replace \einfalg

This commit is contained in:
Maximilian Keßler 2022-02-16 01:10:56 +01:00
parent 59332e34e4
commit 09d70ffb3d
2 changed files with 9 additions and 34 deletions

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@ -1,10 +1,10 @@
\documentclass[10pt,ngerman,a4paper, fancyfoot, git]{mkessler-script}
\course{Algebra I}
\lecturer{algebra}
\author{}
\lecturer{Prof.~Dr.~Jens Franke}
\author{Josia Pietsch}
\usepackage{}
\usepackage{algebra}
\begin{document}
@ -86,7 +86,7 @@ Fields which are not assumed to be algebraically closed have been renamed (usual
\begin{enumerate}
\item Consider a sequence $M_0'' \subset M_1'' \subset \ldots \subset M''$. Then $p\inv M_i''$ yields a strictly ascending sequence.
If $M$ is generated by $S, |S| < \omega$, then $M''$ is generated by $p(S)$.
\item Because of 1. we can replace $M'$ by $f(M')$ and assume $0 \to M' \xrightarrow{f} M \xrightarrow{p} M'' \to 0$ to be exact. The fact about finite generation follows from \einfalg.
\item Because of 1. we can replace $M'$ by $f(M')$ and assume $0 \to M' \xrightarrow{f} M \xrightarrow{p} M'' \to 0$ to be exact. The fact about finite generation follows from EInführung in die Algebra.
If $M', M''$ are Noetherian, $N \se M$ a submodule, then $N' \coloneqq f\inv(N)$ and $N''\coloneqq p(N)$ are finitely generated. Since $0 \to N' \to N \to N'' \to 0$ is exact, $N$ is finitely generated.
@ -1446,7 +1446,7 @@ Recall the definition of a normal field extension in the case of finite field ex
\end{proposition}
\begin{proof}
Let $x \in Q(A)$ be integral over $A$. Then there is a normed polynomial $P \in A[T]$ with $P(x) = 0$.
In \einfalg it was shown that $A[T]$ is a UFD and that the prime elements of $A[T]$ are the elements which are irreducible in $Q(A)[T]$ and for which the $\gcd$ of the coefficients is $\sim 1$. % TODO reference
In EInführung in die Algebra it was shown that $A[T]$ is a UFD and that the prime elements of $A[T]$ are the elements which are irreducible in $Q(A)[T]$ and for which the $\gcd$ of the coefficients is $\sim 1$. % TODO reference
The prime factors of a normed polynomial are all normed up to multiplicative equivalence. We may thus assume $P$ to be irreducible in $Q(A)[T]$.
But then $\deg P = 1$ as $x$ is a zero of $P$ in $Q(A)$, hence $P(T) = T - x$ and $x \in A$ as $P \in A[T]$.
@ -1461,7 +1461,7 @@ Recall the definition of a normal field extension in the case of finite field ex
\end{remark}
\begin{remark}
A finite field extension of $\Q$ is called an \vocab{algebraic number field} (ANF). If $K$ is an ANF, let $\cO_K$ (the \vocab[Ring of integers in an ANF]{ring of integers in $K$}) be the integral closure of $\Z$ in $K$.
One can show that this is a finitely generated (hence free, by results of \einfalg % EINFALG
One can show that this is a finitely generated (hence free, by results of EInführung in die Algebra % EINFALG
) abelian group.
We have $\cO_{\Q} = \Z$ by the proposiiton.
\end{remark}

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@ -1,5 +1,8 @@
\ProvidesPackage{algebra}[2022/02/10 - Style file for notes of Algebra I]
\RequirePackage{mkessler-math}
\RequirePackage[english, numberall]{mkessler-fancythm}
\RequirePackage{hyperref}
\RequirePackage[english, index]{mkessler-vocab}
@ -9,21 +12,12 @@
\RequirePackage[utf8x]{inputenc}
\RequirePackage{babel}
\RequirePackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
% Kopf- und Fußzeilen
\RequirePackage{scrlayer-scrpage, lastpage}
\setkomafont{pageheadfoot}{\large\textrm}
\lohead{\head}
\rohead{\Namen}
\cfoot*{\thepage{}/\pageref{LastPage}}
% Position des Titels
\RequirePackage{titling}
\setlength{\droptitle}{-1.0cm}
\RequirePackage[normalem]{ulem}
\RequirePackage{pdflscape}
\RequirePackage{longtable}
@ -40,27 +34,8 @@
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\trdeg}{trdeg}
\DeclareMathOperator{\hght}{ht}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\mSpec}{mSpec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Ob}{Ob}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Alg}{\mathfrak{Alg}}
\DeclareMathOperator{\Var}{\mathfrak{Var}}
\DeclareMathOperator{\op}{{}^{\text{op}}}
\newcommand{\Wlog}{W.l.o.g. }
%\newcommand{\wlog}{w.l.o.g. }
%\RequirePackage{ebgaramond}
%\RequirePackage{ebgaramond-maths}
\title{\textbf{Algebra 1}}
\newcommand{\Namen}{}
\author{Lecturer: \textsc{Prof. Dr. Jens Franke}\\\small{Notes: \textsc{Josia Pietsch}}}
\newcommand{\head}{Algebra 1}
\subtitle{Summer semester 2021, University Bonn}
\date{\today}
\newcommand{\einfalg}{Einführung in die Algebra}
\newcommand{\fk}{\ensuremath\mathfrak{k}}
\newcommand{\fl}{\ensuremath\mathfrak{l}}
\newcommand{\fs}{\ensuremath\mathfrak{s}}