From bff3b9eef2fb51aa65d3e5c711a6d0788988402b Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 23 Apr 2021 16:34:54 +0200 Subject: [PATCH] large update --- .gitignore | 3 + lecture.cls | 2 +- presentation.cls | 265 +++++ sose2021/algebra/alg1.xopp | Bin 0 -> 535336 bytes sose2021/funktheo/blatt1.xopp | Bin 0 -> 602197 bytes sose2021/seminar/folien.pdf | Bin 0 -> 282835 bytes sose2021/seminar/notizen.xopp | Bin 0 -> 446247 bytes sose2021/seminar/output.pdf | Bin 0 -> 972439 bytes sose2021/seminar/output_reduced.pdf | Bin 0 -> 761948 bytes sose2021/seminar/output_reduced.tex | 1030 ++++++++++++++++++++ sose2021/seminar/serre_p_adic_numbers.xopp | Bin 0 -> 1690435 bytes sose2021/seminar/skript.pdf | Bin 0 -> 229558 bytes sose2021/seminar/skript_short.pdf | Bin 0 -> 197822 bytes sose2021/seminar/skript_short.tex | 742 ++++++++++++++ sose2021/seminar/tobeamer.py | 13 + sose2021/tut-ana/praesenz-1.xopp | Bin 0 -> 1018834 bytes ws2020/algebra/lectures/notes.xopp | Bin 231661 -> 931389 bytes ws2020/algebra/lectures/notes2.xopp | Bin 0 -> 688399 bytes ws2020/theo/uebungen/vl19.pdf | Bin 105269 -> 105292 bytes ws2020/theo/uebungen/vl19.tex | 2 +- ws2020/theo/uebungen/vl22.xopp | Bin 0 -> 326468 bytes ws2020/theo/uebungen/vl23.xopp | Bin 0 -> 237105 bytes ws2020/theo/uebungen/vl24.xopp | Bin 0 -> 370727 bytes ws2020/tut-ana/praesenz12.xopp | Bin 709910 -> 610538 bytes ws2020/tut-ana/praesenz13.xopp | Bin 0 -> 873336 bytes ws2020/tut-ana/praesenz14.xopp | Bin 0 -> 1099921 bytes ws2020/wtheo/uebungen | 2 +- 27 files changed, 2056 insertions(+), 3 deletions(-) create mode 100644 presentation.cls create mode 100644 sose2021/algebra/alg1.xopp create mode 100644 sose2021/funktheo/blatt1.xopp create mode 100644 sose2021/seminar/folien.pdf create mode 100644 sose2021/seminar/notizen.xopp create mode 100644 sose2021/seminar/output.pdf create mode 100644 sose2021/seminar/output_reduced.pdf create mode 100644 sose2021/seminar/output_reduced.tex create mode 100644 sose2021/seminar/serre_p_adic_numbers.xopp create mode 100644 sose2021/seminar/skript.pdf create mode 100644 sose2021/seminar/skript_short.pdf create mode 100644 sose2021/seminar/skript_short.tex create mode 100644 sose2021/seminar/tobeamer.py create mode 100644 sose2021/tut-ana/praesenz-1.xopp create mode 100644 ws2020/algebra/lectures/notes2.xopp create mode 100644 ws2020/theo/uebungen/vl22.xopp create mode 100644 ws2020/theo/uebungen/vl23.xopp create mode 100644 ws2020/theo/uebungen/vl24.xopp create mode 100644 ws2020/tut-ana/praesenz13.xopp create mode 100644 ws2020/tut-ana/praesenz14.xopp diff --git a/.gitignore b/.gitignore index 518718e..44d02ef 100644 --- a/.gitignore +++ b/.gitignore @@ -12,3 +12,6 @@ *.dat *.autosave.xopp *.xopp~ +*.nav +*.out +*.snm diff --git a/lecture.cls b/lecture.cls index b035bff..a1f08c6 100644 --- a/lecture.cls +++ b/lecture.cls @@ -63,7 +63,7 @@ \theoremstyle{definition} \newmdtheoremenv{satz}{Satz}[section] \newmdtheoremenv{lemma}[satz]{Lemma} -\newmdtheoremenv{korrolar}[satz]{Korrolar} +\newmdtheoremenv{korollar}[satz]{Korollar} \newmdtheoremenv{definition}[satz]{Definition} \newtheorem{bsp}[satz]{Beispiel} diff --git a/presentation.cls b/presentation.cls new file mode 100644 index 0000000..2ab1f07 --- /dev/null +++ b/presentation.cls @@ -0,0 +1,265 @@ +\ProvidesClass{presentation} +\LoadClass[notheorems]{beamer} + +\RequirePackage[utf8]{inputenc} +\RequirePackage[T1]{fontenc} +\RequirePackage{textcomp} +\RequirePackage[german]{babel} +\RequirePackage{amsmath, amssymb, amsthm} +\RequirePackage{mdframed} +\RequirePackage{fancyhdr} +\RequirePackage{geometry} +\RequirePackage{import} +\RequirePackage{pdfpages} +\RequirePackage{transparent} +\RequirePackage{xcolor} +\RequirePackage{array} +\RequirePackage{tikz} +\RequirePackage{pgfplots} +%\RequirePackage[nobottomtitles]{titlesec} +\RequirePackage{listings} +\RequirePackage{mathtools} +\RequirePackage{forloop} +\RequirePackage{totcount} +\RequirePackage{calc} +\RequirePackage{wasysym} +\RequirePackage{environ} + +\usetikzlibrary{quotes, angles} +\pgfplotsset{ + compat=1.15, + default 2d plot/.style={% + grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2} + }, +} + +% beamer specific proof envs without title or qed symbol + +% proof beginning (suppressing qed symbol) +\NewEnviron{proofb} +{ + \begin{proof} + \renewcommand{\qedsymbol}{} + \BODY + \end{proof} +} + +% proof intermediate (suppressing both) +\NewEnviron{proofi} +{ + \renewcommand{\proofname}{\hskip-\labelsep\spacefactor3000 } + \begin{proof} + \renewcommand{\qedsymbol}{} + \BODY + \end{proof} +} + +% proof end (suppressing title) +\NewEnviron{proofe} +{ + \renewcommand{\proofname}{\hskip-\labelsep\spacefactor3000 } + \begin{proof} + \BODY + \end{proof} +} + + +% PARAGRAPH no indent but skip +\setlength{\parskip}{3mm} +\setlength{\parindent}{0mm} + +\theoremstyle{plain} +\newtheorem{satz}{Satz}[section] +\newtheorem{lemma}[satz]{Lemma} +\newtheorem{korollar}[satz]{Korollar} + +\theoremstyle{definition} +\newtheorem{definition}[satz]{Definition} + +\theoremstyle{remark} +\newtheorem{bsp}[satz]{Beispiel} +\newtheorem{bem}[satz]{Bemerkung} +\newtheorem{aufgabe}{Aufgabe} + +% enable aufgaben counting +%\regtotcounter{aufgabe} + +% temporary calculation counter +\newcounter{var} + +\newcommand{\N}{\mathbb{N}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\C}{\mathbb{C}} + +% HEADERS + +%\pagestyle{fancy} + +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} +\pdfsuppresswarningpagegroup=1 + +% horizontal rule +\newcommand\hr{ + \noindent\rule[0.5ex]{\linewidth}{0.5pt} +} + +% punkte tabelle +%\newcommand{\punkte}[1][1]{ +% \newcounter{k} +% \setcounter{k}{#1} +% \@punkten{\value{k}}{\totvalue{aufgabe}} +% \setcounter{k}{#1-1} +% \setcounter{aufgabe}{\value{k}} +% \vspace{5mm} +%} +% +%\def\@punkten#1#2{ +% \newcounter{n} +% % create a temporary calculation counter +% \setcounter{var}{#2-#1+1} +% \begin{tabular}{|c|*{\value{var}}{m{1cm}|}m{1cm}|@{}m{0cm}@{}} +% \hline +% Aufgabe +% \forloop{n}{#1}{\not{\value{n} > #2}}{ +% & \centering A\then +% } +% & \centering $\sum$ & \\[5mm] \hline +% Punkte +% \forloop{n}{#1}{\not{\value{n} > #2}}{ +% & +% } +% & & \\[5mm] \hline +% \end{tabular} +%} + +% code listings, define style +\lstdefinestyle{mystyle}{ + commentstyle=\color{gray}, + keywordstyle=\color{blue}, + numberstyle=\tiny\color{gray}, + stringstyle=\color{black}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +% activate my colour style +\lstset{style=mystyle} + +% better stackrel +\let\oldstackrel\stackrel +\renewcommand{\stackrel}[2]{% + \oldstackrel{\mathclap{#1}}{#2} +}% + +% integral d sign +\makeatletter \renewcommand\d[2][]{\ensuremath{% + \,\mathrm{d}^{#1}#2\@ifnextchar^{}{\@ifnextchar\d{}{\,}}}} +\makeatother + +% contradiction +\newcommand{\contr}{\text{\Large\lightning}} + +% disjoint unions: provides cupdot and bigcupdot +\makeatletter +\def\moverlay{\mathpalette\mov@rlay} +\def\mov@rlay#1#2{\leavevmode\vtop{% + \baselineskip\z@skip \lineskiplimit-\maxdimen + \ialign{\hfil$\m@th#1##$\hfil\cr#2\crcr}}} +\newcommand{\charfusion}[3][\mathord]{ + #1{\ifx#1\mathop\vphantom{#2}\fi + \mathpalette\mov@rlay{#2\cr#3} + } + \ifx#1\mathop\expandafter\displaylimits\fi} +\makeatother + +\newcommand{\cupdot}{\charfusion[\mathbin]{\cup}{\cdot}} +\newcommand{\bigcupdot}{\charfusion[\mathop]{\bigcup}{\cdot}} + +\ExplSyntaxOn + +% S-tackrelcompatible ALIGN environment +% some might also call it the S-uper ALIGN environment +% uses regular expressions to calculate the widest stackrel +% to put additional padding on both sides of relation symbols +\NewEnviron{salign} +{ + \begin{align} + \lec_insert_padding:V \BODY + \end{align} +} +% starred version that does no equation numbering +\NewEnviron{salign*} +{ + \begin{align*} + \lec_insert_padding:V \BODY + \end{align*} +} + +% some helper variables +\tl_new:N \l__lec_text_tl +\seq_new:N \l_lec_stackrels_seq +\int_new:N \l_stackrel_count_int +\int_new:N \l_idx_int +\box_new:N \l_tmp_box +\dim_new:N \l_tmp_dim_a +\dim_new:N \l_tmp_dim_b +\dim_new:N \l_tmp_dim_needed + +% function to insert padding according to widest stackrel +\cs_new_protected:Nn \lec_insert_padding:n + { + \tl_set:Nn \l__lec_text_tl { #1 } + % get all stackrels in this align environment + \regex_extract_all:nnN { \c{stackrel}{(.*?)}{(.*?)} } { #1 } \l_lec_stackrels_seq + % get number of stackrels + \int_set:Nn \l_stackrel_count_int { \seq_count:N \l_lec_stackrels_seq } + \int_set:Nn \l_idx_int { 1 } + \dim_set:Nn \l_tmp_dim_needed { 0pt } + % iterate over stackrels + \int_while_do:nn { \l_idx_int <= \l_stackrel_count_int } + { + % calculate width of text + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 1 }$} + \dim_set:Nn \l_tmp_dim_a {\box_wd:N \l_tmp_box} + % calculate width of relation symbol + \hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 2 }$} + \dim_set:Nn \l_tmp_dim_b {\box_wd:N \l_tmp_box} + % check if 0.5*(a-b) > minimum padding, if yes updated minimum padding + \dim_compare:nNnTF + { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } > { \l_tmp_dim_needed } + { \dim_set:Nn \l_tmp_dim_needed { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } } + { } + \quad + % increment list index by three, as every stackrel produces three list entries + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + \int_incr:N \l_idx_int + } + % replace all relations with align characters (&) and add the needed padding + \regex_replace_all:nnN + { 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a/sose2021/seminar/skript_short.tex b/sose2021/seminar/skript_short.tex new file mode 100644 index 0000000..755aa5f --- /dev/null +++ b/sose2021/seminar/skript_short.tex @@ -0,0 +1,742 @@ +\documentclass{../../lecture} + +\usepackage[]{tikz-cd} + +\begin{document} + +\stepcounter{section} +\section{p-adische Zahlen} + +\subsection{Der Ring $\Z_p$ und sein Quotientenkörper $\Q_p$} + +Sei $p \in \N$ eine im Folgenden fest gewählte Primzahl. + +Genauso wie eine natürliche Zahl $m \in \N$ bezüglich der Basis $10$ dargestellt werden kann, +ist das auch bezüglich der Basis $p$ möglich. Jede natürliche Zahl besitzt also eine +$p$-adische Entwicklung der Form +\[ +m = a_0 + a_1 p + \ldots + a_n p^{n} +\] wobei die Koeffizienten $a_i$ in $\{0, 1, \ldots, p-1\} $ liegen. Die Darstellung ist damit eindeutig. + +\begin{bsp}[] + Diese Darstellung finden wir durch sukzessives Dividieren mit Rest. Für $n = 216$ erhalten + wir für $p = 5$ + \begin{salign*} + 216 &= 1 + 3 \cdot 5 + 3 \cdot 5^2 + 1 \cdot 5^3 + .