From ea848f022756d4e2feb6b04e1f75480e3a2d0b92 Mon Sep 17 00:00:00 2001 From: henrydatei Date: Wed, 6 Feb 2019 21:42:38 +0000 Subject: [PATCH] GEO fertig :smile: --- .../TeX_files/Kommutative_Ringe/Ideale.tex | 1 + .../symmetrische_Polynome.tex | 100 +++++++++++++++++- 3. Semester/GEO/Vorlesung GEO.pdf | Bin 825639 -> 834920 bytes .../tex/latex/mathoperators/mathoperators.sty | 1 + 4 files changed, 101 insertions(+), 1 deletion(-) diff --git a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Ideale.tex b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Ideale.tex index 108c24c..8496e4b 100644 --- a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Ideale.tex +++ b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Ideale.tex @@ -163,6 +163,7 @@ Sei $R$ ein Ring. (Die meisten Beweise finden sich wieder im LAAG 1+2 Skript!) \end{example} \begin{proposition} + \proplbl{2_2_13} Jedes echte Ideal $I \properideal R$ ist in einem maximalen Ideal von $R$ enthalten. \end{proposition} \begin{proof} diff --git a/3. Semester/GEO/TeX_files/Kommutative_Ringe/symmetrische_Polynome.tex b/3. Semester/GEO/TeX_files/Kommutative_Ringe/symmetrische_Polynome.tex index efac8e5..571e6e0 100644 --- a/3. Semester/GEO/TeX_files/Kommutative_Ringe/symmetrische_Polynome.tex +++ b/3. Semester/GEO/TeX_files/Kommutative_Ringe/symmetrische_Polynome.tex @@ -21,4 +21,102 @@ Sei $R$ ein Ring, $n\in\natur$, $R[\uline{x}]=R[x_1,...,x_n]$. Erinnerung: universelle Eigenschaft von $R[\uline{x}]$: Ist $\phi: R\to R'$ ein Ringhomomorphismus, $r_1,...,r_n\in R'$, so gibt es einen eindeutig bestimmten Ringhomomorphismus $\phi_r:R[\uline{x}]\to R'$ mit $\phi_r\vert_R=\phi$ und $\phi_r(x_i)=r_i\quad\forall i$. Ist $R$ ein Teilring von $R'$ und $\phi=\id_R$, so schreiben wir auch $f(r_1,...,r_n)$ für $\phi_r(f)$. -\end{remark} \ No newline at end of file +\end{remark} + +\begin{definition}[symmetrisches Polynom] + Die Gruppe $S_n$ wirkt auf $R[\uline{x}]$ durch Permutation der Variablen: + \begin{align} + f(x_1,...,x_n)^\sigma = f(x_{1^\sigma},...,x_{n^\sigma})\notag + \end{align} + wobei $f\in R[\uline{x}]$ und $\sigma\in S_n$. Ein Polynom $f\in R[\uline{x}]$ heißt \begriff{symmetrisch}, wenn $f=f^\sigma$ $\forall\sigma\in S_n$. Wir schreiben $S$ für die Menge der symmetrischen Polynome. +\end{definition} + +\begin{remark} + \begin{enumerate}[label=(\alph*)] + \item $S$ ist ein Teilring von $R[\uline{x}]$, der $R$ und $s_1,...,s_n$ enthält. + \item Insbesondere ist $f(s_1,...,s_n)$ für alle $f\in R[\uline{x}]$ symmetrisch. + \end{enumerate} +\end{remark} + +\begin{lemma} + \proplbl{2_10_6} + Für $k=n$ ist $s_k(x_1,...,x_{n-1},0)$ das $k$-te elementarsymmetrische Polynom in $x_1,...,x_{n-1}$. +\end{lemma} +\begin{proof} + \begin{align} + s_k(x_1,...,x_n) = \sum_{1\ge i_1<\dots< i_k\ge n} x_{i_1}\cdots x_{i_k}\notag + \end{align} + Ist $i_k=n$, so ist $\phi_{x_1,...,x_{n-1},0}(x_{i_1}\cdots x_{i_k})=0$, somit + \begin{align} + s_k(x_1,...,x_{n-1},0) &= \sum_{1\ge i_1<\dots< i_k\ge