(jzF7&tB^ndq zT2VeBpK?OF+zFc_>M^7w;fp^%+`esPEIb^;rw$*_FN|1ku6M_Z_?!FN>)q=eV#xpa z&kNkrTdyEims~5txmjJ}e<3Y?m;a`3_NaYu(L2A=AK|5YeC-sBSa0{c3${i@XzuAT-$5H;_+xK^`-yXyB#($o$+ney)ipj%LgvfQpuCEW_cW7ld z+uhyM3fSScdbqDyIV^hh0<^fhL!0?D0KPxgo7x;V@!HMdQ0Cum-jXv);@ 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LFbX9lMNdWw;uU0_ diff --git a/sose2020/la/uebungen/la2.tex b/sose2020/la/uebungen/la2.tex index 537d50b..8b8709d 100644 --- a/sose2020/la/uebungen/la2.tex +++ b/sose2020/la/uebungen/la2.tex @@ -136,7 +136,7 @@ Damit folgt $a^2 = 2$, aber $a \in \Z$ $\contr$. \end{proof} - Beh.: $2$ und $1 \pm \sqrt{-3} $ sind irreduzibel. + Beh.: $\pm 2$ und $\pm (1 \pm \sqrt{-3})$ sind irreduzibel. \begin{proof} Es gilt zunächst $\delta (2) = \delta (1 \pm \sqrt{-3}) = 4$. @@ -150,13 +150,17 @@ $y \in \Z[\sqrt{-3}]^{\times }$. Für $1 \pm \sqrt{-3} $ analog. + + Da $2 \; \widehat{=} -2$ und + $1 \pm \sqrt{-3} \;\widehat{=} -(1 \pm \sqrt{-3})$, folgt die Behauptung. \end{proof} Beh.: $2$ ist kein Primelement in $\Z[\sqrt{-3} ]$. \begin{proof} Es gilt $2 \mid 4 = (1 + \sqrt{-3})(1 - \sqrt{-3})$. Aber - $2 \nmid (1 \pm \sqrt{-3})$, da $1 \pm \sqrt{-3} $ irreduzibel. Also - ist $2$ kein Primelement. + $2 \nmid (1 \pm \sqrt{-3})$, da $1 \pm \sqrt{-3} $ irreduzibel und $2 \neq \pm 1$ und + $2 \neq 1 \pm \sqrt{-3}$. + Also ist $2$ kein Primelement. \end{proof} \item Beh.: $\text{GGT}(4, 2+2\sqrt{-3}) = \emptyset$. \begin{proof} @@ -171,12 +175,12 @@ \begin{align*} \delta^{-1}(1) &= \{\pm 1 \} \\ \delta^{-1}(4) &= \{\pm 2, \pm (1 \pm \sqrt{-3}) \} \\ - \delta ^{-1}(16) &= \{\pm 4, \pm (2 \pm \sqrt{-3}) \} + \delta ^{-1}(16) &= \{\pm 4, \pm (2 \pm 2\sqrt{-3}) \} .\end{align*} Damit folgt, dass alle möglichen gemeinsamen Teiler gegeben sind durch: \[ - T = \{\pm 1, \pm 2, \pm (1 \pm \sqrt{-3}), \pm 4, \pm (2 \pm \sqrt{-3})\} + T = \{\pm 1, \pm 2, \pm (1 \pm \sqrt{-3}), \pm 4, \pm (2 \pm 2\sqrt{-3})\} .\] Es gilt weiter $4 \nmid (2 \pm 2\sqrt{-3})$, da $\forall a, b \in \Z$ gilt: \[ @@ -188,7 +192,7 @@ \[ 4 = (2 \pm 2 \sqrt{-3} ) (a + \sqrt{-3} b) = \underbrace{2a \mp 6b}_{=4} + \sqrt{-3} (\underbrace{2b \pm 2a}_{= 0}) - \implies 2a = \mp 2b \implies = \mp \underbrace{8b}_{\in \Z} = 4 \quad \contr + \implies 2a = \mp 2b \implies \mp \underbrace{8b}_{\in \Z} = 4 \quad \contr .\] Analog folgt wieder $\pm (2 \pm 2 \sqrt{-3}) \nmid \pm 4$. Damit folgt $\pm 4$, $\pm (2 \pm 2 \sqrt{-3}) \not\in \text{GGT}(4, 2 + 2 \sqrt{-3})$.