\end{salign*} +\end{bsp} + +Um nun auch negative und sogar gebrochene Zahlen darstellen zu können, gehen wir zu unendlichen +Reihen über, wir betrachten also Objekte der Form +\[ +\sum_{i=0}^{\infty} a_i p^{i} = a_0 + a_1 p + a_2 p^2 + \ldots +\] mit $0 \le a_i < p$ für $i \in \N_0$. + +\begin{bem}[] + $\sum_{i=0}^{\infty} a_i p^{i}$ ist rein formal gemeint, d.h. bezeichnet einfach + die Folge der Partialsummen + \[ + s_n = \sum_{i=0}^{n-1} a_i p^{i} \in \Z, \quad n \in \N + .\] +\end{bem} + +Um nun die ganzen $p$-adischen Zahlen zu definieren, betrachten wir +die Folgen der Restklassen +\[ +\overline{s}_{n} = s_n \; \text{mod } p^{n} \in \Z / p^{n} \Z +.\] +Zwischen den Ringen $\Z / p^{n} \Z$ existieren kanonische Projektionen +\begin{salign*} + \phi_{n}\colon \Z / p^{n+1}\Z &\to \Z / p^{n} \Z \\ + \overline{a} &\mapsto a \; \text{mod } p^{n} +,\end{salign*} d.h. es entsteht eine Folge +\[ + \Z / p \Z \xleftarrow{\phi_1} \Z / p^2 \Z \xleftarrow{\phi_2} \Z / p^{3} \Z \xleftarrow{\phi_3} \ldots +.\] Ein solches System wird projektives System genannt, genauer: + +\begin{definition} + Ein projektives System ist + eine Folge von Mengen $(D_n)_{n \in \N}$ und eine Folge + von Abbildungen $(p_n)_{n \in \N}$ mit $p_{n}\colon D_{n+1} \to D_n$ + \[ + D_1 \xleftarrow{p_1} D_2 \leftarrow \ldots \leftarrow D_{n} \xleftarrow{p_{n}} D_{n+1} \leftarrow \ldots + .\] + Die Teilmenge + \[ + D = \varprojlim \; (D_n, p_n) = + \left\{ (a_n)_{n \in \N} \in \prod_{n=1}^{\infty} D_n \mid p_n(a_{n+1}) = a_n \forall n \in \N \right\} + \] heißt projektiver Limes des Systems. +\end{definition} + +\begin{bem}[] + Falls die $D_n$ Ringe und die $p_n$ Ringhomomorphismen sind, wird + $\varprojlim \; (D_n, p_n)$ zum Teilring des Produktrings $\prod_{n=1}^{\infty} D_n $ + (leicht nachzurechnen). +\end{bem} + +\begin{definition}[Ganze $p$-adische Zahlen] + Der projektive Limes des Systems $(\Z / p^{n} \Z, \phi_n)$ + \[ + \Z_p \coloneqq \varprojlim \; (\Z / p^{n} \Z, \phi_n) + \] heißt der Ring der ganzen $p$-adischen Zahlen. +\end{definition} + +Notation: Setze im Folgenden $A_n \coloneqq \Z / p^{n} \Z$. Außerdem bezeichne +$\pi_n\colon \Z_p \to A_n$ die kanonische Projektion. + +%\begin{definition}[Ganze $p$-adische Zahlen] +% Eine ganze $p$-adische Zahl ist eine formale unendliche Reihe +% \[ +% \sum_{i=0}^{\infty} a_i p^{i} = a_0 + a_1 p + a_2 p^2 + \ldots +% \] mit $0 \le a_i < p$ für $i \in \N_0$. Die Menge dieser formalen Reihen +% wird mit $\Z_p$ bezeichnet. +%\end{definition} + +%Womit können wir jetzt die ganzen Zahlen $\Z$ in den $p$-adischen Zahlen $\Z_p$ identifizieren? +%Wie kann +%also beispielsweise $-1$ in $\Z_p$ dargestellt werden? +%Dazu stellen wir folgendes fest +% +%\begin{lemma} +% Sei $a \in \Z$. +% Die Restklasse $a \; \text{mod } p^{n} \in \Z / p^{n} \Z$ wird in eindeutiger +% Darstellung durch +% \[ +% a \equiv a_0 + a_1 p + a_2 p^2 + \ldots + a_{n-1} p^{n-1} \; (\text{mod } p^{n}) +% \] gegeben, wobei $0 \le a_i < p$ für $i \in \{0, \ldots, n-1\} $. +% \label{le-eind-rest} +%\end{lemma} +% +%\begin{proof} +% Per Induktion. Für $n = 1$ ist offenbar $a \equiv a_0 \; (\text{mod } p) $ mit $0 \le a_0 < p$. +% Sei nun die Behauptung für $n-1$ gezeigt. Dann ex. eine eindeutige Darstellung +% \begin{salign*} +% a &\equiv a_0 + a_1 p + a_2 p^2 + \ldots + a_{n-2} p^{n-2} \; (\text{mod } p^{n-1}) +% \intertext{Also} +% a &= a_0 + a_1 p + a_2 p^2 + \ldots + a_{n-2} p^{n-2} + g p^{n-1} +% \intertext{ +% für ein $g \in \Z$. Sei $0 \le a_{n-1} < p$, s.d. $g \equiv a_{n-1} \; (\text{mod } p) $, also +% $g = a_{n-1} + h p$ für $h \in \Z$. $a_{n-1}$ ist also eindeutig durch $a$ bestimmt und es +% folgt +% } +% a &= a_0 + \ldots + a_{n-2} p^{n-2} + a_{n-1} p^{n-1} + h p^{n} +% \end{salign*} +%\end{proof} +% +%Jede ganze Zahl $a$ definiert nun eine Folge von Restklassen $\overline{s_n} = \overline{a} \in \Z / p^{n} \Z$ +%für $n \in \N$, die nach \ref{le-eind-rest} von der Gestalt +%\begin{salign*} +% s_1 &\equiv a_0 \; (\text{mod } p) \\ +%s_2 &\equiv a_0 + a_1p \; (\text{mod } p^2) \\ +%&\;\;\vdots +%\end{salign*} +%sind mit eindeutig bestimmten Koeffizienten $a_0, a_1, \ldots \in \{0, \ldots, p-1\} $. Die Zahlenfolge +%\[ +%s_n = a_0 + a_1 p + a_2 p^2 + \ldots + a_{n-1} p^{n-1} +%\] definiert nun eine ganze $p$-adische Zahl $\sum_{i=0}^{\infty} a_i p^{i} \in \Z_p$, die +%wir die $p$-adische Entwicklung von $a$ nennen. +% +%\begin{bsp} +% Was ist jetzt die $p$-adische Entwicklung von $-1$? Es ist +% \begin{salign*} +% -1 &= (p-1) + (p-1)p + \ldots + (p-1)p^{n-1} - p^{n} \\ +% \text{also }-1 &\equiv (p-1) + (p-1)p + \ldots + (p-1)p^{n-1} \; (\text{mod } p^{n}) +% .\end{salign*} +% Es ist also $-1 = (p-1) + (p-1)p + (p-1)p^2 + \ldots \in \Z_p$ die $p$-adische Entwicklung von $-1$. +% \label{bsp-minus1} +%\end{bsp} + +%Um $\Z_p$ eine algebraische Struktur zu geben, könnten wir mit diesen Reihen mit Überträgen +%rechnen, so wie wir es von der Basis $10$ gewohnt sind. Einfacher wird es jedoch, wenn wir +%eine ganze $p$-adische Zahl $x = \sum_{i=0}^{\infty} a_i p^{i}$ mit +%der Folge der Restklassen $\overline{s_n} \in \Z / p^{n} \Z$ der Partialsummen identifizieren. +% +%Dazu benötigen wir noch eine Vorüberlegung und einige Begriffe. +% +%\begin{definition} +% Ein projektives System ist +% eine Folge von Mengen $(D_n)_{n \in \N}$ und eine Folge +% von Abbildungen $(p_n)_{n \in \N}$ mit $p_{n}\colon D_{n+1} \to D_n$ +% \[ +% D_1 \xleftarrow{p_1} D_2 \leftarrow \ldots \leftarrow D_{n} \xleftarrow{p_{n}} D_{n+1} \leftarrow \ldots +% .\] +% Die Teilmenge +% \[ +% D = \varprojlim \; (D_n, p_n) = +% \left\{ (a_n)_{n \in \N} \in \prod_{n=1}^{\infty} D_n \mid p_n(a_{n+1}) = a_n \forall n \in \N \right\} +% \] heißt projektiver Limes des Systems. +%\end{definition} +% +%\begin{bem}[] +% Falls die $D_n$ Ringe und die $p_n$ Ringhomomorphismen sind, wird +% $\varprojlim \; (D_n, p_n)$ zum Teilring des Produktrings $\prod_{n=1}^{\infty} D_n $ +% (leicht nachzurechnen). +%\end{bem} + + +%\begin{satz} +% Ordnet man jeder ganzen $p$-adischen Zahl +% \[ +% x = \sum_{i=0}^{\infty} a_i p^{i} +% \] die Folge $(\overline{s}_n)_{n \in \N}$ der Restklassen +% \[ +% \overline{s}_n = \sum_{i=0}^{n-1} a_i p^{i} \; (\text{mod } p^{n}) \in A_n +% \] zu, so erhält man eine Bijektion +% \[ +% \Z_p \xrightarrow{\sim} \varprojlim \; (A_n, \phi_n) +% .\] +%\end{satz} +% +%\begin{proof} +% Die Zuordnung ist wohldefiniert, da +% \[ +% s_{n+1} = a_0 + a_1 p + \ldots + a_{n} p^{n} +% \equiv a_0 + a_1 p + \ldots + a_{n-1} p^{n-1} \; (\text{mod } p^{n}) +% = s_n +% .\] Die Bijektivität folgt direkt aus \ref{le-eind-rest} +%\end{proof} + +%\begin{bem}[] +% Die Koeffizienten der $p$-adischen Entwicklung von $a \in \Z$ ergeben sich durch die Kongruenzen +% \[ +% a \equiv a_0 + a_1 p + \ldots + a_{n-1} p^{n-1} \; (\text{mod } p^{n}) +% \] mit $0 \le a_i < p$. Bei der Identifizierung $\Z_p = \varprojlim \; (A_n, \phi_n)$ geht +% $a \in \Z$ daher über in +% \[ +% (a \; \text{mod } p, a \; \text{mod } p^2, a \; \text{mod } p^{3} , \ldots ) \in +% \prod_{n=1}^{\infty} A_n +% .\] +% $\Z$ wird so zum Teilring von $\varprojlim \; (A_n, \phi_n)$. +% \label{bem-z-ident} +%\end{bem} +% +%\begin{bsp}[$\ref{bsp-minus1}$ fortgesetzt] +% Mit \ref{bem-z-ident} folgt also +% \[ +% -1 = (p-1, p^2-1, p^{3}-1, \ldots) \in \varprojlim \; (A_n, \phi_n) +% .\] +%\end{bsp} + +%Wir identifizieren nun im Folgenden stets $\Z_p = \varprojlim \; (A_n, \phi_n)$. $\pi_n$ bezeichne +%den kanonischen Projektionshomomorphismus $\pi_n\colon \Z_p \to A_n$. + +\begin{bem} + \begin{enumerate} + \item Per Definition ist $x \in \Z_p$ also ein Element $x \in \prod_{n=1}^{\infty} \Z / p^{n}\Z $ + mit der ,,Kompatibilitätsbedingung'': + \[ + x_{n+1} \equiv x_n \text{ (mod } p^{n}) + .\] + \item Die Inklusion + \[ + \Z \hookrightarrow \Z_p, a \mapsto (a \; \text{mod } p, a \; \text{mod } p^2, \ldots) + \] ist ein injektiver Ringhomomorphismus. Damit wird $\Z$ zum Teilring von $\Z_p$. + \item $\Z_p$ erbt als Teilring nun also die komponentenweise Addition + und Multiplikation des Produktrings + $\prod_{n=1}^{\infty} A_n $, d.h. für $(a_n)_{n \in \N}, (b_n)_{n \in \N} \in \Z_p$ gilt + \[ + (a_n)_{n \in \N} + (b_n)_{n \in \N} = (a_n + b_n)_{n \in \N} + \quad + (a_n)_{n \in \N} \cdot (b_n)_{n \in \N} = (a_n \cdot b_n)_{n \in \N} + .