replace \einfalg

2021_Algebra_I.tex

@@ -1,10 +1,10 @@
 \documentclass[10pt,ngerman,a4paper, fancyfoot, git]{mkessler-script}
 
 \course{Algebra I}
-\lecturer{algebra}
+\lecturer{Prof.~Dr.~Jens Franke}
-\author{}
+\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}

algebra.sty

@@ -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}}