n} \phi_{x_1,...,x_{n-1},0}(x_{i_1}\cdots x_{i_k}) \notag \\ + &= \sum_{1\ge i_1<\dots< i_k\ge n-1} x_{i_1}\cdots x_{i_k} \notag + \end{align} +\end{proof} + +\begin{definition} + Sei $f=\sum_\alpha a_\alpha x^\alpha\in R[\uline{x}]$. Wir definieren den \begriff{Totalgrad} von $f$ als + \begin{align} + \deg(f) = \max\left\lbrace \sum_{i=1}^n \alpha_i\mid a_\alpha\neq 0\right\rbrace \notag + \end{align} + und das \begriff{Gewicht} von $f$ als + \begin{align} + \w(f) = \max\left\lbrace \sum_{i=1}^n i\alpha_i\mid a_\alpha\neq 0\right\rbrace \notag + \end{align} +\end{definition} + +\begin{example} + \begin{enumerate}[label=(\alph*)] + \item $\deg(s_k) = k$ + \item $\w(f) = d\Rightarrow \deg(f(s_1,...,s_n))\le d$ + \item $f=x_1+x_2\Rightarrow \deg(f) = 1$, $\w(f)=2$ \\ + $f(s_1,s_2) = s_1+s_2$, $\deg(f(s_1,s_2)) = 2$ + \end{enumerate} +\end{example} + +\begin{theorem}[Hauptsatz über symmetrische Polynome] + Sei $f\in R[\uline{x}]$ symmetrisch vom Grad $\deg(f)=d$. Dann ist $f(x_1,...,x_n)=g(s_1,...,s_n)$ für ein eindeutig bestimmtes Polynom $g\in R[y_1,...,y_n]$ das vom Gewicht $\w(g)\le d$ ist. +\end{theorem} +\begin{proof}[Existenz] + Induktion nach $n$: \\ + \emph{$n=1$:} klar \\ + \emph{$n>1$:} Induktion nach $d$: \\ + \emph{$d=0$:} klar \\ + \emph{$d>0$:} Mit $\phi = \phi_{x_1,...,x_{n-1},0}$ ist $\deg(f)\le d$ \\ + $\xRightarrow[\propref{2_10_6}]{n\text{-I.H.}} \phi(f)=g_1(\phi(s_1),...,\phi(s_{n-1}))$, $\w(g_1)\le d$ \\ + $\Rightarrow f_1 = f(x_1,...,x_n)-g_1(s_1,...,s_{n-1})\in S$, $\deg(f_1)\le\max\{\deg(f),\w(g_1)\}\le d$, $f_1(x_1,...,x_{n-1},0) = \phi(f_1) = \phi(f) - g_1(\phi(s_1),...,\phi(s_{n-1})) = 0$ \\ + $\Rightarrow x_n\mid f_1$ in $R[x_1,...,x_n]\xRightarrow{f_1\in S} x_i\mid f_1$ für alle $i\xRightarrow{x_i\text{ prim}} s_n=x_1\cdots x_n\mid f_1$ \\ + $\Rightarrow f_1 = s_n\cdot f_2$ mit $f_2\in S$, $\deg(f_2)\le d-n$ \\ + $\xRightarrow{d\text{-I.H.}} f_2=g_2(s_1,...,s_n)$, $\w(g_2)\le d-n$ \\ + Mit $g=g_1+y_ng_2$ ist + \begin{align} + g(s_1,...,s_n) &= g_1(s_1,...,s_n)+s_ng_2(s_1,...,s_n) \notag \\ + &= f-f_1 + s_nf_2 = f \notag \\ + \w(g) &\le \max\{\underbrace{\w(g_1)}_{\le d},n+\underbrace{\w(g_2)}_{\le d-n}\} \le d\notag + \end{align} +\end{proof} + +\begin{conclusion} + Der Ring $S$ der symmetrischen Polynome in $x_1,...,x_n$ ist isomorph zum Ring $R[x_1,...,x_n]$ selbst. +\end{conclusion} +\begin{proof} + $\phi_{s_1,...,s_{n}}:R[x]\to S$ ist surjektiv (Existenz) und injektiv (Eindeutigkeit) +\end{proof} + +\begin{definition}[Diskriminante] + Die \begriff{Diskriminante} von $f_{allg}$ ist + \begin{align} + \Delta = \prod_{i2_F|27^(EmLU^~jYZ!WmMAw_(%$d%~yv&Gde>n6;n#F+vGL zVD6gkMH`H6?Us}2(cgE7(fbNzZe(+Ga%Ev{3T19&Z(?c+ID^auhs*^5hs*^6x6B0t ztKk7Qm(lqH69YCfHJ8D10x6fb=K~Xe$U}>!+oWi*ZUgk8>4O?Gwpv?GEx9T3vVVQg znISnB4kb#Ky(yYTBsH4LneSZZ9EMbv6{-Gx^2O_ue}2VOWoxV8dpSwz@r?Gc#*8SIyn}?I~PdHU69Lm#a&-WQ;01`)2WexoNI{@f$IF z2~YLx<(a?x27h#WbAuoEo5g1N2VOP)OX%d3&Y+*4U%xte@%rTV6AF_g6%CUTwNk8l 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