\] + %\item Versieht man $A_n$ mit der diskreten Topologie (d.h. alle Teilmengen sind offen) + % und $\prod_{n=1}^{\infty} A_n $ mit der Produkttopologie (die von den Urbildern der + % kanonischen Projektionen $\pi_n$ erzeugt wird), wird $\Z_p$ zu einem + % topologischen Ring. + \end{enumerate} +\end{bem} + +%\begin{satz}[von Tychonoff] +% Ist $(X_i)_{i \in I}$ eine Familie kompakter topologischer Räume, dann ist +% auch das kartesische Produkt $\prod_{i \in I}^{} X_i $ kompakt bezüglich der +% Produkttopologie. +% \label{satz-tycho} +%\end{satz} +% +%\begin{proof} +% Der Satz ist äquivalent zum Auswahlaxiom. Ein Beweis findet sich beispielsweise +% in Klaus Jänich: \textit{Topologie}. +%\end{proof} +% +%\begin{korollar}[] +% $\Z_p$ ist kompakt. \label{kor-compact} +%\end{korollar} +% +%\begin{proof} +% Nach \ref{satz-tycho} ist $\prod_{n=1}^{\infty} A_n $ kompakt. Außerdem ist +% \[ +% \Z_p = \bigcap_{n \in \N} +% \left\{ x \in \prod_{n=1}^{\infty} A_n \mid \phi_{n}(\pi_{n+1}(x)) = \pi_n(x)\right\} +% = \bigcap_{n \in \N} f_n^{-1}(\{0\}) +% \] mit $f_n \colon \prod_{n=1}^{\infty} A_n \to A_n, x \mapsto \phi_{n}(\pi_{n+1}(x)) - \pi_n(x)$. +% Es ist $f_n$ stetig, da $\pi_n$ per Definition der Produkttopologie und $\phi_n$ als +% Abbildung zwischen diskreten Räumen stetig sind. Da $\{0\} \subseteq A_n$ abgeschlossen, folgt +% die Behauptung. +%\end{proof} + +\begin{lemma} + Es ist $\pi_n$ surjektiv und $\text{ker } \pi_n = p^{n} \Z_p$ $\forall n \in \N$. + Insbesondere gilt + \[ + \Z_p / p^{n} \Z_p \stackrel{\sim }{=} \Z / p^{n} \Z = A_n + .\] + \label{le-kanproj} +\end{lemma} +\begin{proof} + Die Surjektivität ist klar. Z.z.: $\text{ker } \pi_n = p^{n} \Z_p$. Sei dazu $x \in \Z_p$. + Dann ist $p^{n} x_n \equiv 0 \; (\text{mod } p^n)$. Also $\pi_n(p^n x) = 0$. Damit + $p^{n} \Z_p \subseteq \text{ker } \pi_n$. + + Sei nun $x = (x_m)_{m \in \N} \in \text{ker } \pi_n$ + und $m \ge n$. + Wegen Kompatibilität folgt + \[ + x_m \equiv x_n \; (\text{mod } p^{n}) \equiv 0 \; (\text{mod } p^{n}) + .\] Also folgt $x_m \in p^{n} A_m$. + + Es ist (nachrechnen) + \begin{salign*} + A_{m-n} = \Z / p^{m-n} \Z \stackrel{\sim }{=} p^{n} \Z / p^{m} \Z = p^{n}A_m + .\end{salign*} + Das heißt es ex. ein eindeutiges $y_{m-n} \in A_{m-n}$, s.d. + $p^{n}y_{m-n} \equiv x_m \; (\text{mod } p^{m}) $. + Es bleibt zu zeigen, dass $p^{n}y = x$. + + Z.z.: $x = p^{n} y$. Für $m \le n$ ist $x_m = 0 = p^{n} y_m$. Für $m > n$ ist wegen Kompatibilität + \[ + p^{n} y_m = p^{n} y_{m+n-n} \equiv x_{m+n} \; (\text{mod } p^{m+1}) \equiv x_m \; (\text{mod } p^{m}) + .\] Also $x = p^{n}y$. + Insgesamt folgt also $\text{ker } \pi_n = p^{n}\Z_p$. Die behauptete Isomorphie folgt jetzt direkt aus + dem Homomorphiesatz. +\end{proof} + +\begin{lemma}[] + Für $u \in \Z_p$ sind äquivalent + \begin{enumerate}[(i)] + \item $u \in \Z_p^{\times }$ + \item $p \nmid u$ + \item $0 \neq u_1 \in \Z / p \Z$ + \end{enumerate} +\end{lemma} +\begin{proof} + (ii)$\iff$(iii) ist klar wegen Kompatibilität. b.z.z. (i) $\iff$ (ii). Sei dazu + $u = (\overline{u_n})_{n\in \N} \in \Z_p^{\times }$. + Dann $\exists v = (\overline{v_n})_{n \in \N} \in \Z_p$ + mit $uv = 1$ insb. $\overline{u_1 v_1} \equiv 1 \; (\text{mod } p) $ also + insbesondere $p \nmid u_1 \implies \overline{u_1} \neq 0$. + + Sei umgekehrt $\overline{u_1} \neq 0$. Wegen Kompatibilität folgt damit $p \nmid u_n$ $\forall n \in \N$, denn + ang. $p \mid u_n$ für ein $n \in \N$. Dann folgt + \[ + 0 \equiv u_n \; (\text{mod } p) \equiv u_1 \; (\text{mod } p) \quad \contr + .\] + Da $p$ prim folgt insbesondere $(p^{n}, u_n) = 1$. Also ex. nach euklid. Alg. $a, b \in \Z$, s.d. + $1 = a p^{n} + b u_n$, also $1 = \overline{bu_n}$ mit $\overline{b} \in A_n$. Also + $\overline{u_n} \in A_n^{\times }$ und damit + $v \coloneqq (\ldots \overline{u_n}^{-1}, \overline{u_{n-1}}^{-1}, \ldots, \overline{u_1}^{-1}) = u^{-1} + \in \Z_p$. +\end{proof} + +\begin{lemma}[] + Für $x \in \Z_p \setminus \{0\} $ ex. $n \in \N_0$ und $u \in \Z_p^{\times }$, s.d. + \[ + x = p^{n} u + .\] Diese Darstellung ist eindeutig. + \label{le-decomp} +\end{lemma} +\begin{proof} + \begin{enumerate}[(i)] + \item Existenz: + Sei $x \in \Z_p \setminus \{0\} $. Da $x \neq 0$ ex. wegen Kompatibilität ein + $n \in \N_0$ maximal, s.d. + $x_n = \pi_n(x) = 0$. Also ist $x \in \text{ker } \pi_n$, insbesondere ex. nach + \ref{le-kanproj} ein $u \in \Z_p$ mit $x = p^{n}u$. Ang.: $p \mid u$, dann + ist $\pi_1(u) = 0$ also ex. wieder nach \ref{le-kanproj} ein $v \in \Z_p$ mit $u = pv$. Dann + ist aber + \[ + \pi_{n+1}(x) = \pi_{n+1}(p^{n}u) = \pi_{n+1}(p^{n+1}v) = 0 + .\] Widerspruch zur Maximalität von $n$. + \item Eindeutigkeit: Sei $x = p^{n} u = p^{m} v$ mit $u, v \in \Z_p^{\times }$ und $n, m \in \N_0$. + Sei o.E. $n \ge m$. Es ist $\pi_n(x) = \pi_n(p^{n}) \pi_n(u) = 0$ also + auch $0 = \pi_n(x) = \pi_n(p^{m}) \pi_{n}(v)$. Da $v \in \Z_p^{\times }$ ist + $\pi_n(v) \in A_n^{\times}$, also kein Nullteiler. Also folgt + $\pi_n(p^{m}) = 0$ und damit $p^{m} \equiv 0 \; (\text{mod } p^{n}) $, also + $m \ge n$. Insgesamt also $m = n$. + + Nun gilt weiter $x = p^{n} u = p^{n} v$, also $p^{n}(u-v) = 0$. Ang. $u-v \neq 0$. Dann + ist nach (i) $u - v = p^{k} w$ mit $k \in \N_0$ und $w \in \Z_p^{\times }$. Also + $0 = p^{n}(u-v) = p^{n+k} w$. Da $w \in \Z_p^{\times }$ also kein Nullteiler, folgt + $0 = p^{n+k} \in \Z$ $\contr$. + \end{enumerate} +\end{proof} + +\begin{definition}[$p$-Bewertung] + Für $x \in \Z_p \setminus \{0\} $ sei $x = p^{n} u$ mit $u \in \Z_p^{\times }$. Dann setze + \[ + v_p(x) \coloneqq n + \] und setze $v_p(0) \coloneqq \infty$. $v_p(x)$ heißt die $p$-Bewertung von $x$. +\end{definition} + +\begin{bem}[] + Wegen \ref{le-decomp} ist die $p$-Bewertung wohldefiniert. + Per Konvention setze $n + \infty = \infty$ und $\infty > n$ für $n \in \N_0$. Es ist + leicht nachzurechnen, dass für $x, y \in \Z_p$ gilt + \[ + v_p(xy) = v_p(x) + v_p(y), \quad v_p(x+y) \ge \min (v_p(x), v_p(y)) + .\] +\end{bem} + +%\begin{lemma}[Eigenschaften der $p$-Bewertung] +% Für $x, y \in \Z_p$ gilt +% \[ +% v_p(xy) = v_p(x) + v_p(y), \quad v_p(x+y) \ge \min (v_p(x), v_p(y)) +% .\] +%\end{lemma} +% +%\begin{proof} +% Nachrechnen. +%\end{proof} + +\begin{korollar}[] + $\Z_p$ ist nullteilerfrei. +\end{korollar} + +\begin{proof} + Seien $x, y \in \Z_p$ mit $xy = 0$. Dann folgt + \[ + \infty = v_p(0) = v_p(xy) = v_p(x) + v_p(y) + .\] Also $v_p(x) = \infty$ oder $v_p(y) = \infty$, also $x = 0$ oder $y = 0$. +\end{proof} + +%\begin{lemma}[Topologie auf $\Z_p$] +% Die Topologie auf $\Z_p$ wird induziert durch die Metrik +% \[ +% d(x, y) = \exp(-v_p(x-y)) +% .\] $\Z_p$ ist vollständig und $\Z$ ist dicht in $\Z_p$. +%\end{lemma} +% +%\begin{bem}[Bälle] +% Es sei im Folgenden stets +% \[ +% B(x, r) = \{ y \in \Z_p \mid d(x,y) < r \} \text{ und } +% \overline{B(x,r)} = \{y \in \Z_p \mid d(x,y) \le r\} +% .\] Da $d(x,y) \in \{ \exp(n) \mid n \in \Z\}$ gilt +% \[ +% \overline{B(x, e^{-n})} = \{ y \in \Z_p \mid v_p(x-y) \ge n\} +% = \{ y \in \Z_p \mid v_p(x-y) > n-1\} = B(x, e^{-(n-1)}) +% .\] +%\end{bem} +% +%\begin{proof}[Beweis des Lemmas (Skizze)] +% Grobe Beweisschritte +% \begin{itemize} +% \item $d(\cdot , \cdot )$ ist eine Metrik. +% \item Die offenen Mengen $V \subseteq \Z_p$ bezüglich der Produkttopologie sind von der Form +% \[ +% V = \bigcup_{v \in V} (v + p^{n_v} \Z_p) +% .\] +% \item Es ist $v + p^{n} \Z_p = B(v, e^{(-(n-1))})$. +% \item $v + p^{n} \Z_p$ offen bezüglich der Produkttopologie, da $p^{n} \Z_p = \text{ker } \pi_n +% = \pi_n^{-1}(\{0\})$. +% \end{itemize} +% Z.z.: $\Z_p$ vollständig. Da $\Z_p$ nach \ref{kor-compact} +% kompakt ist, hat jede Folge in $\Z_p$ eine konvergente Teilfolge. Insbesondere hat +% also jede Cauchy-Folge eine konvergente Teilfolge und damit konvergiert jede Cauchy-Folge +% in $\Z_p$. +%\end{proof} + +\begin{definition} + Der Quotientenkörper der ganzen $p$-adischen Zahlen $\Z_p$ + heißt Körper der $p$-adischen Zahlen + \[ + \Q_p \coloneqq Q(\Z_p) + .\] +\end{definition} + +\begin{bem} + \begin{enumerate}[] + \item Ein Element $x = \frac{a}{b} \in \Q_p^{\times}$ mit $a, b \in \Z_p$, $b \neq 0$ + kann eindeutig als $x = p^{r}w$ mit $r \in \Z$ und $w \in \Z_p^{\times }$ dargestellt werden, + denn nach \ref{le-decomp} ist + \[ + x = \frac{a}{b} = \frac{p^{n}u}{p^{m}v} = p^{n-m} \underbrace{u v^{-1}}_{\in \Z_p^{\times }} + .\] + Damit setzt sich die Definition von $v_p$ auf $\Q_p$ fort. Es gilt + $v_p(x) \ge 0 \iff x \in \Z_p$. + \item Nach (1) ist also $\Q_p = \Z_p[p^{-1}]$. + \end{enumerate} +\end{bem} + +%\begin{lemma}[Topologie auf $\Q_p$] +% $\Q_p$ mit der von $\Z_p$ geerbten Metrik $d(x,y) = \exp(-v_p(x-y))$ ist +% lokal kompakt und enthält $\Z_p$ als offenen Teilring. $\Q$ ist dicht in $\Q_p$. +%\end{lemma} +% +%\begin{proof} +% Da $x \in \Z_p \iff v_p(x) \ge 0 \iff v_p(x) > -1$ folgt $\Z_p = \overline{B(0, 1)} = B(0, e)$, +% also $\Z_p$ offen. +% Da $\Z_p$ kompakt, folgt, dass +% $B(x, e)$ kompakt $\forall x \in \Q_p$, also $\Q_p$ lokal kompakt. Außerdem ist +% $\Z$ dicht in $\Z_p$, d.h. für $x \in \Q_p$ mit $x = p^{k} u$ und $k \in \Z$, $u \in \Z_p^{\times }$ ex. +% eine Folge $(y_n)_{n \in \N} \subseteq \Z$ mit $y_n \xrightarrow{n \to \infty} u$. Dann +% setze $z_n \coloneqq p^{k} y_n \in \Q$. Dann folgt direkt +% $z_n = p^{k} y_n \xrightarrow{n \to \infty} p^{k} u = x$. +%\end{proof} + +\begin{bem}[] + \begin{enumerate}[] + \item $\Q_p$ kann auch als Vervollständigung von $\Q$ bezüglich der $p$-adischen + Metrik $d(\cdot , \cdot )$ definiert werden (analog zu $\R$ als + Vervollständigung von $\Q$ bezüglich $|\cdot |$). + \item Es ist leicht nachzurechnen, dass $d(\cdot , \cdot )$ die ultrametrische Ungleichung + (auch starke Dreiecksungleichung) erfüllt, d.h. + \[ + d(x, z) \le \max(d(x,y), d(y,z)) + \] für $x, y, z \in \Q_p$. Damit folgt das eine Folge + $(a_n)_{n \in \N} \subseteq \Q_p$ genau dann konvergiert, wenn + $\lim_{n \to \infty} (u_{n+1} - u_n) = 0$ (was in $\R$ bezüglich $|\cdot |$ falsch ist). + \end{enumerate} +\end{bem} + +\subsection{$p$-adische Gleichungen} + +\begin{lemma}[] + Sei $D_1 \leftarrow D_2 \leftarrow \ldots$ ein projektives System und + $D = \varprojlim \; (D_n, p_n)$ sein inverser Limes. Falls $D_n \neq \emptyset$ und endlich + folgt $D \neq \emptyset$. + \label{le-projlim} +\end{lemma} + +\begin{proof} + Sei zunächst $p_{n}\colon D_{n+1} \to D_n$ surjektiv. Dann ex. für alle $x_{n} \in D_{n}$ + ein $x_{n+1} \in D_{n+1}$, s.d. $p_{n}(x_{n+1}) = x_n$. Da $D_1 \neq \emptyset$ folgt + $D \neq \emptyset$ induktiv. + + Im Allgemeinen bezeichne für $m,n \in \N$: + \[ + D_{n,m} \coloneqq (p_{n} \circ \ldots \circ p_{n+m-1})(D_{n+m}) + .