From 27ec35a62168b0cab245203cc47fa95f24ceb771 Mon Sep 17 00:00:00 2001 From: Valentin Brandl Date: Tue, 15 Jan 2019 15:40:03 +0100 Subject: [PATCH] Add solution for dima 11 --- school/di-ma/uebung/11/11_1.tex | 110 ++++++++++++++++++++++++++++++++ school/di-ma/uebung/11/11_2.tex | 66 +++++++++++++++++++ school/di-ma/uebung/11/11_3.tex | 90 ++++++++++++++++++++++++++ 3 files changed, 266 insertions(+) create mode 100644 school/di-ma/uebung/11/11_1.tex create mode 100644 school/di-ma/uebung/11/11_2.tex create mode 100644 school/di-ma/uebung/11/11_3.tex diff --git a/school/di-ma/uebung/11/11_1.tex b/school/di-ma/uebung/11/11_1.tex new file mode 100644 index 0000000..d163d4e --- /dev/null +++ b/school/di-ma/uebung/11/11_1.tex @@ -0,0 +1,110 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{cancel} +\usepackage{wasysym} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {11} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 11.1} + +\begin{enumerate}[1.] + + \item Sei $p$ prim: + + \begin{align*} + (a + b)^p &\equiv a^p + b^p &\mod p + \end{align*} + + \begin{align*} + (a + b)^p &\equiv \sum\limits^p_{k=0} \binom{p}{k} a^k b^{p-k} &\mod p \\ + &\equiv a^p + b^p + \sum\limits^{p-1}_{k=1} \binom{p}{k} a^k b^{p-k} &\mod p \\ + * &\equiv a^p + b^p &\mod p + \end{align*} + + $*$: $\binom{p}{k} = p * \frac{(p-1)!}{k!(p-k)!}$ + + $p$ prim $\Rightarrow$ $ggT(k!(p-k!), p) = 1$ $\Rightarrow$ $\frac{(p-1)!}{k!(p-k)!} \in \mathbb{Z}_p$ + + $\Rightarrow$ $p | \binom{k}{p}$ für $1 \leq k \leq p-1$ + + $\Rightarrow$ $\sum\limits_{k=1}^{p-1} \binom{p}{k} a^k b^{p-k} \equiv 0 \mod p$ + + q.e.d. + + + \item Sei $a,b \in \mathbb{N}, c = ggT(a,b)$ + + \begin{align*} + \varphi(a * b) &= \varphi(a) * \varphi(b) * \frac{c}{\varphi(c)} + \end{align*} + + Zu zeigen: $\varphi(a) * \varphi(b) = \varphi(ggT(a,b)) * \varphi(kgV(a,b))$ + + Seien $P'$ die gemeinsamen Primteiler von $a$ und $b$ und $A$ und $B$ die Mengen der disjunkten Primteiler von + $a$ und $b$. + + \begin{align*} + \varphi(a) * \varphi(b) &= \prod\limits_{p \in P'} p^{(a_p - 1)(b_p -1)} (p-1)^2 * \prod\limits_{p \in A} + p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \\ + &= \prod\limits_{p \in P'} p^{(min(a_p,b_p)-1)(max(a_p,b_p)-1)} (p-1)^2 * \prod\limits_{p \in A} + p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \\ + &= \prod\limits_{p \in P'} p^{min(a_p,b_p)-1} (p-q) * \left( + \prod\limits_{p \in P'} p^{max(a_p,b_p)-1} (p-1) * \prod\limits_{p \in A} + p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \right) \\ + &= \varphi(ggT(a,b)) * \varphi(kgV(a,b)) \\ + \\ + \varphi(a * b) &= \varphi(a) * \varphi(b) * \frac{c}{\varphi(c)} \\ + &= \varphi(ggT(a,b)) * \varphi(kgV(a,b)) * \frac{ggT(a,b)}{\varphi(ggT(a,b))} \\ + &= \varphi(kgV(a,b)) * ggT(a,b) \\ + &= \prod\limits_{p \in P'} p^{max(a_p,b_p)-1} (p-1) * \prod\limits_{p \in A} + p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) * \prod\limits_{p \in P'} + p^{min(a_p,b_p)} \\ + &= \prod\limits_{p \in P'} p^{(a_p+b_p-1)} (p-1) * \prod\limits_{p \in A} p^{a_p-1} (p-1) * + \prod\limits_{p \in B} p^{b_p-1} (p-1) \\ + &= \varphi(a*b) + \end{align*} + + q.e.d. + +\end{enumerate} + + +\end{document} diff --git a/school/di-ma/uebung/11/11_2.tex b/school/di-ma/uebung/11/11_2.tex new file mode 100644 index 0000000..77f70b9 --- /dev/null +++ b/school/di-ma/uebung/11/11_2.tex @@ -0,0 +1,66 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{cancel} +\usepackage{wasysym} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {11} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 11.2} + +Sei $p$ prim. + +\begin{align*} + f(x) = x^p - x \in \mathbb{F}_p[x] +\end{align*} + +Nullstellen berechnen: + +\begin{align*} + x^p - 1 &\equiv 0 &\mod p \\ + x (x^{p-1}-1) &\equiv 0 &\mod p \\ + \Rightarrow (x \equiv 0 \mod p) \lor (x^{p-1} \equiv 1 \mod p) +\end{align*} + +Gilt wegen Satz von Fermat $\forall x \in \{1,...,p-1\}$ + +$\Rightarrow$ $\forall x \in \mathbb{F}_p$ gilt: $x$ ist Nullstelle von $x^p - x$ + +\end{document} diff --git a/school/di-ma/uebung/11/11_3.tex b/school/di-ma/uebung/11/11_3.tex new file mode 100644 index 0000000..312257f --- /dev/null +++ b/school/di-ma/uebung/11/11_3.tex @@ -0,0 +1,90 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{cancel} +\usepackage{wasysym} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {11} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 11.3} + +\begin{enumerate}[1.] + + \item + \begin{alignat*}{6} + (2&x^3 && + 1&) / (3x^2 + x + 1) = \frac{2}{3}x - \frac{2}{9} \\ + -(2&x^3 + \frac{2}{3}&x^2 + \frac{2}{3}&x) \\ \cline{1-4} + (& -\frac{2}{3}&x^2 - \frac{2}{3}&x + 1) \\ + -(& -\frac{2}{3}&x^2 - \frac{2}{9}&x - \frac{2}{9}) \\\cline{1-4} + (-\frac{4}{9}&x + \frac{11}{9}) + \end{alignat*} + + Quotient: $\frac{2}{3}x - \frac{2}{9}$ + + Rest: $- \frac{4}{9}x + \frac{11}{9}$ + + \item + \begin{align*} + f(x) = 8x + 11 \\ + g(x) = 12x^2 + 9x + 11 \\ + f,g \in \mathbb{F}_{17}[x] + \end{align*} + + \begin{alignat*}{3} + (12&x^2 + 9&x + 11&) / (8x+11) = 10x + 15 \\ + -(12&x^2 + 8x &) \\ \cline{1-3} + && x + 11& \\ + - (&& x + 12&) \\ \cline{1-3} + && 16& + \\ + \\ + \\ + 8&x + 11& / (16) = 9x + 6 \\ + -(8&x) & \\ \cline{1-2} + & 11& \\ + -(&11) \\ \cline{1-2} + 0 + \end{alignat*} + + $\Rightarrow$ $ggT(f,g) = 16 = 1 * (12x^2 + 9x + 11) - (8x+11)(10x+15)$ + +\end{enumerate} + +\end{document}