From b824e1bf9bde157153d25ad6be2be39522e5313e Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 15 Jan 2020 15:54:16 +0100 Subject: [PATCH] update la10 --- ws2019/la/uebungen/la10.pdf | Bin 132190 -> 132311 bytes ws2019/la/uebungen/la10.tex | 16 ++++++---------- 2 files changed, 6 insertions(+), 10 deletions(-) diff --git a/ws2019/la/uebungen/la10.pdf b/ws2019/la/uebungen/la10.pdf index 365e9a3d484d2ed671c1d89eda356650ccde83cf..9e4e36986fd56344957db9e8ed6918e6f27a2800 100644 GIT binary patch delta 36062 zcmV(zK<2;RhzQq-2(ai1135S|lVLI_f8||UbKAxhe%G&H->8YX+51I_9+W!iB#t|& zqe*6JyEBvtQII9drliD4+yCCP00``YX8};8E=gZ(m<0E7_FTSmTYrB1>IXKOU+>Y; zd_+fo+xzNx@AGe%8ad)L*Q4Xpk>o`2k!6A~V@Aib(d+RWA>_&JVt##cF}=Cjf1d~; z#(!Uu{fU%n%;^4v8z;^9uV>5r`X-+)$~RtS%bRSm|MvL1y|0h=KJGD?kdD}D0_uox zeDv;e@AX?cn!#7!jVOF%N52*?UXG-IR}4H}jDFnvN4?}|=^|XZri61lGDZOP*%&<9t4Ol(N8_E^3e{V;)WUF`j z_xu7`3gbY#dW;ibV>`|+fUSiQ|$V8@IMw8HN8XmM(Se>EKxMFtVX8WqwC zU$Rnc2$AFvBJsP(PpBG%AP!`8A^|H@is3)&kV`O06vz*ZOc<3ObQNFFO3)d|&N3UF 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zKr9$^f}tER%)i}91EDCg4DS8~BR+7i0gQ%R!TpC|>?s&;0~6CA{K>|dM$*ZCmC5h6 zcR(c)wv?grHE8Le%AY5e4ymTPExA-H7c3oC?TA=%t3K$ql&Sjqo24w(`Nx)ysQ!3v jDVqdTf4vcP=%oAyYJ*{nm#zc?9||-zISM5uMNdWwN%piq diff --git a/ws2019/la/uebungen/la10.tex b/ws2019/la/uebungen/la10.tex index 0569565..de0b60f 100644 --- a/ws2019/la/uebungen/la10.tex +++ b/ws2019/la/uebungen/la10.tex @@ -122,18 +122,14 @@ $\implies \text{ker } \left.p\right|_{K[x]_{< n}} = \{0\} $. Sei nun $A \in K[x] / f K[x]$. Dann ex. ein $g \in K[x]$ mit $A = g + f K[x]$. - Definiere nun rekursiv durch Polynomdivision: - \[ - g_k := q_{k+1} \cdot f + g_{k+1} - .\] - mit $g_1 := g$. Dabei gilt nach VL $\text{deg}(g_{k+1}) < \text{deg}(g_k)$. Wähle - das erste $g_k$, sodass gilt $\text{deg}(g_k) < n$. Damit folgt + Teile nun $g$ durch $f$ mit Rest. Dann ex. $q, r \in K[x]$, s.d. \begin{align*} - &g = g_1 = q_1 \cdot f + \ldots + q_{k} \cdot f + g_k = (q_1 + \ldots + q_k) \cdot f + g_k \\ - \implies &g - g_k = \underbrace{(q_1 + \ldots + q_k) \cdot f}_{\in f K[x]} + &g = q \cdot f + r \\ + \implies &g- r = \underbrace{q \cdot f}_{\in f K[x]}\\ + \implies& g \sim r .\end{align*} - $\implies$ $g \sim g_k$ und wegen $\text{deg}(g_k) < n$ folgt $g_k \in K[x]_{< n}$.\\ - $\implies \left.p\right|_{K[x]_{