\] Da $D_{n+m} \neq \emptyset$ folgt $D_{n,m} \neq \emptyset$ und da + $D_k$ endlich folgt $\# p_k(D_{k+1}) \le \# D_{k+1}$ $\forall k \in \N$. D.h. $\#D_{n,m}$ + ist monoton fallend in $m$ bei festem $n$. + Da $D_{n,m} \neq \emptyset$ wird die Folge stationär, d.h. + es ex. ein $m_0 \in \N$, s.d. $D_{n, m_0} = D_{n, m}$ $\forall m \ge m_0$. + Sei $E_n$ dieser Grenzwert. + + Es ist leicht nachzurechnen, dass $p_n(E_{n+1}) = E_n$. + %Beh.: $p_{n}(E_{n+1}) = E_n$ $\forall n \in \N$. Sei dazu $n \in \N$. Nun ex. ein + %$m_0 \in \N$, s.d. $E_{n+1} = D_{n+1, m_0}$ und + %$E_n = D_{n, m_0} = D_{n, m_0+1}$. Damit folgt + %\begin{salign*} + % p_{n}(E_{n+1}) + % &= p_{n}(D_{n+1, m_0}) \\ + % &= p_{n}((p_{n+1} \circ \ldots \circ p_{n+m_0})(D_{n+1+m_0})) \\ + % &= (p_{n} \circ p_{n+1} \circ \ldots \circ p_{n+m_0})(D_{n+m_0+1}) \\ + % &= D_{n, m_0+1} \\ + % &= E_n + %.\end{salign*} + Also sind die Einschränkungen $p_{n}|_{E_{n+1}}\colon E_{n+1} \to E_n$ surjektiv, + $E_n \neq \emptyset$ und endlich, also + folgt nach der Vorüberlegung $\varprojlim \; (E_n, p_n|_{E_n}) \neq \emptyset$, also + insbesondere $D \neq \emptyset$. +\end{proof} + +\begin{satz}[] + Seien $f^{(i)} \in \Z_p[X_1, \ldots, X_m]$ Polynome in den ganzen $p$-adischen Zahlen. Dann + sind äquivalent: + \begin{enumerate}[(i)] + \item Die $f^{(i)}$ haben eine gemeinsame Nullstelle in $(\Z_p)^{m}$. + \item Für $n \in \N$ haben die Polynome $f^{(i)} \; (\text{mod } p^{n}) $ eine + gemeinsame Nullstelle in $(A_n)^{m}$. + \end{enumerate} + \label{satz-nsequiv} +\end{satz} + +\begin{proof} + Sei $D = \{ \text{gemeinsame NS von } f^{(i)} \text{ in } (\Z_p)^{m} \} \subseteq (\Z_p)^{m}$ + und\\ + $D_n \coloneqq \{ \text{ gemeinsame NS von } f^{(i)} \text{ in } (A_n)^{m} \; (\text{mod } p^{n}) \} $. + Es bezeichne $(\phi_{n})^m\colon (A_{n+1})^{m} \to (A_n)^{m}$ die Abbildung, die $\phi_{n}$ + komponentenweise anwendet. Dann ist $(D_n, (\phi_n)^m)$ ein projektives System + mit $D = \varprojlim \; (D_n, (\phi_n)^m)$. + + Sei nun $D \neq \emptyset$ und $x \in D$. Dann ist $\pi_n(x) \in D_n$ $\forall n \in \N$. Seien umgekehrt + $D_n \neq \emptyset$ $\forall n \in \N$. Da $D_n \subseteq A_n$ endlich folgt mit + \ref{le-projlim} $D \neq \emptyset$. +\end{proof} + +\begin{definition}[] + Ein Element $x = (x_1, \ldots, x_m) \in (\Z_p)^{m}$ (bzw. $(A_n)^{m}$) heißt primitiv, falls ein + $x_i \in \Z_p^{\times}$ (bzw. $\in A_n^{\times}$) ist. +\end{definition} + +%\begin{definition}[] +% Sei $R$ ein Ring. Ein Polynom $f \in R[X_1, \ldots, X_m]$ heißt +% homogen vom Grad $2$, falls in $R[X_1, \ldots, X_m][T]$ gilt +% \[ +% f(TX_1, \ldots, TX_m) = T^{k} f(X_1, \ldots, X_m) +% .\] Ein homogenes Polynom vom Grad $2$ heißt quadratische Form. +%\end{definition} +% +%\begin{bsp}[] +% Das Polynom $f = X^5 + X^3Y^2 + XY^4 \in \Z[X,Y]$ ist homogen, aber $g = X^2 + X + Y^2 \in \Z[X,Y]$ ist nicht +% homogen. +%\end{bsp} +% +%\begin{korollar}[] +% Seien $f^{(i)} \in \Z_p[X_1, \ldots, X_m]$ homogene Polynome. Dann sind äquivalent +% \begin{enumerate}[(i)] +% \item Die $f^{(i)}$ haben eine nichttriviale gemeinsame Nullstelle in $(\Q_p)^{m}$. +% \item Die $f^{(i)}$ haben eine gemeinsame primitive Nullstelle in $(\Z_p)^{m}$. +% \item Für $n \in \N$ haben die Polynome $f^{(i)} \; (\text{mod } p^{n}) $ eine gemeinsame +% primitive Nullstelle. +% \end{enumerate} +%\end{korollar} + +%\begin{proof} +% (i)$\implies$(ii): Sei $x = (x_1, \ldots, x_m)$ eine nichttriviale gemeinsame Nullstelle +% der $f^{(i)}$. Dann setze +% \[ +% k \coloneqq \min(v_p(x_1), \ldots, v_p(x_m)) \text{ und } y = p^{-k} x +% .\] Sei $i \in \{1, \ldots, m\} $, s.d. $k = v_p(x_i)$. Dann ist +% $v_p(y_i) = v_p(p^{-k}) + v_p(x_i) = -k + k = 0$. Also $y_i \in \Z_p^{\times }$ und damit $y$ primitiv. +% Außerdem gilt für ein $n \in \N$ +% \[ +% f^{(i)}(y) = f^{(i)}(p^{-k} x) \quad \stackrel{\text{Homog.}}{=} \quad p^{-nk} f^{(i)}(x) = 0 +% .\] +% (ii)$\implies$(i) ist trivial und (ii) $\iff$ (iii) folgt aus \ref{satz-nsequiv}. +%\end{proof} +% +%\begin{bem}[] +% Die Voraussetzung homogenes Polynom ist notwendig, wie am Beispiel: +% \[ +% f = pX - 1 \in \Z_p[X] +% \] deutlich wird, denn $f(p^{-1}) = 0$, aber im Körper $\Q_p$ hat das lineare Polynom $f$ maximal +% eine Nullstelle und $p^{-1} \not\in \Z_p$. +%\end{bem} + +Wir möchten nun betrachten, unter welchen Umständen eine Lösung $\; (\text{mod } p^{n}) $ zu einer +echten Lösung in $\Z_p$ entwickelt werden kann. Dazu verwenden wir die $p$-adische Version +des Newton Verfahrens. + +\begin{lemma}[Henselsches Lemma] + Sei $f = a_mX^{m} + \ldots + a_0 \in \Z_p[X]$ und $f' = a_m m X^{m-1} + \ldots + a_1 \in \Z_p[X]$ seine + Ableitung. Weiter sei $x \in \Z_p$, s.d. $f(x) \equiv 0 \; (\text{mod } p^{n}) $ für ein $n \in \N$ + und $v_p(f'(x)) = k$ mit $0 \le 2k < n$. Dann existiert ein $y \in \Z_p$, s.d. + \[ + f(y) \equiv 0 \; (\text{mod } p^{n+1}), v_p(f'(y)) = k \text{ und } y \equiv x \; (\text{mod } p^{n-k}) + .\] + \label{le-hensel} +\end{lemma} + +\begin{proof} + Nach Voraussetzung ist $f(x) = p^{n}b$ und $f'(x) = p^{k}c$ mit $b \in \Z_p$ und $c \in \Z_p^{\times }$. + Dann setze $z \coloneqq -b c^{-1}$ und $y \coloneqq x + p^{n-k}z$. Damit erfüllt + $y \equiv x \; (\text{mod } p^{n-k}) $. Der binomische Lehrsatz + liefert + \begin{salign*} + a_i y^{i} = a_i \sum_{j=0}^{i} \binom{i}{j} x^{i-j} (p^{n-k} z)^{j} + = a_i x^{i} + a_i i x^{i-1} p^{n-k} z + p^{2n-2k} z^2 R_i + \end{salign*} für $R_i \in \Z_p$. Aufsummieren und addieren von $a_0$ liefert mit $R \in \Z_p$ + eine ,,Taylorentwicklung'': + \begin{salign*} + f(y) &= f(x) + p^{n-k} z f'(x) + p^{2n-2k} z^2 R + \intertext{Einsetzen liefert} + f(y) &= p^{n}b - p^{n-k} b c^{-1} p^{k} c + p^{2n-2k} z^2 R \\ + &= p^{2n-2k} z^2 R \\ + &\equiv 0 \; (\text{mod } p^{n+1}) + ,\end{salign*} da $2k < n \implies 2n - 2k \ge n+1$. + Anwenden der ,,Taylorentwicklung'' auf $f'$ liefert + \begin{salign*} + f'(y) &= f'(x) + p^{n-k} z f''(x) + p^{2n-2k} z^2 R \\ + &= p^{k} c + p^{n-k} z f''(x) + p^{2n-2k}z^2R \\ + &= p^{k}(\underbrace{c + p^{n-2k} z f''(x) + p^{2n-3k} z^2 R}_{=: s}) + .\end{salign*} + Es ist $n-2k > 0$ und $2n - 3k > 0$, aber $c \in \Z_p^{\times }$, also $p \nmid s$ und damit + $s \in \Z_p^{\times }$ und $v_p(f'(y)) = k$. +\end{proof} + +Zum Studium der quadratischen Formen benötigen wir noch die auf $m$ Variablen verallgemeinerte Version +des Henselschen Lemmas. + +\begin{satz} + Sei $f \in \Z_p[X_1, \ldots, X_m]$ und $x = (x_i) \in (\Z_p)^{m}$, s.d. + $f(x) \equiv 0 \; (\text{mod } p^{n}) $. Weiter existiere ein + $1 \le j \le m$, s.d. $v_p\left( \frac{\partial f}{\partial X_j}(x) \right) = k$ mit + $0 \le 2k < n$. Dann existiert eine Nullstelle $y \in (\Z_p)^{m}$ von $f$ mit + $y \equiv x \; (\text{mod } p^{n-k}) $. + \label{satz-hensel} +\end{satz} + +\begin{proof} + Sei zunächst $m = 1$. Mit \ref{le-hensel} angewendet auf $x^{(0)} \coloneqq x$, erhält man + $x^{(1)} \in \Z_p$ mit + \[ + f(x^{(1)}) \equiv 0 \; (\text{mod } p^{n+1})\text{, } + v_p(f'(x^{(1)})) = k \text{ und } + x^{(1)} \equiv x^{(0)} \; (\text{mod } p^{n-k}) + .\] Wende \ref{le-hensel} nun auf $x^{(1)}$ und $n+1$ an. Induktiv erhält man eine Folge + $(x^{(q)})_{q \in \N}$ mit den Eigenschaften + \[ + x^{(q+1)} \equiv x^{(q)} \; (\text{mod } p^{n+q-k}) \text{ und } f(x^{(q)}) \equiv 0 \; (\text{mod } p^{n+q}) + .\] Es gilt nun $v_p(x^{(q+1)} - x^{(q)}) \ge n+q-k$, also + $d(x^{(q+1)}, x^{(q)}) \xrightarrow{q\to \infty} 0$. Also ist $x^{(q)}$ eine Cauchy Folge + und konvergiert gegen ein $y \in \Z_p$. Dann gilt + \[ + 0 = \lim_{q \to \infty} f(x^{(q)}) = f(\lim_{q \to \infty} x^{(q)}) = f(y) + \] und $y \equiv x \; (\text{mod } p^{n-k}) $. + + Sei nun $m > 1$. Ersetze in $f$ die Variablen $X_i$ durch $x_i$ für $i \neq j$. + Dann sei $g \in \Z_p[X_j]$ das entstandene Polynom in einer Variablen. Wende nun den Fall für $m = 1$ + auf $g$ an. Dann erhalten wir ein $y_j \in \Z_p$ mit $y_j \equiv x_j \; (\text{mod } p^{n-k}) $ + und $g(y_j) = 0$. Setze nun $y_i \coloneqq x_i$ für $i \neq j$. Dann ist + $y \equiv x \; (\text{mod } p^{n-k}) $ und + \[ + f(y) = f(y_1, \ldots, y_m) = f(x_1, \ldots, x_{j-1}, y_j, x_{j+1}, \ldots, x_m) = g(y_j) = 0 + .