jrpie-notes/algebra
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use fancythm. remove \npr

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Maximilian Keßler 2022-02-16 01:15:50 +01:00
parent 93ce9390d4
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2 changed files with 4 additions and 4 deletions
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4
2021_Algebra_I.tex
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@ -773,7 +773,7 @@ In general, these inequalities may be strict.
\subsection{Transcendence degree} \subsection{Transcendence degree}
\subsubsection{Matroids} \subsubsection{Matroids}
\begin{definition}[Hull operator] \begin{definition}[Hull operator]
\npr
Let $X$ be a set, $\cP(X)$ the power set of $X$. A \vocab{Hull operator} on $X$ is a map $\cP(X) \xrightarrow{\cH} \cP(X)$ such that Let $X$ be a set, $\cP(X)$ the power set of $X$. A \vocab{Hull operator} on $X$ is a map $\cP(X) \xrightarrow{\cH} \cP(X)$ such that
\begin{enumerate} \begin{enumerate}
\item[H1] $\A A \in \cP(X) ~ A \se \cH(A)$. \item[H1] $\A A \in \cP(X) ~ A \se \cH(A)$.
@ -806,7 +806,7 @@ In general, these inequalities may be strict.
Let $L / K$ be a field extension and let $\cH(T)$ be the algebraic closure in $L$ of the subfield of $L$ generated by $K$ and $T$.\footnote{This is the intersection of all subfields of $L$ containing $K \cup T$, or the field of quotients of the sub-$K$-algebra of $L$ generated by $T$.} Let $L / K$ be a field extension and let $\cH(T)$ be the algebraic closure in $L$ of the subfield of $L$ generated by $K$ and $T$.\footnote{This is the intersection of all subfields of $L$ containing $K \cup T$, or the field of quotients of the sub-$K$-algebra of $L$ generated by $T$.}
Then $\cH$ is a matroidal hull operator. Then $\cH$ is a matroidal hull operator.
\end{lemma} \end{lemma}
\begin{proof}\npr \begin{proof}
H1, H2 and F are trivial. For an algebraically closed subfield $K \se M \se L$ we have $\cH(M) = M$. Thus $\cH(\cH(T)) = \cH(T)$ (H3). H1, H2 and F are trivial. For an algebraically closed subfield $K \se M \se L$ we have $\cH(M) = M$. Thus $\cH(\cH(T)) = \cH(T)$ (H3).
Let $x,y \in L$, $T \se L$ and $x \in \cH(T \cup \{y\}) \sm \cH(T)$. We have to show that $y \in \cH(T \cup \{x\}) \sm \cH(T)$. Let $x,y \in L$, $T \se L$ and $x \in \cH(T \cup \{y\}) \sm \cH(T)$. We have to show that $y \in \cH(T \cup \{x\}) \sm \cH(T)$.

4
algebra.sty
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@ -3,7 +3,7 @@
\RequirePackage{mkessler-math} \RequirePackage{mkessler-math}
\RequirePackage[english, numberall]{mkessler-fancythm} \RequirePackage[number in = section]{fancythm}
\RequirePackage{hyperref} \RequirePackage{hyperref}
\RequirePackage[english, index]{mkessler-vocab} \RequirePackage[english, index]{mkessler-vocab}
\RequirePackage{mkessler-hypersetup} \RequirePackage{mkessler-hypersetup}
@ -36,12 +36,12 @@
\DeclareMathOperator{\hght}{ht} \DeclareMathOperator{\hght}{ht}
\newcommand{\Wlog}{W.l.o.g. } \newcommand{\Wlog}{W.l.o.g. }
\newcommand{\fm}{\ensuremath\mathfrak{m}}
\newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}} \newcommand{\Vspec}{\ensuremath V_{\mathbb{S}}}%\Spec}}
\newcommand{\Vs}{\ensuremath V_{\mathbb{S}}}%\Spec}} \newcommand{\Vs}{\ensuremath V_{\mathbb{S}}}%\Spec}}
\newcommand{\Va}{\ensuremath V_{\mathbb{A}}}%\Spec}} \newcommand{\Va}{\ensuremath V_{\mathbb{A}}}%\Spec}}
\newcommand{\Vp}{\ensuremath V_{\mathbb{P}}}%\Spec}} \newcommand{\Vp}{\ensuremath V_{\mathbb{P}}}%\Spec}}
\newcommand{\Pn}{\bP^n}%\Spec}} \newcommand{\Pn}{\bP^n}%\Spec}}
\newcommand{\Span}[1]{\langle#1\rangle} \newcommand{\Span}[1]{\langle#1\rangle}
\newcommand{\npr}{\footnote{Not relevant for the exam.}} \newcommand{\npr}{\footnote{Not relevant for the exam.}}
\newcommand{\limrel}{\footnote{Limited relevance for the exam.}} % may appear in 3x questions \newcommand{\limrel}{\footnote{Limited relevance for the exam.}} % may appear in 3x questions