From ddcd01a5bce6a785c74f1492826adaa87e0631fe Mon Sep 17 00:00:00 2001 From: henrydatei Date: Mon, 4 Feb 2019 14:54:56 +0000 Subject: [PATCH] GEO Kapitel 2.7 --- .../Irreduzibilitaetskriterien.tex | 79 +++++++++++++++++- .../Kommutative_Ringe/Satz_von_Gauss.tex | 2 + 3. Semester/GEO/Vorlesung GEO.pdf | Bin 779888 -> 788714 bytes 3 files changed, 78 insertions(+), 3 deletions(-) diff --git a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Irreduzibilitaetskriterien.tex b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Irreduzibilitaetskriterien.tex index 74fe421..37cfde7 100644 --- a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Irreduzibilitaetskriterien.tex +++ b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Irreduzibilitaetskriterien.tex @@ -19,7 +19,17 @@ Sei $R$ ein faktorieller Ring, $K = \Quot(R)$. \end{proposition} \begin{proof} - to be written ... + Sei $f=g\cdot h$ mit $g=\sum_{i=0}^k b_ix^i$, $h=\sum_{i=0}^l c_ix^i\in R[x]$ und $n=k+l$. + \begin{itemize} + \item $p\nmid a_n = b_k\cdot c_l\Rightarrow p\nmid b_k$ und $p\nmid c_l$ + \item $p\mid a_0 = b_0\cdot c_0$ und $p^2\mid a_0\Rightarrow$ ohne Einschränkung $p\mid b_0$, aber $p\nmid c_0$ + \end{itemize} + Sei $m=\max\{i\in\natur\mid p\mid b_0,...,p\mid b_i\}\in \{0,...,k-1\}$ \\ + $\Rightarrow a_{m+1} = \underbrace{b_0c_{m+1}+b_1c_m+...+b_mc_1}_{=0\mod p} + \underbrace{b_{m+1}c_0}_{\neq 0\mod p}$ \\ + $\Rightarrow p\nmid a_{m+1}$ \\ + $\Rightarrow m+1\ge n\Rightarrow k\ge m+1\ge n = k+l\Rightarrow k=n$ und $l=0$ \\ + $\Rightarrow h\in R\xRightarrow{f\text{ primitiv}} h\in R^\times \subseteq R[x]^\times$ \\ + Somit ist $f$ irreduzibel in $R[x]$ und mit \propref{2_6_10} und $f\notin R$ folgt, dass $f$ irreduzibel in $K[x]$ ist. \end{proof} \begin{example} @@ -31,5 +41,68 @@ Sei $R$ ein faktorieller Ring, $K = \Quot(R)$. \end{example} \begin{proposition}[Reduktionskriterium] - Sei $0 \neq f = \sum_{i=0}^{n} a_i x^i \in R[x]\setminus R$ und $p \in R$ prim mit $p \nmid a_n$. Ist $\bar{f} \in (R[x]/(p))[x]$ -\end{proposition} \ No newline at end of file + Sei $0 \neq f = \sum_{i=0}^{n} a_i x^i \in R[x]\setminus R$ und $p \in R$ prim mit $p \nmid a_n$. Ist $\bar{f} \in \left(\lnkset{R}{(p)} \right)[x]$ irreduzibel, so auch $f$ irreduzibel in $K[x]$. +\end{proposition} +\begin{proof} + Schreibe $f=c\cdot f_0$, $c\in R$ und $f_0\in R[x]$ primitiv. Wenn $p\nmid a_n$, dann folgt $p\nmid c$ und $p\nmid \LC(f_0)$. Sei also ohne Einschränkung $f=f_0$ primitiv. Sei $f=g\cdot h$ mit $g=\sum_{i =0}^k b_ix^i$, $h=\sum_{i =0}^l c_ix^i\in R[x]$ und $k+l=n$. Wenn $p\nmid a_n$, dann $p\nmid b_k$ und $p\nmid c_l$. \\ + $\Rightarrow\bar{f} = \bar{g}\cdot\bar{h}$, wobei $\deg(\bar{f})=n$, $\deg(\bar{g})=k$ und $\deg(\bar{h})=l$ \\ + $\xRightarrow{\bar{f}\text{ irreduzibel}}$ ohne Einschränkung $\bar{g}\in\left(\lnkset{R}{(p)}\right)[x]^\times$, insbesondere $k=0\Rightarrow l=n$ \\ + $\Rightarrow g\in R\xRightarrow{f\text{ primitiv}} g\in R^\times$ \\ + $\Rightarrow f$ irreduzibel in $K[x]$ +\end{proof} + +\begin{example} + $R=\whole$: $f=x^3+3x^2+2x+1$ ist irreduzibel in $\ratio[x]$, denn + \begin{align} + \bar{f} = x^3+x^2+1\in \mathbb{F}_2[x]\notag + \end{align} + hat keine Nullstellen in $\mathbb{F}_2=\{0,1\}$. +\end{example} + +\begin{remark} + Die Nullstellen von $x^n-1$ in $\comp$, also die $n$-Torsion $\comp^\times[n]$ von $\comp^\times$ nennt man die $n$-ten \begriff{Einheitswurzeln}. Diesel bilden eine zyklische Gruppe der Ordnung $n$, erzeugt von $\zeta_n = e^{\frac{2\pi i}{n}}$. Elemente von $\comp^\times$ der Ordnung genau $n$ nennt man \begriff[Eineheitswurzeln!]