\] +\end{proof} + +Aus dem letzten Satz können wir einfache Schlussfolgerungen für quadratische Formen ziehen. + +\begin{korollar}[] + Sei $f \in \Z_p[X_1, \ldots, X_m]$ und $x \in \Z_p$ mit + \[ + f(x) \equiv 0 \; (\text{mod } p) + \] und es sei mind. eine partielle Ableitung + $\frac{\partial f}{\partial X_j}(x) \not\equiv 0 \; (\text{mod } p) $, dann hebt sich $x$ + zu einer echten Nullstelle. + \label{kor-1} +\end{korollar} + +\begin{proof} + Das ist der Fall $n = 1$ und $k = 0$ in \ref{satz-hensel}. +\end{proof} + +\begin{korollar}[] + Sei $p\neq 2$ und $f = \sum_{i=1}^{m} \sum_{j=1}^{m} a_{ij} X_i X_j \in \Z_p[X_1, \ldots, X_m]$ eine + quadratische Form mit $a_{ij} = a_{ji}$ und sei $ p \nmid \text{det}(a_{ij})$. Sei weiter $a \in \Z_p$. Dann + hebt sich jede primitive Lösung der Gleichung $f(x) \equiv a \; (\text{mod } p) $ zu einer + echten Lösung. +\end{korollar} +\begin{proof} + Mit \ref{kor-1} g.z.z., dass mind. eine partielle Ableitung $\; (\text{mod } p) $ nicht verschwindet. + Sei $A = (a_{ij}) \in \Z_p^{m \times m}$. Da $\text{det}(a_{ij}) \not\equiv 0 \; (\text{mod } p) $ + folgt $\text{det}(a_{ij}) \in \mathbb{F}_p^{\times }$ und damit $\text{ker } A = \{0\} $. Es gilt weiter + \[ + \frac{\partial f}{\partial X_i} = 2 \sum_{j=1}^{m} a_{ij}X_j \text{ also } + \begin{pmatrix} \partial_{X_i} f(x) \\ \vdots \\ \partial_{X_m} f(x) \end{pmatrix} + = 2 A x + .\] Da $x$ primitiv ist $x \neq 0 \in \mathbb{F}_p^{m}$ und damit mind. eine partielle Ableitung + $\not\equiv 0 \; (\text{mod } p) $. +\end{proof} + +%\begin{korollar}[] +% Sei $p=2$ und $f = \sum_{i=1}^{m} \sum_{j=1}^{m} a_{ij} X_i X_j \in \Z_2[X_1, \ldots, X_m]$ +% eine quadratische Form mit $a_{ij} = a_{ji}$. Weiter sei $a \in \Z_2$ und $x$ eine primitive Lösung +% der Gleichung $f(x) \equiv a \; (\text{mod } 8) $. Dann hebt sich $x$ zu einer echten Lösung, falls +% nicht alle partiellen Ableitungen $\; (\text{mod } 4) $ verschwinden. Dies ist erfüllt, wenn +% $\text{det}(a_{ij})$. +%\end{korollar} + +% ???? + +\end{document} diff --git a/sose2021/seminar/tobeamer.py b/sose2021/seminar/tobeamer.py new file mode 100644 index 0000000..9f7d0fd --- /dev/null +++ b/sose2021/seminar/tobeamer.py @@ -0,0 +1,13 @@ +with open("folien.tex", "r") as f: + content = f.read() + +edited = "" +for char in content: + edited += char + if char == '.': + edited += "\\pause" + +with open("output.tex", "w+") as f: + f.write(edited) + +print(edited) diff --git a/sose2021/tut-ana/praesenz-1.xopp b/sose2021/tut-ana/praesenz-1.xopp new file mode 100644 index 0000000000000000000000000000000000000000..7d52ad8409f031096b4d500fff2644c9d1d9945b GIT binary patch literal 1018834 zcmV($K;yq3iwFP!000001GK$q(^WA!Lg74@Aksd_w z)1UF1xt_f<7v;asji|?{3YVGtS#9QQ#~kw?|NGO+7iT^`dH4Re*RTHhl|SdcI`jU+ zgIA9qymAa$N6!Z|MibI@1A`8?a812`t9!5*KYple*Es~>kInl z>a9D^zPu(-@zWnXQ-@bVt zZ(RK=-S#iP{r$>sPk(rQ^Uk*qe!ukH+n+D}eC76!?(Hv69_XRIyOFNleRDb7yYl$* 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können, gehen wir zu unendlichen +Reihen über:\pause{} +\[ +\sum_{i=0}^{\infty} a_i p^{i} = a_0 + a_1 p + a_2 p^2 + \ldots +\] mit $0 \le a_i < p$ für $i \in \N_0$.\pause{} + +Dabei ist $\sum_{i=0}^{\infty} a_i p^{i}$ rein formal gemeint,\pause{} d.h. +bezeichnet einfach die Folge der Partialsummen +\[ +s_n = \sum_{i=0}^{n-1} a_i p^{i} = a_0 + a_1 p + a_2 p^2 + \ldots + a_{n-1} p^{n-1} +.\] +\end{frame} + +\begin{frame} + Wir betrachten nun die Folge der Restklassen der Partialsummen + $(\overline{s}_n)_{n \in \N} \in \prod_{m=1}^{\infty} \Z / p^{m} \Z$: + \[ + \overline{s}_{n} = s_n \; \text{mod } p^{n} \in \Z / p^{n} \Z + .\]\pause + Die Folgenelemente $\overline{s}_n$ erfüllen eine ,,Kompatibilitätsbedingung``: + \[ + s_{n+1} = a_0 + \ldots + a_{n} p^{n} \equiv a_0 + \ldots + a_{n-1} p^{n-1} + \; (\text{mod } p^{n}) = s_n + .\]\pause + %Die Folgenelemente $\overline{s}_n$ liegen alle in verschiedenen Ringen.\pause{} + %Allerdings existieren + %zwischen den Ringen $\Z / p^{n} \Z$ kanonische Projektionen + Mit der kanonischen Projektion + \begin{salign*} + \phi_{n}\colon \Z / p^{n+1}\Z &\to \Z / p^{n} \Z \\ + \overline{a} &\mapsto a \; \text{mod } p^{n} + ,\end{salign*}\pause{} gilt also $\phi_n(\overline{s}_{n+1}) = \overline{s}_n$. +\end{frame} + +\begin{frame} + +Mit den $\phi_n$ entsteht eine Folge +\[ + \Z / p \Z \xleftarrow{\phi_1} \Z / p^2 \Z \xleftarrow{\phi_2} \Z / p^{3} \Z \xleftarrow{\phi_3} \ldots +.\]\pause Ein solches System wird projektives System genannt, genauer:\pause{} + +\begin{definition} + Ein projektives System ist + eine Folge von Mengen $(D_n)_{n \in \N}$ und eine Folge + von Abbildungen $(p_n)_{n \in \N}$ mit $p_{n}\colon D_{n+1} \to D_n$ + \[ + D_1 \xleftarrow{p_1} D_2 \leftarrow \ldots \leftarrow D_{n} \xleftarrow{p_{n}} D_{n+1} \leftarrow \ldots + .\pause{}\] + Die Teilmenge + \[ + \varprojlim \; (D_n, p_n) = + \left\{ (a_n)_{n \in \N} \in \prod_{m=1}^{\infty} D_m \mid p_n(a_{n+1}) = a_n \forall n \in \N \right\} + \]\pause heißt projektiver Limes des Systems. +\end{definition} + +\end{frame} + +\begin{frame} + +\begin{bem}[] + Falls die $D_n$ Ringe und die $p_n$ Ringhomomorphismen sind, wird + $\varprojlim \; (D_n, p_n)$ zum Teilring des Produktrings $\prod_{n=1}^{\infty} D_n $ + (leicht nachzurechnen).\pause{} +\end{bem} + +\begin{definition}[Ganze $p$-adische Zahlen] + Der projektive Limes des Systems $(\Z / p^{n} \Z, \phi_n)$ + \[ + \Z_p \coloneqq \varprojlim \; (\Z / p^{n} \Z, \phi_n) + \] heißt der Ring der ganzen $p$-adischen Zahlen. +\end{definition} +\pause{} + +Notation: Setze im Folgenden $A_n \coloneqq \Z / p^{n} \Z$.\pause{} Außerdem bezeichne +$\pi_n\colon \Z_p \to A_n$ die kanonische Projektion. + +\end{frame} + +\begin{frame} + +\begin{bem} + \begin{enumerate} + \item Per Definition ist $x \in \Z_p$ also ein Element $x \in \prod_{n=1}^{\infty} \Z / p^{n}\Z $\pause{} + mit der ,,Kompatibilitätsbedingung``: + \[ + x_{n+1} \equiv x_n \text{ (mod } p^{n}). + \]\par + \pause\item Die Inklusion + \[ + \Z \hookrightarrow \Z_p, a \mapsto (a \; \text{mod } p, a \; \text{mod } p^2, \ldots) + \]\pause{}ist ein injektiver Ringhomomorphismus.\pause{} Damit wird $\Z$ zum Teilring von $\Z_p$.\pause{} + \item $\Z_p$ erbt als Teilring nun also die komponentenweise Addition + und Multiplikation des Produktrings + $\prod_{n=1}^{\infty} A_n $,\pause{} d.h. für $(a_n)_{n \in \N}, (b_n)_{n \in \N} \in \Z_p$ gilt + \[ + (a_n)_{n \in \N} + (b_n)_{n \in \N} = (a_n + b_n)_{n \in \N} + \quad + (a_n)_{n \in \N} \cdot (b_n)_{n \in \N} = (a_n \cdot b_n)_{n \in \N} + .\] + \end{enumerate} +\end{bem} + +\end{frame} + +\begin{frame} + +\begin{bem} + %\item Erinnerung:\pause{} $(X, \mathcal{T})$ mit $\mathcal{T} \subseteq \mathcal{P}(X)$ heißt + % topologischer Raum und $\mathcal{T}$ das System der offenen Teilmengen,\pause{} + % falls endl. Schnitte und beliebige Vereinigungen offener Mengen + % wieder offen sind.\pause{} + Versieht man $A_n$ mit der diskreten Topologie (d.h. alle Teilmengen sind offen)\pause{} + und $\prod_{n=1}^{\infty} A_n $ mit der Produkttopologie (kleinste Topologie, s.d. + die kanonischen Projektionen $\prod_{m=1}^{\infty} A_m \twoheadrightarrow A_n$ stetig sind),\pause{} wird + $\Z_p$ zu einem topologischen Ring. +\end{bem} + +\end{frame} + +\begin{frame} + +\begin{satz}[von Tychonoff] + Ist $(X_i)_{i \in I}$ eine Familie kompakter topologischer Räume, dann ist + auch das kartesische Produkt $\prod_{i \in I}^{} X_i $ kompakt bezüglich der + Produkttopologie.\pause{} + \label{satz-tycho} +\end{satz} + +\begin{proof} + Der Satz ist äquivalent zum Auswahlaxiom.\pause{} Ein Beweis findet sich beispielsweise + in Klaus Jänich: \textit{Topologie}.\pause{} +\end{proof} + +\end{frame} + +\begin{frame} + +\begin{korollar}[] + $\Z_p$ ist kompakt.\pause{} \label{kor-compact} +\end{korollar} + +\begin{proof}[Beweisskizze] + \begin{itemize}[<+->] + \item Nach \ref{satz-tycho} ist $\prod_{n=1}^{\infty} A_n$ kompakt. + \item $\Z_p$ ist abgeschlossen in $\prod_{n=1}^{\infty} A_n$. + \item Als abgeschlossene Teilmenge eines kompakten Raums ist $\Z_p$ kompakt.\qedhere + \end{itemize} +\end{proof} + +%\begin{proof} +% \uncover<+->{Nach \ref{satz-tycho} ist $\prod_{n=1}^{\infty} A_n $ kompakt.} \uncover<+->{Außerdem ist} +% \[ +% \uncover<+->{\Z_p = \bigcap_{n \in \N} +% \left\{ x \in \prod_{m=1}^{\infty} A_m \mid \phi_{n}(\pi_{n+1}(x)) = \pi_n(x)\right\}} +% \uncover<+->{= \bigcap_{n \in \N} f_n^{-1}(\{0\})} +% \] \uncover<+->{mit $f_n \colon \prod_{n=1}^{\infty} A_n \to A_n, x \mapsto \phi_{n}(\pi_{n+1}(x)) - \pi_n(x)$.} +% \uncover<+->{Es ist $f_n$ stetig, da $\pi_n$ per Definition der Produkttopologie und $\phi_n$ als +% Abbildung zwischen diskreten Räumen stetig sind.} \uncover<+->{Da $\{0\} \subseteq A_n$ abgeschlossen, folgt +% die Behauptung.\qedhere} +%\end{proof} + +\end{frame} + +\begin{frame} + +\begin{lemma} + Es ist $\pi_n$ surjektiv und $\text{ker } \pi_n = p^{n} \Z_p$ $\forall n \in \N$.\pause{} + Insbesondere gilt + \[ + \Z_p / p^{n} \Z_p \stackrel{\sim }{=} \Z / p^{n} \Z = A_n + .\pause{}\] + \label{le-kanproj} +\end{lemma} + +\begin{proofb} + Die Surjektivität ist klar.\pause\\[3mm] + + Z.z. $p^{n} \Z_p \subseteq \text{ker } \pi_n$.\pause{} + Sei dazu $x = (\overline{x}_m)_{m \in \N} \in \Z_p$.\pause{} + Dann ist $p^{n} x_n \equiv 0 \; (\text{mod } p^n)$.\pause{} Damit folgt $\pi_n(p^n x) = 0$,\pause{} + also $p^{n} x \in \text{ker } \pi_n$.\pause\\[3mm] + + Z.z.: $\text{ker } \pi_n \subseteq p^{n} \Z_p$.\pause{} + Sei dazu $x = (\overline{x}_m)_{m \in \N} \in \text{ker } \pi_n$. Sei weiter ein + $m \ge n$.\pause{} + Wegen Kompatibilität folgt + \[ + x_m \equiv x_n \; (\text{mod } p^{n}) \equiv 0 \; (\text{mod } p^{n}) +.\pause{}\] Also folgt $\overline{x}_m \in p^{n} A_m$. +\end{proofb} + +\end{frame} + +\begin{frame} + +\begin{proofe} + Es ist + \begin{salign*} + A_{m-n} = \Z / p^{m-n} \Z &\xrightarrow{\; \sim \;} p^{n} \Z / p^{m} \Z = p^{n}A_m \\ + \overline{a} &\longmapsto \overline{p^{n}a} + \end{salign*} ein Gruppenisomorphismus (nachrechnen).\pause{} + Das heißt es ex.\pause{} ein (eindeutiges) $\overline{y}_{m-n} \in A_{m-n}$, \pause{} s.d. + $p^{n}y_{m-n} \equiv x_m \; (\text{mod } p^{m}) $.\pause{} + Setze nun $y \coloneqq (\overline{y}_{m-n})_{m > n}$.\pause{} + Nun bleibt noch zu verifizieren, dass $y \in \Z_p$\pause{} + und $x = p^{n}y$\pause{} (Übungsaufgabe).\pause{} Damit folgt $x \in p^{n} \Z_p$.\pause\\[3mm] + + Die behauptete Isomorphie folgt jetzt direkt aus + dem Homomorphiesatz.\pause{} +\end{proofe} + +\end{frame} + +\begin{frame} + +\begin{lemma}[] + Für $u \in \Z_p$ sind äquivalent + \begin{enumerate}[<+->][(i)] + \item $u \in \Z_p^{\times }$ + \item $p \nmid u$ + \item $0 \neq \overline{u}_1 \in \Z / p \Z$ + \end{enumerate} + \label{le-units} +\end{lemma} + +\end{frame} + +\begin{frame} + +\begin{proof} + (ii)$\iff$(iii) ist klar wegen $\text{ker } \pi_1 = p \Z_p$.\pause{}\\[3mm] + + (i) $\implies$ (iii):\pause{} Sei dazu + $u = (\overline{u_n})_{n\in \N} \in \Z_p^{\times }$.\pause{} + Dann ex. ein $v = (\overline{v_n})_{n \in \N} \in \Z_p$ + mit $uv = 1$ insb.\pause{} $\overline{u}_1 \overline{v}_1 = \overline{1}$,\pause{} also + $\overline{u}_1 \neq 0$.\pause{}\\[3mm] + + (iii) $\implies$ (i):\pause{} Sei umgekehrt $\overline{u_1} \neq 0$.\pause{} + Wegen Kompatibilität folgt damit $p \nmid u_n$ $\forall n \in \N$, denn + ang.\pause{} $p \mid u_n$ für ein $n \in \N$.\pause{} Dann folgt + \[ + 0 \equiv u_n \; (\text{mod } p) \equiv u_1 \; (\text{mod } p) \quad \contr + .\pause{}\] + Da $p$ prim folgt insbesondere $(p^{n}, u_n) = 1$.\pause{} Also ex. $a, b \in \Z$,\pause{} s.d. + $1 = a p^{n} + b u_n$, also $1 = \overline{bu_n}$ mit $\overline{b} \in A_n$.\pause{} Also + $\overline{u_n} \in A_n^{\times }$ und damit + $v \coloneqq (\overline{u}_1^{-1}, \overline{u}_2^{-1}, \ldots) = u^{-1} \in \Z_p$. +\end{proof} + +\end{frame} + +\begin{frame} + \begin{bsp} + Für $p = 2$ ist\pause{} + \[ + 7 = (\overline{7}, \overline{7}, \ldots) = + (\overline{1}, \overline{3}, \overline{7}, \overline{7}, \ldots) \in \Z_2 + ,\]\pause d.h. nach \ref{le-units} ist $\frac{1}{7} \in \Z_2$.\pause{} Die ersten $6$ Folgenelemente sind + \[ + \frac{1}{7} + = (\overline{1}, \overline{3}, \overline{7}, \overline{7}, \overline{23}, \overline{55}, \ldots) + .\] + \end{bsp} +\end{frame} + +\begin{frame} + +\begin{lemma}[] + Für $x \in \Z_p \setminus \{0\} $ ex.\pause{} $n \in \N_0$ und $u \in \Z_p^{\times }$, \pause{} s.d. + \[ + x = p^{n} u + .\pause{}\] Diese Darstellung ist eindeutig. + \label{le-decomp} +\end{lemma} + +%\begin{proofb} +% \begin{enumerate}[(i)] +% \end{enumerate} +%\end{proofb} + +\end{frame} + +\begin{frame} + \begin{proof} + \begin{enumerate}[(i)] + \item Existenz: + Sei $x \in \Z_p \setminus \{0\} $.\pause{} Da $x \neq 0$ ex.\pause{} wegen Kompatibilität ein + maximales $n \in \N_0$,\pause{} s.d. + $\overline{x}_n = \pi_n(x) = 0$,\pause{} denn + sei $\overline{x}_n = 0$ und $\overline{x}_{n+1} \neq 0$,\pause{} + dann ist $\forall m \ge n+1$:\pause{} + \[ + x_m \equiv x_{n+1} \; (\text{mod } p^{n+1}) \not\equiv 0 \; (\text{mod } p^{n+1}) + .\]\pause + Falls $n = 0$, dann setze $u = x \in \Z_p^{\times }$.\pause{} + Sonst ist $x \in \text{ker } \pi_n$, + insbesondere ex.\pause{} nach + \ref{le-kanproj} ein $u \in \Z_p$ mit $x = p^{n}u$.\pause{} Jetzt bleibt zu verifizieren, + dass $u \in \Z_p^{\times }$ (Übungsaufgabe).\pause + %Ang.\pause{}: $p \mid u$, dann + %ist $\pi_1(u) = 0$ also ex.\pause{} wieder nach \ref{le-kanproj} ein $v \in \Z_p$ mit $u = pv$. + %\pause{} Dann ist aber + %\[ + %\pi_{n+1}(x) = \pi_{n+1}(p^{n}u) = \pi_{n+1}(p^{n+1}v) = 0 + %.\pause{}\] Widerspruch zur Maximalität von $n$.\pause{} Also $p \nmid u$ und damit + %$u \in \Z_p^{\times}$. + \item Eindeutigkeit:\pause{} Man verwende, dass Einheiten in Ringen keine Nullteiler sind.\pause{} + Die Details sind Übungsaufgabe. + \end{enumerate} + \end{proof} +\end{frame} + +%\begin{frame} +% +%\begin{proofe} +% +%\begin{enumerate}[(i)] +% \setcounter{enumi}{1} +%\end{enumerate} +% +%\end{proofe} +% +%\end{frame} + +%\begin{frame} +% +%\begin{proofe} +% \begin{enumerate}[(i)] +% \setcounter{enumi}{1} +% \item Eindeutigkeit: Sei $x = p^{n} u = p^{m} v$ mit $u, v \in \Z_p^{\times }$ und $n, m \in \N_0$.\pause{} +% Sei o.E. $n \ge m$.\pause{} Es ist $\pi_n(x) = \pi_n(p^{n}) \pi_n(u) = 0$ also +% auch $0 = \pi_n(x) = \pi_n(p^{m}) \pi_{n}(v)$.\pause{} Da $v \in \Z_p^{\times }$ ist +% $\pi_n(v) \in A_n^{\times}$, also kein Nullteiler.\pause{} Also folgt +% $\pi_n(p^{m}) = 0$ und damit $p^{m} \equiv 0 \; (\text{mod } p^{n}) $, also +% $m \ge n$.\pause{} Insgesamt also $m = n$.\pause{} +% +% Nun gilt weiter $x = p^{n} u = p^{n} v$, also $p^{n}(u-v) = 0$.\pause{} Ang.\pause{} $u-v \neq 0$.\pause{} Dann +% ist nach (i) $u - v = p^{k} w$ mit $k \in \N_0$ und $w \in \Z_p^{\times }$.\pause{} Also +% $0 = p^{n}(u-v) = p^{n+k} w$.\pause{} Da $w \in \Z_p^{\times }$ also kein Nullteiler, folgt +% $0 = p^{n+k} \in \Z$ $\contr$.\pause{} +% \end{enumerate} +%\end{proofe} +% +%\end{frame} + +\begin{frame} + +\begin{definition}[$p$-Bewertung] + Für $x \in \Z_p \setminus \{0\} $ sei $x = p^{n} u$ mit $u \in \Z_p^{\times }$.\pause{} Dann setze + \[ + v_p(x) \coloneqq n + \] und setze $v_p(0) \coloneqq \infty$.\pause{} $v_p(x)$ heißt die $p$-Bewertung von $x$.\pause{} +\end{definition} + +\begin{bem}[] + Wegen \ref{le-decomp} ist die $p$-Bewertung wohldefiniert.\pause{} + + Per Konvention setze $n + \infty = \infty$ und $\infty > n$ für $n \in \N_0$.\pause{} + Es lässt sich leicht verifizieren, dass für $x, y \in \Z_p$ gilt: + \[ + v_p(xy) = v_p(x) + v_p(y), \quad v_p(x+y) \ge \min (v_p(x), v_p(y)) + .\]\pause Daraus lässt sich ebenfalls direkt folgern, dass $\Z_p$ nullteilerfrei ist + (Übungsaufgabe). +\end{bem} + +\end{frame} + +%\begin{frame} +% +%\begin{korollar}[] +% $\Z_p$ ist nullteilerfrei.\pause{} +%\end{korollar} +% +%\begin{proof} +% Seien $x, y \in \Z_p$ mit $xy = 0$.\pause{} Dann folgt +% \[ +% \infty = v_p(0) = v_p(xy) = v_p(x) + v_p(y) +% .\pause{}\] Also $v_p(x) = \infty$ oder $v_p(y) = \infty$, also $x = 0$ oder $y = 0$.\pause{} +%\end{proof} +% +%\end{frame} + +\begin{frame} + +Wir können $v_p$ verwenden, um eine Metrik auf $\Z_p$ zu definieren:\pause{} +\[ + d(x, y) \coloneqq \exp(-v_p(x-y)) +\] mit der Konvention $\exp(-\infty) = 0$.\pause{} + +\begin{bem}[Bälle] + \uncover<+->{ + Es sei im Folgenden stets + \begin{salign*} + B(x, r) &= \{ y \in \Z_p \mid d(x,y) < r \} \text{ und }\\ + \overline{B(x,r)} &= \{y \in \Z_p \mid d(x,y) \le r\} + .\end{salign*}} + \uncover<+->{Da $v_p(x) \in \N_0$ gilt} + \begin{salign*} + \uncover<+->{\overline{B(x, e^{-n})} &= \{ y \in \Z_p \mid v_p(x-y) \ge n\} \\} + \uncover<+->{&= \{ y \in \Z_p \mid v_p(x-y) > n-1\} \\} + \uncover<+->{&= B(x, e^{-(n-1)}).} + \end{salign*} +\end{bem} + +\end{frame} + +\begin{frame} + +\begin{lemma}[Topologie auf $\Z_p$] + Die Topologie auf $\Z_p$ wird induziert durch die Metrik $d(\cdot , \cdot )$.\pause{} + $\Z_p$ ist vollständig.\pause% und $\Z$ ist dicht in $\Z_p$.\pause{} +\end{lemma} + +\begin{proof}[Beweisskizze] + \renewcommand{\qedsymbol}{} + \begin{itemize}[<+->] + \item $d(\cdot , \cdot )$ ist eine Metrik. + \item Die offenen Mengen $V \subseteq \Z_p$ bezüglich der Produkttopologie sind von der Form + \[ + V = \bigcup_{v \in V} (v + p^{n_v} \Z_p) + .\] + \item Es ist $v + p^{n} \Z_p = B(v, e^{-(n-1)})$. + \item $B(v, e^{-(n-1)}) = v + p^{n} \Z_p$ offen bezüglich der Produkttopologie. + %, da $p^{n} \Z_p = \text{ker } \pi_n + %= \pi_n^{-1}(\{0\})$.\qedhere + \qedhere + \end{itemize} +\end{proof} + +%\begin{proofb} +% (Skizze). Zu zeigen ist hier +% \begin{itemize} +% \item +% \end{itemize} +% $d(\cdot , \cdot )$ ist eine Metrik (nachrechnen).\pause{} +% Sei nun +% \[ +% S \coloneqq \{ \pi_n^{-1}(B) \mid B \subseteq A_n, n \in \N\} +% .\pause{}\] +% Die offenen Mengen von $\Z_p$ bezüglich der Produkttopologie sind dann per Definition +% gegeben als $\langle S \rangle$, wobei mit $\langle \ldots \rangle$ die durch +% endliche Schnitte und beliebige Vereinigungen erzeugten Mengen gemeint sind.\pause{} +% +% Für $D \in S$ ist $D = \pi_n^{-1}(B) = (A_1, \ldots, A_{n-1}, B, A_{n+1}, \ldots) \subseteq \Z_p$ +% wobei $B \subseteq A_n$ für ein $n \in \N$.\pause{} Für $U \in \langle S \rangle$ ist dann +% $U = (B_1, \ldots, B_n, A_{n+1}, \ldots)$ mit $B_n \subsetneq A_n$ für $n \in \N_0$.\pause{} +% Sei nun $0 \in U$.\pause{} Dann ist $p^{n} \Z_p \subseteq U$.\pause{} +%\end{proofb} +% +%\end{frame} +% +%\begin{frame} +% +%\begin{proofi} +% Für beliebiges $V \subseteq \Z_p$ offen +% ex.\pause{} nun für $v \in V$ ein $n_v \in \N_0$, \pause{} s.d. $v + p^{n_v} \Z_p \subseteq V$.\pause{} Also folgt +% \[ +% V = \bigcup_{v \in V} (v + p^{n_v} \Z_p) +% .\pause{}\] +% Nun ist aber +% \[ +% a \in v + p^{n} \Z_p \iff v_p(a - v) \ge n \iff a \in \overline{B(v, e^{-n})} +% = B(v, e^{-(n-1)}) +% .\pause{}\] +% Also folgt +% \[ +% V = \bigcup_{v \in V} B(v; e^{-(n-1)}) +% \] also $V$ auch offen bezüglich $d(\cdot, \cdot )$.\pause{} Umgekehrt +% sei $U$ offen bezüglich $d(\cdot , \cdot )$.\pause{} Dann ist +% $U$ Vereinigung von offenen (bezüglich $d(\cdot , \cdot )$) Bällen.\pause{} +% Da $p^{n} \Z_p = \pi_n^{-1}(\{0\})$, sind diese auch offen bezüglich der Produkttopologie.\pause{} +%\end{proofi} + +\end{frame} + +\begin{frame} + +\begin{proofe} + Z.z.: $\Z_p$ vollständig.\pause{} Da $\Z_p$ nach \ref{kor-compact} + kompakt ist, hat jede Folge in $\Z_p$ eine konvergente Teilfolge.