{primitive $n$-te Einheitswurzel}. + + \begin{center} + \begin{tikzpicture} + \draw[->] (-1.5,0) --(1.5,0); + \draw[->] (0,-1.5) -- (0,1.5); + \draw (0,0) circle (1); + \draw[fill=black] (0.707,0.707) circle (0.05); + \node at (1.2,0.8) (a) {$e^{\frac{2\pi i}{n}}$}; + \draw[fill=black] (1,0) circle (0.05); + \node at (1.1,-0.2) (b) {1}; + \end{tikzpicture} + \end{center} +\end{remark} + +\begin{definition}[$p$-tes Kreisteilungspolynom] + Für $p\in\natur$ prim ist + \begin{align} + \Phi_p = \frac{x^p-1}{x-1} = x^{p-1} + x^{p-2} + ... + x + 1\in\whole[x]\notag + \end{align} + das $p$-te \begriff{Kreisteilungspolynom}. +\end{definition} + +\begin{remark} + Die Nullstellen von $\Phi_p$ sind genau die primitiven Einheitswurzeln: + \begin{align} + \Phi_p = \prod_{k=1}^{p-1} (x-\zeta_p^k)\notag + \end{align} +\end{remark} + +\begin{proposition} + Für $p\in\natur$ prim ist $\Phi_p$ irreduzibel in $\ratio[x]$. +\end{proposition} +\begin{proof} + Das Polynom $\Phi_p(x+1)$ ist ein Eisensteinpolynom zur Primzahl $p$, also irreduzibel nach \propref{2_6_2}. Da $f(x)\mapsto f(x+1)$ ein Automorphismus von $\ratio[x]$ ist, ist damit auch $\Phi_p(x)$ irreduzibel. +\end{proof} + +\begin{example} + \proplbl{2_7_10} + Sei $f=2x^7-100\in \ratio[x]\Rightarrow$ Multiplikation mit Einheit $2^{-1}\Rightarrow x^7-50\Rightarrow$ \person{Eisenstein}. \\ + \textbf{Achtung:} $2x^7-100\in\real[x]$ ist nicht irreduzibel! +\end{example} + +\begin{*anmerkung} + In \propref{2_7_10} ist \person{Eisenstein} im ersten Schritt nicht anwendbar, da $2\mid 100$, aber auch $2\mid \LC(f)=2$ und $5\mid 100$, aber auch $5\mid 100^2$. +\end{*anmerkung} \ No newline at end of file diff --git a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Satz_von_Gauss.tex b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Satz_von_Gauss.tex index f155b25..96205f9 100644 --- a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Satz_von_Gauss.tex +++ b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Satz_von_Gauss.tex @@ -26,6 +26,7 @@ Sei $R$ ein faktorieller Ring, $K= \Quot(R)$ und $p \in R$ prim. \end{remark} \begin{definition} + \proplbl{2_6_2} Für $f = \sum_{i \ge 0} a_i x^i \in K[x]$. Sei $v_p(f) := \min_{i \ge 0} v_p(a_i) \in \whole \cup \{\infty\}$. \end{definition} @@ -120,6 +121,7 @@ Sei $R$ ein faktorieller Ring, $K= \Quot(R)$ und $p \in R$ prim. \end{remark} \begin{theorem}[Satz von \person{Gauss}] + \proplbl{2_6_10} Sei $R$ faktorieller Ring und $K = \Quot(R)$. Dann ist auch $R[x]$ faktoriell. Ein $f \in R[x]$ ist genau dann prim, wenn \begin{enumerate}[label=(\alph*)] \item $f$ ist ein Primelement von $R$ \textbf{oder} diff --git a/3. Semester/GEO/Vorlesung GEO.pdf b/3. 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