\pause{} Insbesondere hat + also jede Cauchy-Folge eine konvergente Teilfolge und damit konvergiert jede Cauchy-Folge + in $\Z_p$.%\pause{} + +% Z.z.\pause{}: $\Z$ dicht in $\Z_p$.\pause{} Sei $x = (x_n)_{n \in \N} \in \Z_p$.\pause{} Setze $y_n \in \Z$, \pause{} s.d. +% $y_n \equiv x_n \; (\text{mod } p^{n}) $.\pause{} Dann ist für $n \in \N$ fest, +% $y_n \equiv x_m \; (\text{mod } p^{m}) $ $\forall m \le n$, also +% $v_p(y_n - x) \ge n$.\pause{} Also +% \[ +% d(y_n, x) = \exp(-v_p(y_n - x)) \le \exp(-n) \xrightarrow{n \to \infty} 0 +% .\pause{}\] +\end{proofe} + +\end{frame} + +\begin{frame} + +\begin{bem} + Es ist leicht nachzurechnen, dass $d(\cdot , \cdot )$ die ultrametrische Ungleichung + (auch starke Dreiecksungleichung) erfüllt, \pause{} d.h. + \[ + d(x, z) \le \max(d(x,y), d(y,z)) + \] für $x, y, z \in \Z_p$.\pause{} Damit folgt das eine Folge + $(u_n)_{n \in \N} \subseteq \Z_p$ genau dann konvergiert,\pause{} wenn + $\lim_{n \to \infty} (u_{n+1} - u_n) = 0$.\pause +\end{bem} + +\begin{definition} + Der Quotientenkörper der ganzen $p$-adischen Zahlen $\Z_p$ + heißt Körper der $p$-adischen Zahlen + \[ + \Q_p \coloneqq Q(\Z_p) + .\] +\end{definition} + +\end{frame} + +\begin{frame} + +\begin{bem} + \begin{enumerate} + \item Ein Element $x \in \Q_p \setminus \{0\}$,\pause{} + kann eindeutig als\pause{} + \[ + x = p^{r}w + \] dargestellt werden, für ein $r \in \Z$ und $w \in \Z_p^{\times }$.\pause{} + + Damit setzt sich die Definition von $v_p$ und $d(\cdot , \cdot )$ + auf $\Q_p$ fort.\pause{} Es gilt + \[ + x \in \Z_p \iff v_p(x) \ge 0 \iff v_p(x) > -1 \iff x \in B(0, e) + .\] + + \pause\item Nach (1) ist also $\Q_p = \Z_p[p^{-1}]$. + \end{enumerate} +\end{bem} + +\end{frame} + +%\begin{frame} +% +%\begin{lemma}[Topologie auf $\Q_p$] +% $\Q_p$ mit der von $\Z_p$ geerbten Metrik $d(x,y) = \exp(-v_p(x-y))$ ist +% lokal kompakt, \pause{} d.h. jedes Element $x \in \Q_p$ besitzt eine kompakte Umgebung. +% $\Q_p$ enthält $\Z_p$ als offenen Teilring.\pause{} %$\Q$ ist dicht in $\Q_p$.\pause{} +%\end{lemma} +% +%\begin{proof} +% Da $x \in \Z_p \iff v_p(x) \ge 0 \iff v_p(x) > -1$ folgt $\Z_p = \overline{B(0, 1)} = B(0, e)$, +% also $\Z_p$ offen.\pause{} +% Da $\Z_p$ kompakt, folgt, dass +% $B(x, e)$ kompakt $\forall x \in \Q_p$, also $\Q_p$ lokal kompakt.\pause{} +% %Außerdem ist +% %$\Z$ dicht in $\Z_p$, \pause{} d.h. für $x \in \Q_p$ mit $x = p^{k} u$ und $k \in \Z$, $u \in \Z_p^{\times }$ ex.\pause{} +% %eine Folge $(y_n)_{n \in \N} \subseteq \Z$ mit $y_n \xrightarrow{n \to \infty} u$.\pause{} Dann +% %setze $z_n \coloneqq p^{k} y_n \in \Q$.\pause{} Dann folgt direkt +% %$z_n = p^{k} y_n \xrightarrow{n \to \infty} p^{k} u = x$.\pause{} +%\end{proof} +% +%\end{frame} + +\begin{frame} + +\begin{enumerate} + \setcounter{enumi}{2} + \item $\Q_p$ kann auch als Vervollständigung von $\Q$ bezüglich der $p$-adischen + Metrik $d(\cdot , \cdot )$ definiert werden.\pause{} + Somit ist auch $\Q$ dicht in $\Q_p$.\pause{} Man kann ebenfalls zeigen, + dass $\Q_p$ lokal kompakt ist. +\end{enumerate} + +\end{frame} + +\subsection{$p$-adische Gleichungen} + +\begin{frame} + \frametitle{$p$-adische Gleichungen} + Wir wollen nun Gleichungen in den ganzen $p$-adischen Zahlen untersuchen.\pause{} Also + Gleichungssysteme der folgenden Art\pause{} + \begin{salign*} + f^{(1)}(X_1, \ldots, X_m) &= 0 \\ + \vdots \\ + f^{(r)}(X_1, \ldots, X_m) &= 0 + \end{salign*} mit Polynomen $f^{(i)} \in \Z_p[X_1, \ldots, X_m]$. +\end{frame} + +\begin{frame} + +\begin{lemma}[] + Sei $D_1 \leftarrow D_2 \leftarrow \ldots$ ein projektives System und + $D = \varprojlim \; (D_n, p_n)$ sein projektiver Limes.\pause{} Falls $D_n \neq \emptyset$ und endlich + folgt $D \neq \emptyset$.\pause{} + \label{le-projlim} +\end{lemma} + +\begin{proof}[Beweisskizze] + \begin{itemize}[<+->] + \item Zeige Aussage für $p_n$ surjektiv. + \item Betrachte + \[ + D_{n,m} \coloneqq (p_{n} \circ \ldots \circ p_{n+m-1})(D_{n+m}) + . \] + \item Zeige, dass $D_{n,m}$ monoton fallende, nicht leere Folge mit Grenzwert $E_n$ ist. + \item Folgere, dass + $\varprojlim \; (D_n, p_n) \supseteq \varprojlim \; (E_n, p_n|_{E_n}) \neq \emptyset$. + \qedhere + \end{itemize} +\end{proof} + +%\begin{proofb} +% +% Sei zunächst $p_{n}\colon D_{n+1} \to D_n$ surjektiv.\pause{} Dann ex.\pause{} für alle $x_{n} \in D_{n}$ +% ein $x_{n+1} \in D_{n+1}$, \pause{} s.d. $p_{n}(x_{n+1}) = x_n$.\pause{} Da $D_1 \neq \emptyset$ folgt +% $D \neq \emptyset$ induktiv.\pause{}\\[2mm] +% +% Im Allgemeinen bezeichne für $m,n \in \N$: +% Da $D_{n+m} \neq \emptyset$ folgt $D_{n,m} \neq \emptyset$\pause{} und da +% $D_n$ endlich folgt $\# p_n(D_{n+1}) \le \# D_{n+1}$ $\forall n \in \N$.\pause{} D.h. $\#D_{n,m}$ +% ist monoton fallend in $m$ bei festem $n$.\pause{} +% Da $D_{n,m+1} \subseteq D_{n, m}$ wird die Folge stationär,\pause{} d.h. +% es ex.\pause{} ein $m_0 \in \N$,\pause{} s.d. $D_{n, m_0} = D_{n, m}$ $\forall m \ge m_0$.\pause{} +% Sei $E_n$ dieser Grenzwert.\pause{} +%\end{proofb} + +\end{frame} + +%\begin{frame} +% +%\begin{proofe} +% Nun ist $p_{n}(E_{n+1}) = E_n$ (nachrechnen).\pause{} +% %Beh.\pause{}: $p_{n}(E_{n+1}) = E_n$ $\forall n \in \N$.\pause{} Sei dazu $n \in \N$.\pause{} Nun ex.\pause{} ein +% %$m_0 \in \N$,\pause{} s.d. $E_{n+1} = D_{n+1, m_0}$ und +% %$E_n = D_{n, m_0} = D_{n, m_0+1}$.\pause{} Damit folgt +% %\begin{salign*} +% % p_{n}(E_{n+1}) +% % &= p_{n}(D_{n+1, m_0}) \\ +% % &= p_{n}((p_{n+1} \circ \ldots \circ p_{n+m_0})(D_{n+1+m_0})) \\ +% % &= (p_{n} \circ p_{n+1} \circ \ldots \circ p_{n+m_0})(D_{n+m_0+1}) \\ +% % &= D_{n, m_0+1} \\ +% % &= E_n +% %.\pause{}\end{salign*} +% +% Also sind die Einschränkungen $p_{n}|_{E_{n+1}}\colon E_{n+1} \to E_n$ surjektiv,\pause{} +% $E_n \neq \emptyset$ und endlich,\pause{} also +% folgt nach der Vorüberlegung $\varprojlim \; (E_n, p_n|_{E_n}) \neq \emptyset$, \pause{} also +% insbesondere $D \neq \emptyset$.\pause{} +%\end{proofe} +% +%\end{frame} +% +\begin{frame} + +\begin{definition}[] + Ein Element $x = (x_1, \ldots, x_m) \in (\Z_p)^{m}$ (bzw.\pause{} $\in (A_n)^{m}$) heißt primitiv, falls ein + $x_i \in \Z_p^{\times}$ (bzw.\pause{} $\in A_n^{\times}$) ist. +\end{definition} + +\begin{satz}[] + Seien $f^{(i)} \in \Z_p[X_1, \ldots, X_m]$ Polynome in den ganzen $p$-adischen Zahlen.\pause{} Dann + sind äquivalent: + \begin{enumerate}[(i)] + \item Die $f^{(i)}$ haben eine gemeinsame Nullstelle in $(\Z_p)^{m}$.\pause{} + \item Für alle $n \in \N$ haben die Polynome $f^{(i)} \; (\text{mod } p^{n}) $ eine + gemeinsame Nullstelle in $(A_n)^{m}$.\pause{} + \end{enumerate} + Falls die $f^{(i)}$ homogen sind\pause{} und die Lösungen in (i) und (ii) primitiv, dann ist dies + äquivalent zu\pause + \begin{enumerate}[(i)] + \setcounter{enumi}{2} + \item Die $f^{(i)}$ haben eine nichttriviale gemeinsame Nullstelle in $(\Q_p)^{m}$. + \end{enumerate} + \label{satz-nsequiv} +\end{satz} + +\end{frame} + +\begin{frame} + +\begin{proof} + (i) $\implies$ (ii) ist trivial.\pause\\[3mm] + + (ii) $\implies$ (i): Betrachte + \[ + D := \{ \text{gemeinsame NS von } f^{(i)} \text{ in } (\Z_p)^{m} \} + \]\pause und + \[ + D_n \coloneqq \{ \text{gemeinsame NS von } f^{(i)} \text{ in } (A_n)^{m} \; (\text{mod } p^{n}) \} + .\]\pause + Es bezeichne $(\phi_{n})^m\colon (A_{n+1})^{m} \to (A_n)^{m}$ die Abbildung, die $\phi_{n}$ + komponentenweise anwendet.\pause{} Dann ist $(D_n, (\phi_n)^m)$ ein projektives System + mit $D = \varprojlim \; (D_n, (\phi_n)^m)$.\pause{} + + Sei nun $D_n \neq \emptyset$ $\forall n \in \N$.\pause{} Da $D_n \subseteq (A_n)^{m}$ endlich folgt mit + \ref{le-projlim} $D \neq \emptyset$.\pause{}\\[3mm] + + (i) $\implies$ (iii) klar\pause{} und (iii) $\implies$ (i): ,,Runterskalieren`` der Nullstelle. Details + sind Übungsaufgabe. +\end{proof} + +\end{frame} + +%\begin{frame} + +%\begin{definition}[] +% Sei $R$ ein Ring.\pause{} Ein Polynom $f \in R[X_1, \ldots, X_m]$ heißt +% homogen vom Grad $k$, falls in $R[X_1, \ldots, X_m][T]$ gilt +% \[ +% f(TX_1, \ldots, TX_m) = T^{k} f(X_1, \ldots, X_m) +% .\pause{}\] Ein homogenes Polynom vom Grad $2$ heißt quadratische Form.\pause{} +%\end{definition} +% +%\begin{bsp}[] +% Das Polynom $f = X^5 + X^3Y^2 + XY^4 \in \Z[X,Y]$ ist homogen, aber $g = X^2 + X + Y^2 \in \Z[X,Y]$ ist nicht +% homogen.\pause{} +%\end{bsp} + +%\end{frame} + +%\begin{frame} +% +%\begin{korollar}[] +% Seien $f^{(i)} \in \Z_p[X_1, \ldots, X_m]$ homogene Polynome.\pause{} Dann sind äquivalent +% \begin{enumerate}[(i)] +% \item Die $f^{(i)}$ haben eine nichttriviale gemeinsame Nullstelle in $(\Q_p)^{m}$.\pause{} +% \item Die $f^{(i)}$ haben eine gemeinsame primitive Nullstelle in $(\Z_p)^{m}$.\pause{} +% \item Für $n \in \N$ haben die Polynome $f^{(i)} \; (\text{mod } p^{n}) $ eine gemeinsame +% primitive Nullstelle.\pause{} +% \end{enumerate} +%\end{korollar} +% +%\begin{proof} +% (i)$\implies$(ii): ,,Runterskalieren`` der Nullstelle in $\Q_p$.\pause{} Details sind Übungsaufgabe.\pause{} +% (ii)$\implies$(i) ist trivial und (ii) $\iff$ (iii) folgt aus \ref{satz-nsequiv}. +%\end{proof} + +%\end{frame} + +\begin{frame} + +%\begin{bem}[] +% Die Voraussetzung homogenes Polynom ist notwendig,\pause{} wie am Beispiel: +% \[ +% f = pX - 1 \in \Z_p[X] +% \] deutlich wird,\pause{} denn $f(p^{-1}) = 0$,\pause{} +% aber im Körper $\Q_p$ hat das lineare Polynom $f$ maximal +% eine Nullstelle und $p^{-1} \not\in \Z_p$.\pause{} +%\end{bem} + +Wir möchten nun betrachten, unter welchen Umständen eine Lösung $\; (\text{mod } p^{n}) $ zu einer +echten Lösung in $\Z_p$ entwickelt werden kann.\pause{} Dazu verwenden wir die $p$-adische Version +des Newton Verfahrens. +% +\end{frame} + +\begin{frame} + +\begin{lemma}[Henselsches Lemma] + Sei $f = a_mX^{m} + \ldots + a_0 \in \Z_p[X]$ und $f' = a_m m X^{m-1} + \ldots + a_1 \in \Z_p[X]$ seine + Ableitung.\pause{} Weiter sei $x \in \Z_p$, \pause{} s.d. $f(x) \equiv 0 \; (\text{mod } p^{n}) $ für ein $n \in \N$ + und $v_p(f'(x)) = k$ mit $0 \le 2k < n$.\pause{} Dann existiert ein $y \in \Z_p$,\pause{} s.d. + \[ + f(y) \equiv 0 \; (\text{mod } p^{n+1}), v_p(f'(y)) = k \text{ und } y \equiv x \; (\text{mod } p^{n-k}) + .\]\label{le-hensel} +\end{lemma} + +\end{frame} + +\begin{frame} + +\begin{proofb} + \uncover<+->{Nach Voraussetzung ist $f(x) = p^{n}b$ und $f'(x) = p^{k}c$ mit + $b \in \Z_p$ und $c \in \Z_p^{\times }$.} + \uncover<+->{Dann sei $z \in \Z_p$ und $y \coloneqq x + p^{n-k}z$.} + \uncover<+->{Damit erfüllt + $y \equiv x \; (\text{mod } p^{n-k}) $.} + \uncover<+->{Der binomische Lehrsatz liefert} + \begin{salign*} + \uncover<+->{a_i y^{i}&=} + \uncover<+->{a_i \sum_{j=0}^{i} \binom{i}{j} x^{i-j} (p^{n-k} z)^{j}\\} + \uncover<+->{&= a_i x^{i} + a_i i x^{i-1} p^{n-k} z + p^{2n-2k} z^2 R_i} + \end{salign*} \uncover<+->{für $R_i \in \Z_p$.} + \uncover<+->{Aufsummieren und addieren von $a_0$ liefert mit $R \in \Z_p$} + \begin{salign*} + \uncover<+->{f(y) &=}\uncover<+->{f(x) + }\uncover<+->{p^{n-k} z f'(x) +}\uncover<+->{p^{2n-2k} z^2 R} + \end{salign*} + \uncover<+->{eine ,,Taylorentwicklung``.} +\end{proofb} + +\end{frame} + +\begin{frame} + +\begin{proofi} + ,,Taylorentwicklung``: + \begin{salign*} + \uncover<+->{f(y) &= f(x) + p^{n-k} z f'(x) + p^{2n-2k} z^2 R} + \intertext{\uncover<+->{Einsetzen von $f(x) = p^{n}b$ und $f'(x) = p^{k}c$ liefert:}} + \uncover<+->{f(y)&= p^{n}b - p^{n-k} z p^{k} c + p^{2n-2k} z^2 R \\} + \uncover<+->{&= p^{n} (b + zc) + p^{2n-2k} z^2 R} + \intertext{\uncover<+->{Also mit $z \coloneqq -bc^{-1}$ folgt}} + \uncover<+->{f(y) &= p^{2n-2k} z^2 R \\} + \uncover<+->{&\equiv 0 \; (\text{mod } p^{n+1}),} + \end{salign*}\uncover<+->{denn da $2k < n$ folgt $2n - 2k \ge n+1$.} +\end{proofi} + +\end{frame} + +\begin{frame} + +\begin{proofe} + \uncover<+->{Anwenden der ,,Taylorentwicklung`` auf $f'$ liefert} + \begin{salign*} + \uncover<+->{f'(y) &= f'(x) + p^{n-k} z f''(x) + p^{2n-2k} z^2 R \\} + \uncover<+->{&= p^{k} c + p^{n-k} z f''(x) + p^{2n-2k}z^2R \\} + \uncover<+->{&= p^{k}(\underbrace{c + p^{n-2k} z f''(x) + p^{2n-3k} z^2 R}_{=: s}).} + \end{salign*} + \uncover<+->{Es ist $n-2k > 0$ und $2n - 3k > 0$, aber $c \in \Z_p^{\times }$,} + \uncover<+->{also $p \nmid s$} + \uncover<+->{und damit $s \in \Z_p^{\times }$ und $v_p(f'(y)) = k$.\qedhere} +\end{proofe} + +\end{frame} + +\begin{frame} + +Zum Studium der quadratischen Formen benötigen wir noch die auf $m$ Variablen verallgemeinerte Version +des Henselschen Lemmas.\pause{} + +\begin{satz} + Sei $f \in \Z_p[X_1, \ldots, X_m]$ und $x = (x_i) \in (\Z_p)^{m}$, \pause{} s.d. + $f(x) \equiv 0 \; (\text{mod } p^{n}) $.\pause{} Weiter existiere ein + $1 \le j \le m$, \pause{} s.d. $v_p\left( \frac{\partial f}{\partial X_j}(x) \right) = k$ mit + $0 \le 2k < n$.\pause{} Dann existiert eine Nullstelle $y \in (\Z_p)^{m}$ von $f$ mit + $y \equiv x \; (\text{mod } p^{n-k})$. + \label{satz-hensel} +\end{satz} + +\end{frame} + +\begin{frame} + +\begin{proofb} + Sei zunächst $m = 1$.\pause{} Mit \ref{le-hensel} angewendet auf $x^{(0)} \coloneqq x$, erhält man + $x^{(1)} \in \Z_p$ mit + \[ + f(x^{(1)}) \equiv 0 \; (\text{mod } p^{n+1})\text{, } + v_p(f'(x^{(1)})) = k \text{ und } + x^{(1)} \equiv x^{(0)} \; (\text{mod } p^{n-k}) + .\pause{}\] Wende \ref{le-hensel} nun auf $x^{(1)}$ und $n+1$ an.\pause{} Induktiv erhält man eine Folge + $(x^{(q)})_{q \in \N}$ mit den Eigenschaften + \[ + x^{(q+1)} \equiv x^{(q)} \; (\text{mod } p^{n+q-k}) \text{ und } f(x^{(q)}) \equiv 0 \; (\text{mod } p^{n+q}) + .\pause{}\] Nun verifziert man leicht, dass $x^{(q)}$ eine Cauchy-Folge ist\pause{} und + gegen ein $y \in \Z_p$ konvergiert\pause{} mit $f(y) = 0$\pause{} und $y \equiv x \; (\text{mod } p^{n-k}) $. + + %Es gilt nun $v_p(x^{(q+1)} - x^{(q)}) \ge n+q-k$, also + %$d(x^{(q+1)}, x^{(q)}) \xrightarrow{q\to \infty} 0$.\pause{} Also ist $x^{(q)}$ eine Cauchy Folge + %und konvergiert gegen ein $y \in \Z_p$.\pause{} Dafür verifiziert man leicht + %Dann gilt + %\[ + % 0 = \lim_{q \to \infty} f(x^{(q)}) = f(\lim_{q \to \infty} x^{(q)}) = f(y) + %\] und $y \equiv x \; (\text{mod } p^{n-k}) $.\pause{} +\end{proofb} + +\end{frame} + +\begin{frame} + +\begin{proofe} + Sei nun $m > 1$.\pause{} Ersetze in $f$ die Variablen $X_i$ durch $x_i$ für $i \neq j$.\pause{} + Dann sei $g \in \Z_p[X_j]$ das entstandene Polynom in einer Variablen.\pause{} Wende nun den Fall für $m = 1$ + auf $g$ an.\pause{} Dann erhalten wir ein $y_j \in \Z_p$ mit $y_j \equiv x_j \; (\text{mod } p^{n-k}) $ + und $g(y_j) = 0$.\pause{} Setze nun $y_i \coloneqq x_i$ für $i \neq j$.\pause{} Dann ist + $y \equiv x \; (\text{mod } p^{n-k}) $ und + \[ + f(y) = f(y_1, \ldots, y_m) = f(x_1, \ldots, x_{j-1}, y_j, x_{j+1}, \ldots, x_m) = g(y_j) = 0 + .\pause{}\] +\end{proofe} + +\end{frame} + +\begin{frame} + +Aus dem letzten Satz können wir einfache Schlussfolgerungen ziehen.\pause{} + +\begin{korollar}[] + Sei $f \in \Z_p[X_1, \ldots, X_m]$ und $x \in \Z_p$ mit + \[ + f(x) \equiv 0 \; (\text{mod } p) + \] und es sei mind.\pause{} eine partielle Ableitung + $\frac{\partial f}{\partial X_j}(x) \not\equiv 0 \; (\text{mod } p) $, dann hebt sich $x$ + zu einer echten Nullstelle.\pause{} + \label{kor-1} +\end{korollar} + +\begin{proof} + Das ist der Fall $n = 1$ und $k = 0$ in \ref{satz-hensel}.\pause{} +\end{proof} + +\end{frame} + +\begin{frame} + \begin{bsp}[] + Es ist $\sqrt{2} \in \Z_7$,\pause{} denn für $f = X^2 - 2 \in \Z_7[X]$ gilt\pause{} + \[ + f(3) = 3^2 - 2 = 7 \equiv 0 \; (\text{mod } 7) + \]\pause und $f'(x) = 2X$ also $f'(3) = 6 \not\equiv 0 \; (\text{mod } 7) $. + \end{bsp} +\end{frame} + +\begin{frame} + +\begin{korollar}[] + Sei $f = \sum_{i=1}^{m} \sum_{j=1}^{m} a_{ij} X_i X_j \in \Z_p[X_1, \ldots, X_m]$\pause{} eine + quadratische Form mit $a_{ij} = a_{ji}$\pause{} und sei $ p \nmid \text{det}(a_{ij})$.\pause{} Weiter + sei $a \in \Z_p$ und $x \in \Z_p$ primitiv.\pause{} + + Für $p \neq 2$ gilt:\pause{} Falls $f(x) \equiv a \; (\text{mod } p) $,\pause{} + hebt sich $x$ zu einer echten Lösung.\pause{} + + Im Fall $p = 2$ gilt:\pause{} Falls $f(x) \equiv a \; (\text{mod } 8) $,\pause{} + hebt sich $x$ zu einer echten Lösung.\pause{} +\end{korollar} + +%\begin{korollar}[] +% Sei $p\neq 2$ und \pause{} Sei weiter $a \in \Z_p$.\pause{} Dann +% hebt sich jede primitive Lösung der Gleichung $f(x) \equiv a \; (\text{mod } p) $ zu einer +% echten Lösung.\pause{} +%\end{korollar} +%\begin{proof} +% Mit \ref{kor-1} g.z.z., dass mind.\pause{} eine partielle Ableitung $\; (\text{mod } p) $ nicht verschwindet.\pause{} +% Sei $A = (a_{ij}) \in \Z_p^{m \times m}$.\pause{} Da $\text{det}(a_{ij}) \not\equiv 0 \; (\text{mod } p) $ +% folgt $\text{det}(a_{ij}) \in \mathbb{F}_p^{\times }$ und damit $\text{ker } A = \{0\} $.\pause{} Es gilt weiter +% \[ +% \frac{\partial f}{\partial X_i} = 2 \sum_{j=1}^{m} a_{ij}X_j \text{ also } +% \begin{pmatrix} \partial_{X_i} f(x) \\ \vdots \\ \partial_{X_m} f(x) \end{pmatrix} +% = 2 A x +% .\pause{}\] Da $x$ primitiv ist $x \neq 0 \in \mathbb{F}_p^{m}$ und damit mind.\pause{} eine partielle Ableitung +% $\not\equiv 0 \; (\text{mod } p) $.\pause{} +%\end{proof} + +%\begin{korollar}[] +% Sei $p=2$ und $f = \sum_{i=1}^{m} \sum_{j=1}^{m} a_{ij} X_i X_j \in \Z_2[X_1, \ldots, X_m]$ +% eine quadratische Form mit $a_{ij} = a_{ji}$.\pause{} Weiter sei $a \in \Z_2$ und $x$ eine primitive Lösung +% der Gleichung $f(x) \equiv a \; (\text{mod } 8) $.\pause{} Dann hebt sich $x$ zu einer echten Lösung, falls +% nicht alle partiellen Ableitungen $\; (\text{mod } 4) $ verschwinden.\pause{} Dies ist erfüllt, wenn +% $\text{det}(a_{ij})$.\pause{} +%\end{korollar} + +\begin{proof} + Folgerungen aus \ref{kor-1}. Beweise sind Übungsaufgaben. +\end{proof} + +\end{frame} + +\begin{frame} + \frametitle{Literatur} + \begin{itemize} + \item Serre J-P. \textit{A Course in Arithmetic}. New York; Heidelberg; Berlin: Springer; 1973. + \item Neukirch J. \textit{Algebraische Zahlentheorie}. 1st ed. Berlin; Heidelberg [u.a.]: Springer; 2007. + \item Schmidt A. \textit{Einführung in die algebraische Zahlentheorie}. Berlin; Heidelberg [u.a.]: Springer; 2007. + \item Jänich K. \textit{Topologie}. 8th ed. Berlin; Heidelberg [u.a.]: Springer; 2005. + \end{itemize} +\end{frame} + +\end{document} diff --git a/sose2021/seminar/serre_p_adic_numbers.xopp b/sose2021/seminar/serre_p_adic_numbers.xopp new file mode 100644 index 0000000000000000000000000000000000000000..09a800f3c00493a63a58c205e006e372a58d429d GIT binary patch literal 1690435 zcmV((K;XY0iwFP!0000014NrwldHg%rQiEk+_a8pO>wjLH zkJ;WA!}gE=^^dhE{_)$kY5m_YT-DoePyam|d-(rtX`24$xBr0u`(ys?Kc=NP>f-#b z+UHFBL{zC7S#UEP@Avj9@SFJeHxK^Fe~-UYsLlR<;d({0kuM9M0?b-abtHug9vJnn*szbJ^zk*kS)C;est5thn61S$oD_ZY3ZmDQT!(!MAye#pBhJz{X z>@3mP^$KwavH}Jho(E;1eRwRhjGw;ak;kRsr_$anOKL?sCKJ2A2y}P`!A0PfqF~4GxOyi-^4AJ1s@Pw@_kIpf zcyCrZ0+S_jXLEhb_C=~#R88D^V(8J$_@4Tt4vFKq=f3!E#WRQ7Ti3;Fbs?Sods0@) zrmj^q8iUuGzg0p_Qc8@LcSLK7<00QjXkJ`iHUdSA z@6Fy=8mc+<%^Va(X9}pj#CY0EdivDgn`N1D>d2$d*Wi&zd~d3tNb>!W=F8I@!xV4b z(VHMD1T0Y9h_51R(#n#EKxQ#zhJSThjxn9%plQ8mR?J9}{r1(DUc=Zae&Dm?PzwFoNUg3_f zkI`(8E(vossX7w=wmhpVLvvf$dNtO8DSonVY|r=Wq1eO=^mzG71iK7-s@QJYYmn6! 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